CERDÁ PEREZ & JIMENO - Teoría de Juegos - 1a

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PEARSON EDUCACIÓN, S.A., Madrid, 2004

Materia: Juegos, teoría de los (Investigación operativa) 519.8

Formato 170# 240 Páginas: 528

Todos los derechos reservados.Queda prohibida, salvo excepción prevista en la Ley, cualquier forma de reproducción,distribución, comunicación pública y transformación de esta obra sin contar con autorizaciónde los titulares de propiedad intelectual. La infracción de los derechos mencionadospuede ser constitutiva de delito contra la propiedad intelectual 5(%/&6 789 : &;/&6 )<'#;* =$"(->6

DERECHOS RESERVADOS5 2004 por PEARSON EDUCACIÓN, S.A.

Ribera del Loira, 2828042 MADRID (España)

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PÉREZ NAVARRO, J.; JIMENO PASTOR, J. L.; CERDÁ TENA, E.

-+:2; <=>?@A>BC?D><Depósito legal: M.

PEARSON PRENTICE HALL es un sello editorial autorizado de PEARSON EDUCACIÓN, S.A.

"EFGHI JKGLIMGNO;Editor: David Fayerman AragónTécnico editorial: Ana Isabel García Borro

"EFGHI KJ HMIKFPPGQR;Director: José Antonio ClaresTécnico: José Antonio Hernán

'GSJTI KJ PFUGJMLN; Equipo de diseño de Pearson Educación, S.A./IVHISGPGQR; Copibook, S.L.

-VHMJSI HIM; Top Printer Plus S.L.L.

IMPRESO EN ESPAÑA - PRINTED IN SPAIN

Este libro ha sido impreso con papel y tintas ecológicos

ISBN: 978-84-832-2799-2

Contenido

Prólogo !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! "#

Capítulo 1. !"#$%& '( #()#(&(*+%,-.* '( /* 0/(1" !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! $

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$!B! ?+0<)1 0& @)(93 01'(3'C<",3 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 65

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FG0(,",")1 D()D+01')1 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! B5

Capítulo 2. 2/(1"& (&+3+-,"& ,"* -*4"#$%,-.* ,"$)5(+% 678 !!!!!!!!!!!!!!!!!!!!!!!! 5$

.!$! %&'()*+,,"-& !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 5$

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Capítulo 3. 2/(1"& (&+3+-,"& ,"* -*4"#$%,-.* ,"$)5(+% 6778 !!!!!!!!!!!!!!!!!!!!!! $>B

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Capítulo 4. 2/(1"& '-*3$-,"& ,"* -*4"#$%,-.* ,"$)5(+% !!!!!!!!!!!!!!!!!!!!!!!!!!! .$Q

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Capítulo 5. 2/(1"& (&+3+-,"& ,"* -*4"#$%,-.* -*,"$)5(+% !!!!!!!!!!!!!!!!!!!!!!!!!! .TB

B!$! %&'()*+,,"-& !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! .TBB!.! ?+0<)1 M3V01"3&)1 01'U'",)1! FO+"2"M(") M3V01"3&) *0 831P !!!!!!!! .EQ

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FG0(,",")1 D()D+01')1 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 66B

Capítulo 6. 2/(1"& '-*3$-,"& ,"* -*4"#$%,-.* -*,"$)5(+% !!!!!!!!!!!!!!!!!!!!!!! 6>6

5!$! %&'()*+,,"-& !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 6>6

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*) 23M)(32 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 6E5

FG0(,",")1 D()D+01')1 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! >RR

Capítulo 7. 2/(1"& #()(+-'"& !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! >RB

T!$! %&'()*+,,"-& !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! >R5

T!.! ?+0<)1 (0D0'"*)1 0& +& &N90() @"&"') *0 0'3D31 !!!!!!!!!!!!!!!!!!!!!!!!!! >$5T!6! ?+0<)1 (0D0'"*)1 0& +& &N90() "&@"&"') *0 0'3D31 !!!!!!!!!!!!!!!!!!!!!!! >.>

vi Contenido

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Capítulo 8. 2/(1"& ,"")(#%+-9"& !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! >B$E!$! %&'()*+,,"-& !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! >B$

E!.! FG09D2)1 *0 G+0<)1 ,))D0(3'"A)1 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! >B>

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E!>! F2 !"#$ !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! >55

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E!5! F2 A32)( *0 HP3D20V !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! >EQ

FG0(,",")1 D()D+01')1 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! BR.

Bibliografía !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! BRB

Índice analítico !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! BRQ

Contenido vii



!

Prólogo

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Prólogo xi

Formas derepresentación

de un juego

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1.1. INTRODUCCIÓN

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2 Teoría de juegos

Muy breve historia de la Teoría de Juegos

8-#.# */"$(0#'+'$# <-# #. "+*()(#"%/ 0# .+ %#/',+ 0# 3-#4/$ */)/ 0($*(&.("+ /*-''# #"PQRR */" .+ &-6.(*+*(2" 0# "#$% &'%()* #+, -.(+($/. 0%'#v/(1) 0# A/" B#-)+"" @C/'4#"$%#'"5 +-"<-# F+@ %'+6+3/$ +"%#'(/'#$ */)/ ./$ 0# ./$ )+%#)>%(*/$ S#')#./

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Tipos de juegos

Y+6# 0($%("4-(' 0/$ %(&/$ 6>$(*/$ 0# 3-#4/$5 / 0(*F/ 0# /%'/ )/0/5 0/$ #"1/<-#$ 6>$(*/$#" #. +">.($($ 0# -" 3-#4/5 *//&#'+%(?/$ @ "/ *//&#'+%(?/$7 !" #. #"1/<-# *//&#'+%(?/ $#

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Formas de representación de un juego 3

Terminología básica

M-"<-# &/$%#'(/')#"%# $# &'#$#"%+'> @ $# #;&.(*+'> */" )>$ 0#%+..# *+0+ -"/ 0# ./$%=')("/$5 + */"%("-+*(2" 0+)/$ -"+ &'()#'+ 0#1("(*(2" 0# .+ %#')("/./4,+ 6>$(*+ <-# $#-%(.(:+ F+6(%-+.)#"%# #" W#/',+ 0# X-#4/$7

Jugadores

8/" ./$ &+'%(*(&+"%#$ #" #. 3-#4/ <-# %/)+" 0#*($(/"#$ */" #. 1(" 0# )+;()(:+' $- -%(.(90+07 8/" 0/$ / )>$7

Acciones de cada jugador

8/" .+$ 0#*($(/"#$ <-# &-#0# %/)+' *+0+ 3-4+0/' #" *+0+ )/)#"%/ #" <-# .# %/<-# 3-94+'7 !. */"3-"%/ 0# +**(/"#$ 0# -" 3-4+0/' #" *+0+ )/)#"%/ 0#. 3-#4/ &-#0# $#' 1("(%/ /("1("(%/7

Resultados del juego

8/" ./$ 0($%("%/$ )/0/$ #" <-# &-#0# */"*.-(' -" 3-#4/7 Y+0+ '#$-.%+0/ ..#?+ +&+'#3+0+$-"+$ */"$#*-#"*(+$ &+'+ *+0+ 3-4+0/'7

Pagos

Y+0+ 3-4+0/' '#*(6# -" &+4/ +. +*+6+' #. 3-#4/5 <-# 0#&#"0# 0# *->. F+@+ $(0/ #. '#$-.%+90/ 0#. 3-#4/7 !. $(4"(1(*+0/ 0# 0(*F/ &+4/ #$ .+ -%(.(0+0 <-# *+0+ 3-4+0/' +%'(6-@# +0(*F/ '#$-.%+0/5 #$ 0#*('5 .+ ?+./'+*(2" <-# &+'+ #. 3-4+0/' %(#"#" .+$ */"$#*-#"*(+$ 0#+.*+":+' -" 0#%#')("+0/ '#$-.%+0/ #" #. 3-#4/7

Estrategias. Perfiles de estrategias

d"+ #$%'+%#4(+ 0# -" 3-4+0/' #$ -" &.+" */)&.#%/ 0# +**(/"#$ */" .+$ <-# =$%# &/0',+&'/&/"#'$# &+'%(*(&+' #" 0(*F/ 3-#4/7 d" &#'1(. 0# #$%'+%#4(+$ #$ -" */"3-"%/ 0# #$%'+%#94(+$5 -"+ &/' *+0+ 3-4+0/'7

Forma estratégica y forma extensiva

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a .+ 1/')+ #$%'+%=4(*+7

M */"%("-+*(2" $# &'#$#"%+" 0/$ 3-#4/$ )-@ $#"*(../$ <-# (.-$%'+" ./$ %=')("/$ ("%'/90-*(0/$7

4 Teoría de juegos

Ejemplo 1.1

$9 X-#4/ P I3-#4/ 0# &+'#$ / "/"#$J7 e/$ ("0(?(0-/$5 + ./$ <-# 0#"/)("+'#)/$X-4+0/' P IXPJ @ X-4+0/' U IXUJ5 #.(4#" 0# )+"#'+ $()-.%>"#+ #"%'# &+'#$ IDJ / "/"#$IBJ7 8( ./$ 0/$ #.(4#" ./ )($)/ XU %(#"# <-# &+4+' + XP .+ *+"%(0+0 0# a #-'/$7 8( ./$

0/$ #.(4#" */$+$ 0($%("%+$ #$ XP #. <-# %(#"# <-# &+4+' a #-'/$ + XU7 D/' %+"%/5 *+0+ -"/F+ 0# %/)+' -"+ 0#*($(2" $(" */"/*#' .+ %/)+0+ &/' #. /%'/5 &#'/ $+6(#"0/ <-# $/"+)6+$ 0#*($(/"#$ */"$(0#'+0+$ */"3-"%+)#"%# .+$ <-# +1#*%+" +. 6(#"#$%+' 0# *+0+ -"/0# #../$7 W/0+ .+ ("1/')+*(2" '#.#?+"%# .+ &/0#)/$ '#$-)(' #" .+ $(4-(#"%# %+6.+E

:.1$("% ;

D B

D a5 .a .a5 a:.1$("% <

B .a5 a a5 .a

69 X-#4/ U7 e/$ 3-4+0/'#$ %/)+" $-$ 0#*($(/"#$ 0# -" )/0/ $#*-#"*(+.7 !" &'(9)#' .-4+' #. X-4+0/' P #.(4# #"%'# 2 5 3 @ 47 8( #.(4# 2 $# %#')("+ #. 3-#4/ @ $# +.*+":+"-"/$ &+4/$ 0# U @ f I0/"0# #. &'()#' "H)#'/ ("0(*+ .+ 4+"+"*(+ 0#. X-4+0/' P @ #.$#4-"0/ .+ 0#. X-4+0/' UJ7 8( #.(4# 3 5 #"%/"*#$ #. X-4+0/' U %(#"# .+ /&/'%-"(0+0 0##.#4(' #"%'# / I+.*+":>"0/$# -"+$ 4+"+"*(+$ 0# R @ `J / , I*/" 4+"+"*(+$ 0# P@ UJ7 g("+.)#"%#5 #" *+$/ 0# <-# #. X-4+0/' P #.(3+ 45 .# %/*+ #. %-'"/ +. X-4+0/' U <-#&-#0# #.#4(' 0# "-#?/ #"%'# .+$ +.%#'"+%(?+$ / @ , &#'/ +.*+":>"0/$# #" #$%# *+$/ -"+$

4+"+"*(+$ &+'+ ./$ 3-4+0/'#$ 0# a @ R */" /5 / 0# P @ T */" , 7 !. $(4-(#"%# >'6/. 0# 3-#4/ "/$ '#*/4# %/0+ .+ ("1/')+*(2" '#.#?+"%#E

I

Jugador 1

2 Jugador 2 Jugador 2

0

C D

i d i d

4 1 5 1

7 2 4 3

!" #. 3-#4/ 0# &+'#$ / "/"#$ #$%> #;&'#$+0/ #" 1/')+ #$%'+%=4(*+E!. */"3-"%/ 0# ./$ 3-4+0/'#$ #$ 5 % hP5 Ui!. */"3-"%/ 0# .+$ +**(/"#$ 0# XP #$ 6P% h75 8 i5 @ 0# XU #$ 6U% h75 8 i7!. */"3-"%/ 0# .+$ #$%'+%#4(+$ 0# XP #$ 9 P% h75 8 i5 @ #. 0# XU #$ 9 U%h75 8 i7

]+@ *-+%'/ &#'1(.#$ 0# #$%'+%#4(+$ <-# $/" I75 7J5 I75 8 J5 I 8 5 7J @ I 8 5 8 J5 *+0+ -"/0# ./$ *-+.#$ ..#?+ + -"/ 0# ./$ '#$-.%+0/$ 0#. 3-#4/7

L/$ &+4/$ <-# '#*(6#" XP @ XU &+'+ *+0+ &#'1(. 0# #$%'+%#4(+$ $/"E

1PI75 7J%aj 1UI75 7J%.a

1PI75 8 J%.aj 1UI75 8 J%a

1PI 8 5 7J%.aj 1UI 8 5 7J%a1PI 8 5 8 J%aj 1UI 8 5 8 J%.a

!" #. 3-#4/ U I#$%> #;&'#$+0/ #" 1/')+ #;%#"$(?+JE

!. */"3-"%/ 0# ./$ 3-4+0/'#$ #$ 5 % hP5 Ui7!. */"3-"%/ 0# .+$ +**(/"#$ 0# XP #$ 6P% h 2 5 3 5 4i5 @ 0# XU #$ 6U% h/5 , i7

!. */"3-"%/ 0# .+$ #$%'+%#4(+$ 0# XP #$ 9 P% h 2 5 3 5 4i5 @ #. 0# XU #$ 9 U% h/:/5 /:, 5,:/5 ,:, i7!. $(4"(1(*+0/ 0# .+$ #$%'+%#4(+$ 0# XU5 &/' #3#)&./ ,:/5 #$ #. $(4-(#"%#E

kX-4+' , $( XP 3-#4+ 3 @ 3-4+' / $( XP 3-#4+ 4l7]+@ PU &#'1(.#$ 0# #$%'+%#4(+$5 *+0+ -"/ 0# ./$ *-+.#$ */"0-*# + -" '#$-.%+0/ 0#.

3-#4/7L/$ &+4/$ 0# XP @ 0# XU $/"E

1PI 2 5 /:/J %1PI 2 5 /:, J%1PI 2 5 ,:/J% 1PI 2 5 ,:, J%U

1PI3 5 /:/J %1PI3 5 /:, J% R

1PI3 5 ,:/J%1PI3 5 ,:, J%P

1PI 45 /:/J %1PI 45 ,:/J%a1PI 45 /:, J%1PI 45 ,:, J%P

1UI 2 5 /:/J %1UI 2 5 /:, J%1UI 2 5 ,:/J% 1UI 2 5 ,:, J%f

1UI3 5 /:/J %1UI3 5 /:, J% `

1UI3 5 ,:/J%1UI3 5 ,:, J%U

1UI 45 /:/J %1UI 45 ,:/J%R

1UI 45 /:, J%1UI 45 ,:, J%T

1.2. FUNCIONES DE UTILIDAD. UTILIDAD ORDINAL

8#+ ; -" */"3-"%/ 0# +.%#'"+%(?+$ &/$(6.#$5 )-%-+)#"%# #;*.-@#"%#$5 #"%'# .+$ <-# 0#6##.#4(' -" +4#"%# I<-# &-#0# $#' -" ("0(?(0-/5 -"+ 1+)(.(+5 -"+ #)&'#$+5 -" #<-(&/ 0#6+./"*#$%/777J7

!" ; $-&/"#)/$ 0#1("(0+ -"+ '#.+*(2" 6("+'(+ V5 ..+)+0+ '#.+*(2" 0# &'#1#'#"*(+5 0#)+"#'+ <-#5 &+'+ < 5 * à ; 5 < V * <-(#'# 0#*(' <-# .+ +.%#'"+%(?+ < #$ &'#1#'(0+ / ("0(1#'#"9%# + .+ +.%#'"+%(?+ *7

M &+'%(' 0# .+ '#.+*(2" 0# &'#1#'#"*(+ V $# 0#1("#" /%'+$ 0/$ '#.+*(/"#$5 0# .+ $(94-(#"%# 1/')+E

6 Teoría de juegos

<= L+ '#.+*(2" 0# &'#1#'#"*(+ #$%'(*%+5 E

< *á < V *5 &#'/ "/ * V <

<-# $# .## k < #$ &'#1#'(0/ + *l7;= L+ '#.+*(2" 0# ("0(1#'#"*(+5 VE

< V * á < V *5 @ %+)6(=" * V <

<-# $# .## k < #$ ("0(1#'#"%# + *=7

8# $-&/"# <-# .+ '#.+*(2" 0# &'#1#'#"*(+ V #$ '+*(/"+.5 #" #. $#"%(0/ <-# '#*/4# .+ $(94-(#"%# 0#1("(*(2"7

Definición 1.1

L+ '#.+*(2" 0# &'#1#'#"*(+ V #$ '+*(/"+. $( ?#'(1(*+ .+$ 0/$ &'/&(#0+0#$ $(4-(#"%#$E

Y/)&.#%(%-0E O < 5 * à ; 5 $# %(#"# <-# < V * / * V < I/ +)6+$J7

W'+"$(%(?(0+0E O < 5 *5 > à ; 5 $( < V * # * V >5 #"%/"*#$5 < V >7

L+ &'/&(#0+0 0# */)&.#%(%-0 $(4"(1(*+ <-#5 0+0+$ 0/$ +.%#'"+%(?+$ *-+.#$<-(#'+ < # *5$/" */)&+'+6.#$ #"%'# $,5 #" #. $#"%(0/ <-# #$ &'#1#'(0+ < 5 #$ &'#1#'(0+ * / $/" ("0(1#9'#"%#$7

M )#"-0/ &-#0# $#' */"?#"(#"%# I&+'+ $()&.(1(*+' #;&'#$(/"#$ @ +'4-)#"%+*(/"#$J

+$(4"+' + *+0+ +.%#'"+%(?+ #" ; -" "H)#'/5 0# )+"#'+ <-# "H)#'/$ )>$ +.%/$ ("0(<-#"+.%#'"+%(?+$ )>$ 0#$#+0+$7 !" #$# *+$/ .+ 1-"*(2" ? <-# +$(4"+ "H)#'/$ + +.%#'"+%(?+$I8.,*+>, () .!+&+($( ()& $1),!) '"6%) ! J &-#0# $#' *-+.<-(#'+ <-# k'#$&#%#l .+$ &'#1#9'#"*(+$ 0#. +4#"%#7 8# 0(*# <-# 0(*F+ 1-"*(2" ? #$ */)&+%(6.# */" 0(*F+$ &'#1#'#"*(+$5 /<-# #$ -"+ '#&'#$#"%+*(2" 0# =$%+$7

Definición 1.2

d"+ 1-"*(2" ? E ; r @ #$ -"+ 8.,*+>, () .!+&+($( -.) %)2%)'),!$ &$ %)&$*+>, ()2%)8)%),*+$ V5 $( &+'+ %/0/ < 5 * à ; 5 < V * á ? I < J n ? I *J7

W+)6(=" $# 0(*# #" #$%# *+$/ <-# ? )(0# .+$ -%(.(0+0#$ I<-# 0(*F/ +4#"%# +%'(6-@# +.+$ +.%#'"+%(?+$ 0# ; J #" -"+ #$*+.+ /'0("+.7

!$ 1>*(. /6$#'?+' <-# $( ? #$ -"+ 1-"*(2" 0# -%(.(0+0 0# -" +4#"%#5 %+)6(=" ./ $#'>A % B I? J5 $(#"0/ .+ 1-"*(2" B E @r @ #$%'(*%+)#"%# *'#*(#"%#7

A#+)/$ +.4-"/$ #3#)&./$ 0# &'#1#'#"*(+$ @ -%(.(0+0#$ /'0("+.#$7

Ejemplo 1.2

8#+ ; % h 65 05 3 i5 0/"0# 65 0 @ 3 $(4"(1(*+"5 '#$&#*%(?+)#"%#5 k8#)+"+ 4'+%($ #"

N#"(0/') #" +4/$%/l5 k_#"+-.% Y.(/ W-'6/0(#$#.l @ kQ7fff #-'/$l7 8-&/"4+)/$ <-#"-#$%'/ +4#"%# &'#1(#'# #$%'(*%+)#"%# 3 + 0 @ 0 + 67

8/" 1-"*(/"#$ 0# -%(.(0+0 /'0("+. */)&+%(6.#$ */" .+$ &'#1#'#"*(+$ #;&'#$+0+$ .+$$(4-(#"%#$E ? %+. <-# ? I 6J% U5 ? I 0J%`5 ? I3 J%V5 @ %/0+$ .+$ 1-"*(/"#$ A % B I? J*/" B #$%'(*%+)#"%# *'#*(#"%#5 */)/ ? . a5 ? ! P5 T? 5 ? T5 %? 5 I? . UJ/ 5 #%*7 I+ .+H.%()+ &/0',+)/$ ..+)+'.+ "/')+.(:+0+5 &/'<-# +$(4"+ -%(.(0+0 f + .+ /&*(2" )#"/$&'#1#'(0+ @ -%(.(0+0 P + .+ )>$ &'#1#'(0+J7

Ejemplo 1.3

8#+ ; % hI < 5 *J à @U E < n f5 * n fi5 #" 0/"0# < # * '#&'#$#"%+" *+"%(0+0#$ '#$&#*%(?+$0# 0/$ 6(#"#$ 6 @ 07 8# %'+%+5 &/' %+"%/5 0# -" */"3-"%/ 0# */"$-)/ &+'+ 0/$ 6(#"#$7

!" ; $# 0#1("# .+ $(4-(#"%# '#.+*(2" 0# &'#1#'#"*(+E

I < 5 *J V I < ñ5 *ñJ á <* n < ñ *ñ

L+$ $(4-(#"%#$ 1-"*(/"#$ 0# -%(.(0+0 $/" */)&+%(6.#$ */" .+$ &'#1#'#"*(+$ 0#1("(0+$5

? I < 5 *J% <*5 A I < 5 *J% #<*! C5 $(#"0/ # b f5 D I < 5 *J%TIU <*.`J

m6$='?#$# <-# .+$ *-'?+$ 0# ("0(1#'#"*(+ "/ $# )/0(1(*+" +. *+)6(+' .+ #$*+.+ 0#-%(.(0+05 @+ <-# $2./ 0#&#"0#" 0# .+$ &'#1#'#"*(+$ $-6@+*#"%#$7 D/' #../5 6+$%+',+ */" .+("1/')+*(2" <-# $-)("($%'+ .+ -%(.(0+0 /'0("+. &+'+ '#$/.?#' #. &'/6.#)+ 0# #.#**(2"2&%()+ 0# -" ?#*%/' I#5 CJ 0# *+"%(0+0#$ 0# 6 @ 05 &/' &+'%# 0# -" */"$-)(0/' <-#0($&/"# &+'+ #../ 0# -" &'#$-&-#$%/ 1(3/7

Ejemplo 1.4

8#+ ; % @5 + ("%#'&'#%+' */)/ &/$(6.#$ '#$-.%+0/$5 &+'+ -" +4#"%#5 <-# $/" &'#)(/$ /*+$%(4/$ #" 0("#'/7

e#1("()/$ #" ; .+ $(4-(#"%# '#.+*(2" 0# &'#1#'#"*(+7 D+'+ < 5 * à ; 5

< V * á < n *

8/" 1-"*(/"#$ 0# -%(.(0+0 */)&+%(6.#$ */" .+ '#.+*(2" 0# &'#1#'#"*(+ 0#1("(0+ .+$

$(4-(#"%#$E? I < J% < 5 A I < J%#< !C5 $(#"0/ # b f5 D I < J% I < !`Ja5 #%*7

!" 0#1("(%(?+5 %/0+$ .+$ 1-"*(/"#$ $/" *'#*(#"%#$7

Condiciones de existencia y de unicidad de una función de utilidad

A#+)/$ #" &'()#' .-4+' -"+ &'/&/$(*(2" <-# "/$ 0+ */"0(*(/"#$ "#*#$+'(+$ <-# 0#6#

*-)&.(' -"+ '#.+*(2" 0# &'#1#'#"*(+ &+'+ <-# &-#0+ $#' '#&'#$#"%+0+ &/' -"+ 1-"*(2" 0#-%(.(0+07

8 Teoría de juegos

Proposición 1.1

d"+ */"0(*(2" "#*#$+'(+ &+'+ <-# -"+ '#.+*(2" 0# &'#1#'#"*(+ V &-#0+ $#' '#&'#$#"%+90+ &/' -"+ 1-"*(2" 0# -%(.(0+0 #$ <-# $#+ '+*(/"+.7

?)#"'!%$*+>,@

A#+)/$ <-# $( #;($%# -"+ 1-"*(2" 0# -%(.(0+0 <-# '#&'#$#"%+ .+$ &'#1#'#"*(+$ V5 #"9%/"*#$ V 0#6# *-)&.(' .+$ &'/&(#0+0#$ 0# */)&.#%(%-0 @ %'+"$(%(?(0+07 !" #1#*%/E

3($EF%G/G1, 7 8#+ .+ 1-"*(2" 0# -%(.(0+0 ? */)&+%(6.# */" .+ '#.+*(2" V7 D+'+*+0+ < 5 * à ; 5 $# %(#"# <-# ? I < J @ ? I *J à @7 D/' %+"%/5 $# %(#"# <-# *-)&.(' <-#? I < J n ? I *J I./ <-# ()&.(*+ <-# < V *J5 / 6(#" <-# ? I *J n ? I < J I./ <-# ()&.(*+<-# * V < J5 &/' ./ <-# .+ '#.+*(2" 0# &'#1#'#"*(+ *-)&.# .+ &'/&(#0+0 0# */)&.#%(%-07

&)#+H/G/v/,#, 7 8#+" < 5 *5 > à ; 7 8-&/"4+)/$ <-# < V *5 * V >7 A#+)/$ <-# #"%/"9*#$ <

V >7 !" #1#*%/E

< V * á ? I < J n ? I *J

* V > á ? I *J n ? I >J

Y/)/ #. /'0#" #" ./$ "H)#'/$ '#+.#$ ?#'(1(*+ .+ &'/&(#0+0 %'+"$(%(?+ $# %(#"# <-#? I < J n ? I >J5 &/' ./ <-# < V >7

Y+6# &'#4-"%+'$# $( $#'> *(#'%/ #. '#*,&'/*/5 #$ 0#*('5 $( %/0+ '#.+*(2" 0# &'#1#'#"*(+'+*(/"+. &-#0# $#' '#&'#$#"%+0+ &/' -"+ 1-"*(2" 0# -%(.(0+07 L+ '#$&-#$%+5 #" 4#"#'+.5 #$

"/5 */)/ ?#'#)/$ &/$%#'(/')#"%# */" -" */"%'+#3#)&./7 8(" #)6+'4/5 $( #. */"3-"%/ ; #$ 1("(%/ $, $# *-)&.# <-# %/0+ '#.+*(2" 0# &'#1#'#"*(+ '+*(/"+. &-#0# $#' '#&'#$#"%+0+&/' -"+ 1-"*(2" 0# -%(.(0+05 */)/ '#*/4# #. $(4-(#"%# %#/'#)+7

Teorema 1.1 AB)"%)#$ () )C+'!),*+$ 5 .,+*+($( () &$ .!+&+($( "%(+,$&9

8#+ ; 1("(%/7 8( .+$ &'#1#'#"*(+$ 0# -" +4#"%# $/6'# ; $/" '+*(/"+.#$ I*/)&.#%+$ @%'+"$(%(?+$J5 #;($%# -"+ 1-"*(2" ? 0# ; #" @ */)&+%(6.# */" %+.#$ &'#1#'#"*(+$5 #$0#*('5 %+. <-# ? I < J n ? I *J á < V *7 M0#)>$5 $( A #$ -"+ 1-"*(2" 0# -%(.(0+0 */)9&+%(6.# */" V5 $# %(#"# <-# A % B I? J5 $(#"0/ B -"+ 1-"*(2" #$%'(*%+)#"%# *'#*(#"%#7

?)#"'!%$*+>,@

e#)/$%'+'#)/$ #. %#/'#)+ #" 0/$ #%+&+$7 !" &'()#' .-4+' $-&/"0'#)/$ <-# "-"*+F+@ ("0(1#'#"*(+ #"%'# 0/$ #.#)#"%/$ 0($%("%/$ 0# ; 7 D/$%#'(/')#"%# #;%#"0#'#)/$ #.'+:/"+)(#"%/ +. *+$/ 4#"#'+.7

<= !. */"3-"%/ ; #$%> 1/')+0/ &/' + #.#)#"%/$7 8-&/"#)/$ <-# #"%'# 0/$ #.#9)#"%/$ 0($%("%/$ *-+.#$<-(#'+ 0# ; $(#)&'# F+@ &'#1#'#"*(+ #$%'(*%+ F+*(+ -"/ 0# #../$7m'0#"+)/$ ./$ #.#)#"%/$ 0# ; 0# .+ $(4-(#"%# 1/')+E

; %

h < P5 < U5 7775 < +i5 0# )+"#'+ <-#E < + < +.P ñ < U < P

e#1("()/$ .+ 1-"*(2" 0# -%(.(0+0 ? I < /J% /5 &+'+ /%P5 U5 7775 +7 8# %'+%+ 0# -"+1-"*(2" 0# -%(.(0+0 */)&+%(6.# */" .+ '#.+*(2" 0# &'#1#'#"*(+ &-#$

< / < I á / b I á ? I < /J b ? I < IJ

8-&/"4+)/$ +F/'+ <-# A #$ *-+.<-(#' 1-"*(2" 0# -%(.(0+0 */)&+%(6.# */" .+ '#.+9*(2" 0# &'#1#'#"*(+7 W(#"# <-# *-)&.('$# <-#

O < / Ç < I 5 < / < I á A I < /J b A I < IJ

D#'/ < / < I /*-''# $( @ $2./ $( / b I7 D/' %+"%/5 $# %(#"# <-#

A I < /J b A I < IJ á / b I

D+'+ *+0+ < à ; &/0#)/$ #;&'#$+' A I < J% B n? I < Jo5 #" 0/"0# B E hP5 U5 7775 +ir @5$(#"0/ B I/J%A I < /J7 B #$ #$%'(*%+)#"%# *'#*(#"%#5 @+ <-#

/ b I ú B I/J%A I < /J b A I < IJ% B I IJ5 O/5 I à hP5 U5 7775 +i5 / Ç I

;= !;%#"0+)/$ #. '+:/"+)(#"%/ +"%#'(/' +. *+$/ 4#"#'+.7!"%'# ./$ + #.#)#"%/$ 0# ; 5 %/)#)/$ -" '#&'#$#"%+"%# 0# *+0+ *.+$# 0# #.#)#"%/$

("0(1#'#"%#$ #"%'# $,78#+" < P5 < U5 7775 < J à ; 5 %+.#$ <-# < J < J .P ñ < P5 0# )+"#'+ <-# &+'+ *+0+

< à ; 5 #;($%# < / */" / à hP5 U5 7775 J i5 ?#'(1(*+"0/ <-# < V < / 7e#1("()/$ ? I < J%? I < /J% /7 ? #$ -"+ 1-"*(2" 0# -%(.(0+0 */)&+%(6.# */" .+ '#.+9

*(2" V7 !" #1#*%/E O < 5 * à ; 5 $#+ < V *5 #"%/"*#$ < V < /5 * V < I5 */" < / V < I5 &/'./ <-#

? I < J% / n I%? I *J

D/' /%'+ &+'%#5 $#+ A *-+.<-(#' /%'+ 1-"*(2" 0# -%(.(0+0 */)&+%(6.# */" V7 e#6#$#'

A I < J%A I < /J n A I < IJ%A I *J

8( < V *5 #"%/"*#$ A I < J%A I < /J%A I < IJ%A I *J

8( < *5 #"%/"*#$ A I < J%A I < /J b A I < IJ%A I *JD+'+ < à ; 5 $#+ < V < /5 &/0#)/$ &/"#' A I < J%A I < /J% B n? I < /Jo% B I/J7

B #$ -"+ 1-"*(2" B E hP5 U5 7775 J ir @ #$%'(*%+)#"%# *'#*(#"%#5 @+ <-#

/ b I ú < / < I ú A I < /J b A I < IJ ú B I/J b B I IJ

A#+)/$ + */"%("-+*(2" -" */"%'+#3#)&./ */" #. *-+. $# 0#)-#$%'+ <-#5 #" 4#"#'+.5 #.'#*,&'/*/ 0# .+ D'/&/$(*(2" P7P "/ $# *-)&.#7

8#+ ; % hI < P5 < UJ à @U E < P n f5 < U n fi7 8# &-#0# ("%#'&'#%+' ; */)/ #. */"3-"%/ 0#

*/"$-)/ 0# 0/$ 6(#"#$ &+'+ -" +4#"%# #*/"2)(*/7 A#)/$ <-# #. */"3-"%/ ; "/ #$ 1("(%/5&/' ./ <-# "/ #$ +&.(*+6.# #. W#/'#)+ P7P7

10 Teoría de juegos

!" ; 0#1("()/$ .+ '#.+*(2" 0# &'#1#'#"*(+ .#;(*/4'>1(*+5 0# .+ $(4-(#"%# 1/')+E

I < P5 < UJVI *P5 *UJ á k $ +.%+#" .+ 0#%#')("+*(2" 0#. /'0#" 0# &'#1#'#"*(+5 %+. */)/ /*-''# */" .+ &'()#'+ .#%'+ 0# -"+&+.+6'+ #" #. /'0#" #" <-# +&+'#*# #" -" 0(**(/"+'(/7 Y-+"0/ .+ *+"%(0+0 0#. 6(#" P #".+$ 0/$ *#$%+$ 0# 6(#"#$ */("*(0#5 #"%/"*#$ #$ .+ *+"%(0+0 0#. $#4-"0/ 6(#" .+ <-# 0#%#'9)("+ #. /'0#" 0# &'#1#'#"*(+ 0#. */"$-)(0/'7

!$ 1>*(. */)&'/6+' <-# .+ '#.+*(2" 0# &'#1#'#"*(+ .#;(*/4'>1(*+ ?#'(1(*+ .+$ &'/&(#0+90#$ 0# */)&.#%(%-0 @ %'+"$(%(?(0+05 &/' ./ <-# #$ -"+ '#.+*(2" '+*(/"+.7

Proposición 1.2

8#+ ; %

hI < P5 < UJ à @

U

E 1(*+ #" ; "/ #$ '#&'#$#"%+6.# )#0(+"%# "("4-9"+ 1-"*(2" 0# -%(.(0+07

?)#"'!%$*+>,@

e#)/$%'+)/$ .+ &'/&/$(*(2" &/' '#0-**(2" +. +6$-'0/7 8-&/"4+)/$ <-# #;($%#? E ; r @5 1-"*(2" 0# -%(.(0+0 */)&+%(6.# */" .+ '#.+*(2" 0# &'#1#'#"*(+ .#;(*/4'>1(*+7

D+'+ *+0+ < P n f5 $# %(#"# <-# I < P5 UJ b I < P5 PJ ú ? I < P5 UJ b ? I < P5 PJ7 Y/)/? I < P5 UJ @ ? I < P5 PJ $/" "H)#'/$ '#+.#$5 #;($%# -" "H)#'/ '+*(/"+. ) I < PJ %+. <-#

? I < P5 UJ b ) I <

PJ b ? I <

P5 PJ

8#+ < ñP */" < P b < ñP7 8# ?#'(1(*+ <-# ) I < PJ b ) I < ñPJ5 @+ <-#

) I < PJ b ? I < P5 PJ b ? I < ñP5 UJ b ) I < ñPJ

D/' %+"%/ %#"#)/$ 0#1("(0+ -"+ 1-"*(2"

) E @ r K

< r ) I < J

#" 0/"0# K #$ #. */"3-"%/ 0# ./$ "H)#'/$ '+*(/"+.#$7 e(*F+ 1-"*(2" #$ ("@#*%(?+5 @+

<-# < P Ç < ñP ú ) I < PJ Ç ) I < ñPJ7 D#'/ #$%/ "/$ ..#?+ + */"%'+0(**(2" @+ <-# #. */"3-"9%/ 0# ./$ "H)#'/$ '#+.#$ I<-# #$ #. 0/)("(/ 0# .+ 1-"*(2"J #$ -" */"3-"%/ ("1("(%/ "/"-)#'+6.#5 )(#"%'+$ <-# #. */"3-"%/ 0# ./$ '+*(/"+.#$ I<-# #$ #. */"3-"%/ 1("+. 0# .+1-"*(2" ) J #$ ("1("(%/ "-)#'+6.#5 ./ *-+. #$ )+%#)>%(*+)#"%# ()&/$(6.# &+'+ -"+ 1-"9*(2" ("@#*%(?+7

M */"%("-+*(2" ?#'#)/$ #" <-= */"0(*(/"#$ $# &-#0# +$#4-'+' .+ #;($%#"*(+ 0# -"+1-"*(2" 0# -%(.(0+0 */)&+%(6.# */" -"+ '#.+*(2" 0# &'#1#'#"*(+ #" */"3-"%/$ 0# */"$-)/0# -" "H)#'/ 0+0/ I1("(%/J 0# 6(#"#$7

8#+ ; %hI < P5 < U5 7775 < +J à @+ E < / n f5 &+'+ *+0+ /%P5 U5 7775 +i78#+ V -"+ '#.+*(2" 0# &'#1#'#"*(+ #" ; 7

Definición 1.3

L+ '#.+*(2" 0# &'#1#'#"*(+ V 0#1("(0+ #" #. */"3-"%/ ; $# 0(*# <-# #$ */"%("-+ $( $#)+"%(#"# #" #. &+$/ + .,)(%#7 !$ 0#*('5 $( ?#'(1(*+ <-# &+'+ %/0+ $-*#$(2" 0# &+'#$hI < +5 *+Ji=+%P <-# *-)&.# < + V *+5 $# %(#"# <-#

< % .()+rä

< + V .()+rä

*+% *

A#+)/$ <-# .+ '#.+*(2" 0# &'#1#'#"*(+ .#;(*/4'>1(*+ 0#1("(0+ +"%#'(/')#"%# "/ ?#'(1(9*+ .+ &'/&(#0+0 0# */"%("-(0+07 !" #1#*%/E

Y/"$(0#'#)/$ .+$ $(4-(#"%#$ $-*#$(/"#$ 0# *#$%+$ 0# 0/$ 6(#"#$E < +% IP/ +5 fJ # *+% If5 PJ7

D+'+ *+0+ + $# %(#"# <-# < + *+7 8(" #)6+'4/5 .()+rä

*+% If5 PJ If5 fJ% .()+rä

< +

!. $(4-(#"%# %#/'#)+5 <-# "/ ?+)/$ + 0#)/$%'+'5 0+ */"0(*(/"#$ $-1(*(#"%#$ <-# +$#94-'+" .+ #;($%#"*(+ 0# -"+ 1-"*(2" 0# -%(.(0+0 */)&+%(6.# */" -"+ '#.+*(2" 0# &'#1#'#"*(+0#1("(0+ #" -" */"3-"%/ 0# */"$-)/7 L+ 0#)/$%'+*(2" 0#. %#/'#)+ $# #"*-#"%'+ #" C+$9Y/.#..5 pF("$%/" @ q'##" IPQQaJ7

Teorema 1.2

8#+ ; Ñ hI < P5 < U5 7775 < +J à @+ E < / n f5 &+'+ *+0+ /% P5 U5 7775 +i -" */"3-"%/ 0# */"$-9)/7 8#+ V -"+ '#.+*(2" 0# &'#1#'#"*(+ #" ; '+*(/"+. @ */"%("-+7 !"%/"*#$ #;($%# -"+1-"*(2" 0# -%(.(0+0 */"%("-+ ? I < P5 < U5 7775 < +J <-# '#&'#$#"%+ + .+ '#.+*(2" 0# &'#1#'#"9

*(+

V7

Proposiciones con significado sobre una función de utilidad ordinal

Y/"*.-@+)/$ #$%+ $#**(2" ("$($%(#"0/ #" <-# $2./ %(#"#" $(4"(1(*+0/ +<-#..+$ &'/&/$(*(/9"#$ +*#'*+ 0# -"+ 1-"*(2" 0# -%(.(0+0 ? *-@+ ?#'0+0 / 1+.$#0+0 "/ $# +.%#'+ +. $-$%(%-(' ? &/' -"+ %'+"$1/')+*(2" #$%'(*%+)#"%# *'#*(#"%# 0# ? 7

A#+)/$ +.4-"/$ #3#)&./$E

K k 6 &'/0-*# )+@/' I/ )#"/'5 / (4-+.J -%(.(0+0 <-# 0l %(#"# $#"%(0/7

K k 0 &'/0-*# a ?#*#$ )>$ -%(.(0+0 <-# 6l "/ %(#"# $#"%(0/7K kL+ 0(1#'#"*(+ 0# -%(.(0+0 #"%'# 0 @ 3 #$ #. 0/6.# <-# .+ <-# F+@ #"%'# 6 @ 3 l "/

%(#"# $#"%(0/7

1.3. UTILIDAD DE VON NEUMANN-MORGENSTERN.ACTITUDES ANTE EL RIESGO

8( 6(#" &+'+ +.4-"+$ +&.(*+*(/"#$ #$ $-1(*(#"%# */" 0($&/"#' 0# -"+ ("1/')+*(2" /'0("+.0# .+$ -%(.(0+0#$ 0# -" +4#"%#5 #" 4#"#'+. 0(*F+ ("1/')+*(2" #$ ("$-1(*(#"%#7 D/' #3#)&./5

$-&/"4+)/$ <-# #" #. !3#)&./ P7U "/$ &'#4-"%+" $( #. +4#"%# &'#1(#'# .+ +.%#'"+%(?+ 0 /-"+ "-#?+ /&*(2" */"$($%#"%# #" .+":+' -"+ )/"#0+ #<-(.(6'+0+ @ 0+' .+ +.%#'"+%(?+ 6 $(

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

Loterías

8-&/"4+)/$ <-# -" +4#"%# 0#6# #.#4(' -"+ #"%'# ?+'(+$ +.%#'"+%(?+$5 $(#"0/ */"/*(0+$ 0#)+"#'+ /63#%(?+ .+$ &'/6+6(.(0+0#$ +$/*(+0+$ + .+$ +.%#'"+%(?+$7 !" #$%+$ */"0(*(/"#$5 #"%#/',+ 0# .+ 0#*($(2" $# 0(*# <-# #$%+)/$ #" -" */"%#;%/ 0# #.#**(2" #" +)6(#"%# 0#'(#$4/7

8#+ ; %h < P5 < U5 7775 < +i -" */"3-"%/ 0# +.%#'"+%(?+$ #" -" +)6(#"%# 0# '(#$4/7 8-&/9"#)/$5 &/' %+"%/5 <-# 0(*F/ */"3-"%/ #$ 1("(%/ I#$%# $-&-#$%/ ./ F+*#)/$ &+'+ #?(%+' .+#;*#$(?+ */)&.(*+*(2" 0#. 0#$+''/../ &/$%#'(/'J7

Definición 1.4

d"+ ./%#',+ $()&.# #" ; #$ -"+ 0($%'(6-*(2" 0# &'/6+6(.(0+0 #" ; 7 !$ 0#*('5 $# 0(*# <-# L #$ -"+ ./%#',+ $()&.# #" ; 5 $(

L%

EI EP5 EU5 7775 E+J à @+E E/ n f5 &+'+ *+0+ /% P5 U5 7775 +5 @

+

;/%P

E/%P

F#" 0/"0# E/ #$ .+ &'/6+6(.(0+0 0# <-# /*-''+ .+ +.%#'"+%(?+ < /5 &+'+ *+0+ /%P5 U5 7775 +7

Ejemplo 1.5

8#+ ; % h 65 05 3 i */)/ #" #. !3#)&./ P7U7L+ ./%#',+ $()&.# LP% IP/ U5 P/ R5 P/ RJ #$ .+ /&*(2" */"$($%#"%# #" /6%#"#' 6 */"

&'/6+6(.(0+0 P/ U5 0 */" &'/6+6(.(0+0 P/ R @ 3 */" &'/6+6(.(0+0 P/ R7L+ ./%#',+ LU% If5 P5 fJ #$ .+ /&*(2" */"$($%#"%# #" /6%#"#' 0 */" $#4-'(0+07

Ejemplo 1.6 AD,$ $2.)'!$ *.$&-.+)%$ ), &$ %.&)!$ () ., *$'+,"9

d" *.(#"%# +&-#$%+ ^ #-'/$ +. "H)#'/ `7 !" #$# *+$/ #. *.(#"%# &+4+ ^ #-'/$ @ /6%(#"#0#. *+$("/ .+ ./%#',+ $(4-(#"%#E k&'#)(/ 0# UP^ #-'/$ $( $+.# #. ` I&'/6+6(.(0+0 %P/ T`J5@ &'#)(/ "-./ $( "/ $+.# #. ` I&'/6+6(.(0+0 %T^/ T`Jl7 e(*F/ 0# /%'+ )+"#'+5 #" #$%#*+$/

; % hkD'#)(/ 0# UP^ #-'/$l5 kD'#)(/ "-./li

!. *+$("/ .# /1'#*# .+ ./%#',+ L% IP/ T`5 T^/ T`J7

8( .+$ +.%#'"+%(?+$ <-# */"$%(%-@#" #. */"3-"%/ ; $/" ?+./'#$ "-)='(*/$5 + &+'%(' 0#./$ ?+./'#$ &/$(6.#$ @ $-$ '#$&#*%(?+$ &'/6+6(.(0+0#$ <-# +$(4"+ -"+ ./%#',+5 $# &-#0#*+.*-.+' #. ?+./' #$&#'+0/5 %+. */)/ $# 0#1("# + */"%("-+*(2"7

Definición 1.58( ; % h < P5 < U5 7775 < +i #$ -" */"3-"%/ 0# ?+./'#$ "-)='(*/$5 ..+)+)/$ 0$&"% )'2)%$("0# .+ ./%#',+ L% I EP5 EU5 7775 E+J +. ?+./' "-)='(*/ - I LJ% $ 4#"#'+. 0# ./%#',+5 ..+)+0+ ./%#',+ */)&-#$%+5 ./$'#$-.%+0/$ <-# $# &-#0#" /6%#"#' $/" ./%#',+$ $()&.#$5 %+. */)/ $# 0#1("# + */"%("-+*(2"7

Definición 1.68#+ ; % h < P5 < U5 7775 < +i7 e+0+$ $ ./%#',+$ $()&.#$ $/6'# ; 5 L I% I E IP5 E IU5 7775 E I+J5 &+'+

I% P5 U5 7775 $5 @ 0+0+ -"+ 0($%'(6-*(2" 0# &'/6+6(.(0+0 IjP5 jU5 7775 j$J5 */" j I n f5

&+'+ *+0+ I%P5 U5 7775 $5 @$

; I%P

j I%P5 .+ ./%#',+ */)&-#$%+ I LP5 LU5 7775 L$j jP5 jU5 7775 j$J

*/"$($%# #" /6%#"#' .+ ./%#',+ $()&.# L I */" &'/6+6(.(0+0 j I5 &+'+ I%P5 U5 7775 $7

e+0+ .+ ./%#',+ */)&-#$%+ I LP5 LU5 7775 L$j jP5 jU5 7775 j$J5 $# &-#0# *+.*-.+' .+ ./%#',+$()&.# $/6'# ; 5 L% I EP5 EU5 7775 E+J <-# 4#"#'+ .+ )($)+ 0($%'(6-*(2" H.%()+ $/6'# ./$

'#$-.%+0/$ 0# ; 5 0# .+ $(4-(#"%# 1/')+E

E/%jP EP/ ! jU E

U/ !ñ! j$ E$

/ 5 &+'+ /% P5 U5 7775 +

!" ./ <-# $(4-# $# $-&/"# <-# #. 0#*($/' $# 1(3+ #;*.-$(?+)#"%# #" .+ 0($%'(6-*(2"H.%()+ 0# &'/6+6(.(0+0#$ $/6'# ./$ '#$-.%+0/$ <-# */"$%(%-@#" #. */"3-"%/ ; 5 0# )+"#'+<-# 0/$ ./%#',+$ */)&-#$%+$ 0($%("%+$ <-# 0+" .-4+' + -"+ )($)+ ./%#',+ $()&.# $/6'# ; $/" #<-(?+.#"%#$7

D/0#)/$ 0#"/%+' L 0#. $(4-(#"%# )/0/E

L%jP LP!jU LU!ñ! j$ L$

Ejemplo 1.7

8#+" #. */"3-"%/ ; @ .+$ ./%#',+$ $()&.#$ LP @ LU 0#1("(0/$ #" #. !3#)&./ P7a7Y/"$(0#'#)/$ .+ ./%#',+ */)&-#$%+ I LP5 LUj T/ R5 P/ RJ7D/' %+"%/5 #" #$%# *+$/ $# %(#"# <-#E

jP%T/ R5 jU%P/ R

EPP% P/ U5 EP

U% P/ R5 EPT% P/ R

EUP%f5 EU

U%P5 EUT%f

L+ ./%#',+ */)&-#$%+ */"$(0#'+0+ &'/0-*# 1("+.)#"%# $/6'# ; ./$ )($)/$ '#$-.%+90/$ <-# .+ ./%#',+ $()&.#

L%jP LP! jU LU% T/ RIP/ U5 P/ R5 P/ RJ!P/ RIf5 P5 fJ% I^/ P^5 `/ P^5 T/ P^J

m6$='?#$# <-# $# ..#4+ + .+ )($)+ ./%#',+ $()&.# L5 + &+'%(' 0# .+ ./%#',+ */)&-#$%+I LT5 LRj P/ U5 P/ UJ5 $(#"0/ LT% IU/ V5 `/ V5 P/ VJ5 LR% IR/ V5 f5 U/ VJ @+ <-# P/ UIU/ V5 `/ V5P/ VJ! P/ UIR/ V5 f5 U/ VJ% I^/ P^5 `/ P^5 T/ P^J% L7

Relación de preferencia sobre el conjunto de las loterías simples en X

8#+ #. */"3-"%/ ; % h < P5 < U5 7775 < +i78#+

L ; %

EI EP5 EU5 7775 E+J à @+E E/ n f5 &+'+ *+0+ /%P5 U5 7775 +5 @

+

;/%P

E/%P

F#. */"3-"%/ 0# %/0+$ .+$ ./%#',+$ $()&.#$ $/6'# #. */"3-"%/ 0# +.%#'"+%(?+$ ; 78# $-&/"# <-# #. 0#*($/' %(#"# -"+ '#.+*(2" 0# &'#1#'#"*(+ V $/6'# L ; 5 <-# *-)&.#

.+$ &'/&(#0+0#$ 0# */)&.#%(%-0 @ %'+"$(%(?(0+07 D/' %+"%/5 $-&/"#)/$ <-# .+ '#.+*(2" 0#&'#1#'#"*(+ V #$ '+*(/"+.7 M */"%("-+*(2" $# 0#1("#" /%'+$ &'/&(#0+0#$ + */"$(0#'+' #" .+'#.+*(2" 0# &'#1#'#"*(+ V $/6'# L ; 7

Definición 1.7

L+ '#.+*(2" 0# &'#1#'#"*(+ V 0#1("(0+ #" L ; $# 0(*# <-# #$ */"%("-+ $( &+'+ %/0/ L5 Lñ5 Lññ à L ; 5 ./$ */"3-"%/$

hj à nf5 PoE I L5 Lñj j5 P . j V LññJi @ hj à nf5 PoE Lññ V I L5 Lñj j5 P . jJi

$/" *#''+0/$7

L+ &'/&(#0+0 0# */"%("-(0+0 $(4"(1(*+ <-# &#<-#\/$ *+)6(/$ #" .+$ &'/6+6(.(0+0#$"/ &'/0-*#" *+)6(/$ #" #. /'0#" #"%'# 0/$ ./%#',+$7

Definición 1.8

L+ '#.+*(2" 0# &'#1#'#"*(+ V 0#1("(0+ #" L ; ?#'(1(*+ #. +;(/)+ 0# ("0#&#"0#"*(+ $(O L5 Lñ5 Lññ à L ; 5 O j à If5 PJ $# %(#"# <-#

L V Lñ á I L5 Lññj j5 P . jJ V I Lñ5 Lññj j5 P . jJ

Función de utilidad esperada de Von Neumann-Morgenstern

M */"%("-+*(2" $# 0#1("# #. */"*#&%/ 0# 1-"*(2" 0# -%(.(0+0 #$&#'+0+ 0# A/" B#-)+""9C/'4#"$%#'"7 D/$%#'(/')#"%# $# #"-"*(+" @ 0#)-#$%'+" 0/$ &'/&/$(*(/"#$ #" .+$ <-# $#

&'#$#"%+" ()&/'%+"%#$ &'/&(#0+0#$ 0# 0(*F+$ 1-"*(/"#$ 0# -%(.(0+0 @ 1("+.)#"%# $# #"-"9*(+ #. %#/'#)+ 0# .+ -%(.(0+0 #$&#'+0+7

Definición 1.9

8# 0(*# <-# .+ 1-"*(2" 0# -%(.(0+0 ? E L ; r @ #$ -"+ 1-"*(2" 0# -%(.(0+0 #$&#'+0+ 0#A/" B#-)+""9C/'4#"$%#'" IAB9CJ $( #;($%#" + "H)#'/$ 1P5 1U5 7775 1+5 +$/*(+0/$'#$&#*%(?+)#"%# + < P5 < U5 7775 < +5 %+.#$ <-# &+'+ *+0+ ./%#',+ L% I EP5 EU5 7775 E+J à L ; $#

?#'(1(*+ <-#? I LJ%1P EP!1U EU!ñ!1+ E+

L+ $(4-(#"%# &'/&/$(*(2" 0+ -"+ */"0(*(2" "#*#$+'(+ @ $-1(*(#"%# &+'+ <-# -"+ 1-"9*(2" */" 0/)("(/ #" L ; <-# %/)+ ?+./'#$ #" @ $#+ -"+ 1-"*(2" 0# -%(.(0+0 #$&#'+0+ 0#A/" B#-)+""9C/'4#"$%#'"7

Proposición 1.3

d"+ 1-"*(2" 0# -%(.(0+0 ? E L ; r

@ #$ -"+ 1-"*(2" 0# -%(.(0+0 #$&#'+0+ 0# A/" B#-9)+""9C/'4#"$%#'" á O LP5 LU5 7775 L$ à L ; 5 OjP5 jU5 7775 j$ à nf5 Po5 */"

$

; I%P

j I%P5 $#

?#'(1(*+ <-#

? A $

; I%P

j I L IB%$

; I%P

j I? I L IJ

?)#"'!%$*+>,@

D'/6#)/$ *+0+ -"+ 0# .+$ 0/$ ()&.(*+*(/"#$7úJ 8#+"

LP% I EPP5 EP

U5 7775 EP+J5 LU% I EU

P5 EUU5 7775 EU

+J5 7775

L$% I E$P 5 E$

U 5 7775 E$+ J à L ; @ jP5 jU5 7775 j$ à nf5 Po5 */"

$

; I%P

j I%P

Y/"$(0#'+)/$ .+ ./%#',+ */)&-#$%+ I LP5 LU5 7775 L$j jP5 jU5 7775 j$J5 <-# #$ #<-(?+9.#"%# + .+ ./%#',+ $()&.# L% I EP5 EU5 7775 E+J5 #" 0/"0# &+'+ *+0+ /%P5 U5 7775 + #$

E/%jP EP/ ! jU E

U/ !ñ!j$ E$

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L% EPIP5 f5 7775 fJ! EUIf5 P5 7775 fJ!ñ! E+If5 f5 7775 PJ% EP LP! EU L

U!ñ! E+ L

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E/ L/B%

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E/? I L/J%+

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E/1/

#" 0/"0# 1/%? I L/J5 &/' ./ <-# ? #$ -"+ 1-"*(2" 0# -%(.(0+0 #$&#'+0+ 0# A/" B#-9

)+""9C/'4#"$%#'"7

M &+'%(' 0# .+ 0#)/$%'+*(2" 0# .+ &'/&/$(*(2" +"%#'(/' ?#)/$ <-#5 &+'+ *+0+ /%P5 U5 7775 +#$ 1/%? I L/J5 &#'/ L/ #$ .+ ./%#',+ <-# +$(4"+ &'/6+6(.(0+0 -"/ + .+ +.%#'"+%(?+ < / @ &'/6+96(.(0+0 *#'/ + .+$ 0#)>$ +.%#'"+%(?+$5 #$ 0#*(' .+ ./%#',+ L/ #$ .+ /&*(2" */"$($%#"%# #"/6%#"#' < / */" $#4-'(0+07

8# &-#0# 0#1("(' .+ $(4-(#"%# 1-"*(2"

1 E ; r @

< / r 1I < /J

0# )+"#'+ <-# 1I < /J%1/%? I L/J7D/' %+"%/5 $( ? #$ -"+ 1-"*(2" 0# -%(.(0+0 #$&#'+0+ 0# A/" B#-)+""9C/'4#"$%#'"5 $#

%(#"# -"+ 1-"*(2" 1 <-# $# &-#0# ("%#'&'#%+' */)/ .+ 1-"*(2" <-# +$(4"+ + *+0+ +.%#'"+%(9?+ 0#. */"3-"%/ ("(*(+. ; $- -%(.(0+0 ?#'(1(*>"0/$# <-#E

O L% I EP5 EU5 7775 E+J à L ; #$ ? I LJ%+

;/%P

E/1I < /J

!" .+ $(4-(#"%# &'/&/$(*(2" ?#)/$ <-# -"+ 1-"*(2" 0# -%(.(0+0 #$&#'+0+ 0# A/" B#-9)+""9C/'4#"$%#'"5 */''#$&/"0(#"%# + -"+ '#.+*(2" 0# &'#1#'#"*(+5 #$ H"(*+5 $+.?/ -"+%'+"$1/')+*(2" +1," &/$(%(?+7

Proposición 1.4

8#+ ? E L ; r @ -"+ 1-"*(2" 0# -%(.(0+0 #$&#'+0+ 0# A/" B#-)+""9C/'4#"$%#'" */9''#$&/"0(#"%# + .+ '#.+*(2" 0# &'#1#'#"*(+ V $/6'# L ; 7

A E L ; r @ #$ /%'+ 1-"*(2" 0# -%(.(0+0 #$&#'+0+ A/" B#-)+""9C/'4#"$%#'" */''#$9&/"0(#"%# + V7á !;($%#" #5 C à @5 */" #bf5 %+.#$ <-# A I LJ%#? I LJ!C5 &+'+ *+0+ L à L ; 7

?)#"'!%$*+>,@

úJ 8#+" ? @ A 1-"*(/"#$ 0# -%(.(0+0 #$&#'+0+ 0# A/" B#-)+""9C/'4#"$%#'" 0#1("(90+$ #" L < 5 */''#$&/"0(#"%#$ + V7

A#+)/$ <-#5 #"%/"*#$5 #;($%#" #5 C à @5 */" # b f5 %+.#$ <-#

A I LJ%#? I LJ!C5 &+'+ *+0+ L à L ;

D/' $#' ? 1-"*(2" .("#+.5 #$ */"%("-+7 !. */"3-"%/ L ; #$ *#''+0/ @ +*/%+0/7 D/' #.%#/'#)+ 0# p#(#'$%'+$$5 #;($%#" )+;

L à LX

? I LJ @ )(" L à L

X

? I LJ5 &/' ./ <-# #;($%#" L 15 L à L ;

5

%+.#$ <-# L 1V L V L5 O L à L ;

8( L V L 15 #"%/"*#$ *-+.<-(#' 1-"*(2" 0# -%(.(0+0 #$ */"$%+"%#5 &/' ./ <-# A I LJ% J 5? I LJ% J ñ5 O L à L ; 5 @ &/' %+"%/ $# *-)&.# .+ ()&.(*+*(2" &+'+ #%P5 C% J . J ñ7

8-&/"4+)/$ <-# L 1 L7 8#+ L à L ; 7 A#)/$ <-# &/0#)/$ #"*/"%'+' a L à nf5 Po5 %+.<-# L V a L L 1! IP . a LJ L7 !" #1#*%/5 %#"0'> <-# $#' ? I LJ%a L? I L 1J! IP . a LJ? I LJ5&/' ./ <-# 6+$%+ 0#$&#3+' a L5 /6%#"(="0/$# <-#

a L%? I LJ . ? I LJ

? I L 1J . ? I LJ

W(#"# <-# *-)&.('$# <-#

A I LJ%A Ia L L 1! IP . a LJ LJ%a LA I L 1J! IP . a LJA I LJ%a LnA I L 1J . A I LJo!A I LJ

8-$%(%-@#"0/ a L &/' #. ?+./' <-# F#)/$ /6%#"(0/ +"%#'(/')#"%#5 <-#0+

A I LJ%? I LJ . ? I LJ

? I L 1J . ? I LJ nA I L 1J . A I LJo!A I LJ

!1#*%-+"0/ /&#'+*(/"#$5 <-#0+

A I LJ%A I L 1J . A I LJ

? I L 1J . ? I LJ ? I LJ .

? I LJnA I L 1J . A I LJo

? I L 1J . ? I LJ !A I LJ

&/' ./ <-# $# *-)&.# .+ ()&.(*+*(2"5 $(#"0/

#%A I L 1J . A I LJ

? I L 1J . ? I LJb f

C%A I LJ .? I LJnA I L 1J . A I LJo

? I L 1J . ? I LJ

õJ 8-&/"4+)/$ +F/'+ <-# #;(%#" #5 C à @5 */" # b f5 %+.#$ <-# A I LJ%#? I LJ! C5&+'+ *+0+ L à L ; 5 $(#"0/ ? -"+ 1-"*(2" 0# -%(.(0+0 #$&#'+0+ 0# A/" B#-)+""9C/'94#"$%#'" */''#$&/"0(#"%# + V7 A#+)/$ <-# A #$ /%'+ 1-"*(2" 0# -%(.(0+0 #$&#'+0+ 0#

A/" B#-)+""9C/'4#"$%#'"78#+"

LP5 LU5 7775 L$ à L ; 5 jP5 jU5 7775 j$ à nf5 Po5 */"$

; I%P

j I%P

8# %(#"# <-#

A A $

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j I L IB%#? A $

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j I L IB!$

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j IC% #C $

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j I? I L IJD!C%

%

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j In#? I L IJ! Co%$

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j IA I L IJ

./ <-# 0#)-#$%'+5 -%(.(:+"0/ .+ D'/&/$(*(2" P7T <-# A #$ -"+ 1-"*(2" 0# -%(.(0+0 #$&#9'+0+ 0# A/" B#-)+""9C/'4#"$%#'"7 M0#)>$5 &/' $#' ? -"+ '#&'#$#"%+*(2" 0# V $#*-)&.# <-#

L V Lñ á ? I LJ n ? I LñJ á #? I LJ!C n #? I LñJ!C á A I LJ n A I LñJ

M */"%("-+*(2" $# #"-"*(+ #. %#/'#)+ 0# #;($%#"*(+ 0# .+ -%(.(0+0 #$&#'+0+ 0# A/"

B#-)+""9C/'4#"$%#'"7 L+ 0#)/$%'+*(2" 0#. %#/'#)+ $# &-#0# */"$-.%+' #" C+$9Y/.#..5pF($%/" @ q'##" IPQQaJ7

Teorema 1.3 B)"%)#$ () &$ .!+&+($( )'2)%$($

8-&/"4+)/$ <-# .+ '#.+*(2" 0# &'#1#'#"*(+ V $/6'# L ; #$ '+*(/"+.5 */"%("-+ @ ?#'(1(9*+ #. +;(/)+ 0# ("0#&#"0#"*(+7 !"%/"*#$ V +0)(%# -"+ '#&'#$#"%+*(2" #" 1/')+ 0#-%(.(0+0 #$&#'+0+ 0# A/" B#-)+""9C/'4#"$%#'"7

!$ 0#*('5 #;($%#" + ?+./'#$ '#+.#$ 1I < PJ5 1I < UJ5 7775 1I < +J %+.#$ <-#

O L5 Lñ à L ; 5 L% I EP5 EU5 7775 E+J V Lñ% I EñP5 EñU5 7775 Eñ+J á

+

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E/1I < /J n

+

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Eñ/1I < /J

Extensión al caso en que X %R

Y-+"0/ .+$ +.%#'"+%(?+$ / '#$-.%+0/$ $/" *+"%(0+0#$ )/"#%+'(+$ &-#0# $#' ("%#'#$+"%# '#9&'#$#"%+' 0(*F+$ *+"%(0+0#$ &/' .+ ?+'(+6.# */"%("-+ < à @7 Y+6# #"%/"*#$ */"$(0#'+'./%#',+$ '#&'#$#"%+0+$ &/' 1-"*(/"#$ 0# 0#"$(0+0 / &/' 1-"*(/"#$ 0# 0($%'(6-*(2"7

M$,5 $#+ -"+ ./%#',+ #" @ *+'+*%#'(:+0+ &/' -"+ 1-"*(2" 0# 0#"$(0+0 B 7 8#+ L ; #.*/"3-"%/ 0# %/0+$ .+$ 1-"*(/"#$ 0# 0#"$(0+0 #" @7 d"+ 1-"*(2" 0# -%(.(0+0 #$&#'+0+ 0#

A/" B#-)+""9C/'4#"$%#'" #" L ; ?#'(1(*+ <-#5 &+'+ B à L ; 5 ? I B J%I !ä

.ä

1I < J B I < J ,< 5

$(#"0/ 1E @r @ %+. <-# 1I < J #$ .+ -%(.(0+0 0# .+ *+"%(0+0 )/"#%+'(+ $#4-'+ < 5 &/' ./ <-#$# $-&/"# <-# .+ 1-"*(2" 1 #$ *'#*(#"%# @ */"%("-+7

8( .+ ./%#',+ #" @ $# *+'+*%#'(:+ &/' -"+ 1-"*(2" 0# 0($%'(6-*(2" M 5 &-#0# '#&'#$#"%+'%+"%/ 0($%'(6-*(/"#$ 0# &'/6+6(.(0+0 0($*'#%+$ */)/ */"%("-+$ #" @7 L ; $#'> #. */"3-"%/0# %/0+$ .+$ 1-"*(/"#$ 0# 0($%'(6-*(2" #" @7 !"%/"*#$ -"+ 1-"*(2" 0# -%(.(0+0 #$&#'+0+ 0#

A/" B#-)+""9C/'4#"$%#'" $# *+'+*%#'(:+ &/'<-# ? IM J%I !ä

.ä

1I < J ,M I < J7

!" *-+.<-(#'+ 0# .+$ '#&'#$#"%+*(/"#$5 $# *-)&.# %+)6(=" #. %#/'#)+ 0# .+ -%(.(0+0#$&#'+0+ $(#)&'# <-# $# *-)&.+" .+$ F(&2%#$($ 0#. W#/'#)+ P7T7

!" ./ <-# $(4-#5 @ )(#"%'+$ "/ +"-"*(#)/$ #;&.,*(%+)#"%# /%'+ */$+5 $-&/"0'#)/$<-# .+$ &'#1#'#"*(+$ 0# ./$ +4#"%#$ *-)&.#" .+$ */"0(*(/"#$ 0#. W#/'#)+ P7T @ <-#5 #"*/"$#*-#"*(+5 .+$ -%(.(0+0#$ 0# .+$ 0($%("%+$ ./%#',+$ $/" -%(.(0+0#$ #$&#'+0+$ 0# A/" B#-9

)+""9C/'4#"$%#'"7

Ejemplo 1.8

8#+ ; % nf5 äo7 Y/"$(0#'#)/$ %'#$ +4#"%#$ *-@+$ 1-"*(/"#$ 0# -%(.(0+0 $/6'# *+"%(0+90#$ "/ "#4+%(?+$ 0# 0("#'/ $/" 1PI < J%R∂ < 5 1UI < J%T < 5 1TI < J% < U7 D+'+ < P% P5

< U%P^ à nf5 äo5 $# */"$(0#'+ .+ ./%#',+ L% IR/ a5 P/ aJ7 A+)/$ + *+.*-.+' #. ?+./' #$&#9'+0/ 0# L @ */)&+'+' .+ -%(.(0+0 #$&#'+0+ */" .+ -%(.(0+0 0#. ?+./' #$&#'+0/7

!. ?+./' #$&#'+0/ #$

- I LJ% Ra#P!P

a# P^%Uf

a %R

D+'+ #. &'()#' +4#"%# $# %(#"# <-# .+ -%(.(0+0 #$&#'+0+ 0# A/" B#-)+""9C/'4#"$%#'"#$ (4-+. +E

? PI LJ%R

a 1PIPJ!

P

a 1PIP^J%

R

a IR ∂ PJ!

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TU

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D+'+ #$%# &'()#' +4#"%#5 .+ -%(.(0+0 0#. ?+./' #$&#'+0/ #$

1Pn- I LJo% 1PIRJ% R∂ R% V

&/' ./ <-# #$%# +4#"%# &'#1(#'# #. ?+./' #$&#'+0/ +"%#$ <-# .+ ./%#',+ L7D+'+ #. $#4-"0/ +4#"%#5 .+ -%(.(0+0 #$&#'+0+ @ .+ -%(.(0+0 0#. ?+./' #$&#'+0/ $/"

? UI LJ%R

a 1UIPJ!

P

a 1UIP^J%

R

a ITJ!

P

a IRVJ%PU

1Un- I LJo%T# R%PU

&/' ./ <-# +. $#4-"0/ +4#"%# .# 0+ (4-+. #. ?+./' #$&#'+0/ <-# .+ ./%#',+ L7D+'+ #. %#'*#' +4#"%# $# %(#"# <-#

? TI LJ%R

a 1TIPJ!

P

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1Tn- I LJo%1TIRJ%RU% P^

&/' ./ <-# #$%# +4#"%# &'#1(#'# .+ ./%#',+ L +"%#$ <-# #. ?+./' #$&#'+0/7

Características de los agentes ante el riesgo

A+)/$ + *.+$(1(*+' .+$ 1-"*(/"#$ 0# -%(.(0+0 #$&#'+0+ 0# A/" B#-)+""9C/'4#"$%#'" 0#./$ 0($%("%/$ +4#"%#$5 *-+"0/ #. */"3-"%/ 0# '#$-.%+0/$ / +.%#'"+%(?+$ #$ ; % @5 <-# ("%#'9&'#%+)/$ */)/ *+"%(0+0#$ 0# 0("#'/7

Suposición básica inicial

8#+ ; % @7 8-&/"#)/$ <-# .+ 1-"*(2" 0# -%(.(0+0 1 0# -" +4#"%# *-+.<-(#'+ #$ #$%'(*%+9)#"%# *'#*(#"%# I&'#1(#'# -"+ '#"%+ )+@/' + -"+ '#"%+ )#"/'J7

8( 1 #$ 0(1#'#"*(+6.#5 .+ $-&/$(*(2" +"%#'(/' $(4"(1(*+ <-# 1ñI < J b f5 &+'+ %/0/ < à @7

Definición 1.10

8#+ ; % @7 e#*()/$ <-# -" +4#"%# #$ $0)%'" $& %+)'1" #" #. ("%#'?+./ n#5 Co $( #.?+./' #$&#'+0/ 0# *-+.<-(#' ./%#',+ #" n#5 Co #$ +. )#"/$ %+" &'#1#'(0/ */)/ 0(*F+./%#',+7 8( .+ ./%#',+ #$ +. )#"/$ %+" &'#1#'(0+ */)/ $- ?+./' #$&#'+0/5 0#*()/$ <-# #$2%"2),'" $& %+)'1" / $#$,!) ()& %+)'1"7 [ $( #$ ("0(1#'#"%# #"%'# +)6+$ /&*(/"#$50#*()/$ <-# #$ ,).!%$& $& %+)'1"=

Teorema 1.4

8# */"$(0#'+ -" +4#"%# */" 1-"*(2" 0# -%(.(0+0 1E n#5 Cor @ #$%'(*%+)#"%# *'#*(#"%# @0/$ ?#*#$ 0(1#'#"*(+6.#7 !. +4#"%# #$E

M?#'$/ +. '(#$4/ #" n#5 Co $( @ $2./ $( 1ññI < J m f5 &+'+ %/0/ < 0# n#5 Co5 #$ 0#*('5 $(1 #$ *2"*+?+ #" n#5 Co7D'/&#"$/ +. '(#$4/ #" n#5 Co $( @ $2./ $( 1ññI < J n f5 &+'+ %/0/ < 0# n#5 Co5 #$ 0#*('5$( 1 #$ */"?#;+ #" n#5 Co7B#-%'+. +. '(#$4/ #" n#5 Co $( @ $2./ $( 1ññI < J%f5 &+'+ %/0/ < 0# n#5 Co5 #$ 0#*('5$( 1 #$ .("#+. #" n#5 Co7

?)#"'!%$*+>,@

8#+" < P5 < U5 7775 < + à n#5 Co *-+.#$<-(#'+ @ L% I EP5 EU5 7775 E+J -"+ ./%#',+ *-+.<-(#'+#" h < P5 < U5 7775 < +i Ñ n#5 Co7 8#+ - I LJ% EP < P! EU < U!ñ! E+ < + #. ?+./' #$&#'+0/ 0#

L @ $#+ ? I LJ%+

;/%P

E/1I < /J .+ -%(.(0+0 #$&#'+0+ 0# A/" B#-)+""9C/'4#"$%#'"7 !"

?('%-0 0# .+ 0#1("(*(2" 0# */"*+?(0+05 1 #$ *2"*+?+ #" n#5 Co $ ( @ $2./ $(1I EP < P! EU < U!ñ! E+ < +J I<-# #$ &'#*($+)#"%# .+ -%(.(0+0 0#. ?+./' #$&#'+0/J #$)+@/' / (4-+. <-# EP1I < PJ! EU1I < UJ!ñ! E+1I < +J I<-# #$ &'#*($+)#"%# .+ -%(.(0+0#$&#'+0+ 0# .+ ./%#',+ LJ7 8( 1 #$ 0/$ ?#*#$ 0(1#'#"*(+6.#5 -" */"/*(0/ %#/'#)+ 0#.*>.*-./ 0(1#'#"*(+. +$#4-'+ <-# .+ 1-"*(2" 1 #$ *2"*+?+ #" n#5 Co $( @ $2./ $(1ññI < J m f5 &+'+ %/0/ < 0# n#5 Co7 D/' %+"%/5 1ññI < J m f &+'+ %/0/ < 0# n#5 Co $( @ $2./$( &+'+ #. +4#"%# #. ?+./' #$&#'+0/ 0# *-+.<-(#' ./%#',+ #$ +. )#"/$ %+" &'#1#'(0/ */)/.+ ./%#',+5 #$ 0#*(' #$ +?#'$/ +. '(#$4/ #" n#5 Co7

M">./4+)#"%#5 1ññI < J n f &+'+ %/0/ < 0# n#5 Co $( @ $2./ $( #. +4#"%# #$ &'/&#"$/ +.'(#$4/ #" n#5 Co7

D/' /%'+ &+'%#5 .+ 1-"*(2" 1 #$ *2"*+?+ @ */"?#;+ I#$ 0#*('5 +1," / .("#+.J #" n#5 Co$( @ $2./ $( 1ññI < J%f &+'+ %/0/ < 0# n#5 Co5 @ +0#)>$ .+ 1-"*(2" 1 *2"*+?+ @ */"?#;+#" n#5 Co $( @ $2./ $( 1I EP < P! EU < U!ñ! E+ < +J5 <-# #$ .+ -%(.(0+0 0#. ?+./' #$&#'+0/0# L5 #$ (4-+. + EP1I < PJ! EU1I < UJ!ñ! E+1I < +J5 <-# #$ .+ -%(.(0+0 #$&#'+0+ 0# L7D/' %+"%/5 1ññI < J%f &+'+ %/0/ < 0# n#5 Co $( @ $2./ $( #. +4#"%# #$ ("0(1#'#"%# #"%'# #.?+./' #$&#'+0/ 0# *-+.<-(#' ./%#',+ L @ 0(*F+ ./%#',+7 !" */"*.-$(2"5 1ññI < J%f &+'+%/0/ < 0# n#5 Co $( @ $2./ $( #. +4#"%# #$ "#-%'+. +. '(#$4/ #" n#5 Co7

E6')%0$*+>, <=<@

<= L+ 1-"*(2" 0# -%(.(0+0 0# -" +4#"%# "#-%'+. +. '(#$4/ #$ .("#+.7;= L+ 1-"*(2" 0# -%(.(0+0 0# -" +4#"%# +?#'$/ I&'/&#"$/J +. '(#$4/ #$ *2"*+?+ I*/"9

?#;+J7

Ejemplo 1.9

8#+ ; % nf5 äo7 Y/"$(0#'#)/$ ./$ %'#$ +4#"%#$ 0#. !3#)&./ P7V5 *-@+$ 1-"*(/"#$ 0#-%(.(0+0 $/"

1PI < J%R∂ < 5 1UI < J%T < 5 1TI < J% < U

!" #. !3#)&./ P7V F#)/$ ?($%/ .+$ *+'+*%#',$%(*+$ 0# *+0+ -"/ 0# ./$ +4#"%#$ 1'#"%#+. '(#$4/ &+'+ -"+ ./%#',+ 0+0+7 A#+)/$ <-# $# */"1(')+" &+'+ *-+.<-(#' ./%#',+7

!. &'()#' +4#"%# &'#1#',+ #. ?+./' #$&#'+0/ +"%#$ <-# .+ ./%#',+7

1ñPI < J% U∂ <

ú 1ññPI < J%. P∂ < T

a f #" nf5 äo7 !$ +?#'$/ +. '(#$4/ #" nf5 äo

M. $#4-"0/ +4#"%# .# 0+6+ (4-+. #. ?+./' #$&#'+0/ <-# .+ ./%#',+7

1ñUI < J% T ú 1ññUI < J% f #" nf5 äo7 !$ "#-%'+. +. '(#$4/ #" nf5 äo

!. %#'*#' +4#"%# &'#1#',+ .+ ./%#',+ +"%#$ <-# #. ?+./' #$&#'+0/7

1ñTI < J% U < ú 1ññTI < J%U b f #" nf5 äo7 !$ &'/&#"$/ +. '(#$4/ #" nf5 äo

Prima de riesgo

!" %=')("/$ ("%-(%(?/$5 #$ .+ *+"%(0+0 <-# -" +4#"%# +?#'$/ +. '(#$4/ #$%> 0($&-#$%/ +&+4+' &+'+ .(6'+'$# 0#. '(#$4/7

Definición 1.11

e+0/ -" +4#"%# */" 1-"*(2" 0# -%(.(0+0 0#. 0("#'/ 1I < J5 @ 0+0+ -"+ ./%#',+ L $/6'# -"*/"3-"%/ 0# '#$-.%+0/$ h < P5 < U5 7775 < +i Ñ @5 */" ?+./' #$&#'+0/ < f5

$9 L.+)+)/$ )-.+0$&),!) *+)%!" () " + .+ *+"%(0+0 0# 0("#'/ >f %+. <-#1I >fJ%? I LJ7

69 L.+)+)/$ 2%+#$ () %+)'1" () " + .+ *+"%(0+0 o% < f . >f I?+./' #$&#'+0/)#"/$ #<-(?+.#"%# *(#'%/J7

A($-+.(:+*(2" 4'>1(*+ Ig(4-'+ P7PJE

Utilidad

Numerario

x

u x( )

a h z a a b

Figura 1.1 Equivalente cierto y prima de riesgo.

L #$ .+ ./%#',+ IP/ U5 P/ UJ */" &'#)(/$ #! ' @ # . '7

- I LJ% I#!'J/ U! I# . 'J/ U%# #$ #. ?+./' #$&#'+0/7

L+ -%(.(0+0 #$&#'+0+ 0# L #$ ? I LJ%P

U 1I#! 'J!

P

U 1I# . 'J% 1I >J7

> #$ #. #<-(?+.#"%# *(#'%/ 0# L5 @ # . > #$ .+ &'()+ 0# '(#$4/ 0# L7

Ejemplo 1.10

8#+ ; % nf5 äo7 Y/"$(0#'#)/$ ./$ %'#$ +4#"%#$ 0#. !3#)&./ P7V5 *-@+$ 1-"*(/"#$ 0#-%(.(0+0 $/" 1PI < J%R∂ < 5 1UI < J% T < 5 1TI < J% < U @ .+ ./%#',+ L% IR/ a5 P/ aJ &+'+ < P% P5

< U%P^7Y+.*-.#)/$ $-$ #<-(?+.#"%#$ *(#'%/$ @ $-$ &'()+$ 0# '(#$4/7

D+'+ #. &'()#' +4#"%# <-# #$ +?#'$/ +. '(#$4/ %#"#)/$ <-# ? PI LJ%^5R78- #<-(?+.#"%# *(#'%/ $#'> #. ?+./' > %+. <-# 1PI >J%? PI LJ% ^5R7 D#'/

1PI >J% R∂ > ú R∂ >% ^5R ú >%A^5R

R BU

%U5a^

<-# #$ #. #<-(?+.#"%# *(#'%/ 0# L7L+ &'()+ 0# '(#$4/ 0# L #$ o%- I LJ . >% R . U5a^%P5RR7

D+'+ #. $#4-"0/ +4#"%#5 <-# #$ "#-%'+. +. '(#$4/5 %#"#)/$ <-# ? UI LJ%PU78- #<-(?+.#"%# *(#'%/ $#'> #. ?+./' > %+. <-# 1UI >J%? UI LJ%PU7 D#'/

1UI >J%T >% PU ú >%R5 <-# #$ #. #<-(?+.#"%# *(#'%/ 0# L7L+ &'()+ 0# '(#$4/ 0# L &+'+ #. $#4-"0/ +4#"%# #$ o%- I LJ . >%R . R% f7

D+'+ #. %#'*#' +4#"%#5 <-# &'#1(#'# .+ ./%#',+ +"%#$ <-# #. ?+./' #$&#'+0/5 %#"#)/$<-# ? TI LJ% aU7

8- #<-(?+.#"%# *(#'%/ $#'> #. ?+./' > %+. <-# 1TI >J%? TI LJ% aU7 D#'/

1TI >J% >U%aU ú >%∂ aU% `5U

<-# #$ #. #<-(?+.#"%# *(#'%/7L+ &'()+ 0# '(#$4/ 0# L &+'+ #. %#'*#' +4#"%# #$

o%- I LJ . >%R . `5U%.T5U a f

E6')%0$*+>, <=;@

<= !. #<-(?+.#"%# *(#'%/ 0# L &+'+ -" +4#"%# "#-%'+. +. '(#$4/ #$ (4-+. +. ?+./' #$&#'+0/7!. #<-(?+.#"%# *(#'%/ 0# L &+'+ -" +4#"%# +?#'$/ I&'/&#"$/J +. '(#$4/ #$ )#"/' /(4-+. I)+@/' / (4-+.J <-# #. ?+./' #$&#'+0/7

;= L+ &'()+ 0# '(#$4/ 0# L &+'+ -" +4#"%# "#-%'+. +. '(#$4/ #$ "-.+7L+ &'()+ 0# '(#$4/ 0# L &+'+ -" +4#"%# +?#'$/ I&'/&#"$/J +. '(#$4/ #$ )+@/' / (4-+.+ *#'/ I)#"/' / (4-+. + *#'/J7

F= B/ #$ *+$-+.(0+0 <-# ./$ &+4/$ <-# -" *.(#"%# '#+.(:+ &/' -"+ &2.(:+ 0# $#4-'/$ I0#+-%/)2?(.5 0# ("*#"0(/$5 #%*7J $# ..+)#" &'()+$7

Medidas de Arrow-Pratt de aversión al riesgo

L+$ +"%#'(/'#$ 0#1("(*(/"#$ F+" &#')(%(0/ *.+$(1(*+' + ./$ +4#"%#$ 0# )+"#'+ *-+.(%+%(?+#" %'#$ *+%#4/',+$ '#1#'#"%#$ + $-$ &'#1#'#"*(+$ */" '#.+*(2" +. '(#$4/7 L+$ 0#1("(*(/"#$$(4-(#"%#$ ("%#"%+" *-+"%(1(*+' .+$ (0#+$ +"%#'(/'#$5 0# )/0/ <-# $#+ &/$(6.# */)&+'+' .++?#'$(2" / &'/&#"$(2" +. '(#$4/ 0# 0/$ +4#"%#$ *-+.#$<-(#'+7 L+ (0#+ ("%-(%(?+ #$ <-# .+*-'?+%-'+ 0# .+ 1-"*(2" 0# -%(.(0+0 0# -" +4#"%#5 )#0(0+ &/' $- 0#'(?+0+ $#4-"0+5 "/$("1/')+ 0#. 4'+0/ 0# +?#'$(2" +. '(#$4/ 0# 0(*F/ +4#"%#7

Definición 1.12

8#+ 1 -"+ 1-"*(2" 0# -%(.(0+0 0#. 0("#'/ */''#$&/"0(#"%# + -" +4#"%#7 8# $-&/"# <-# 1#$ 0/$ ?#*#$ 0(1#'#"*(+6.# #" n#5 Co7

$9 !. */#1(*(#"%# 0# M''/Z9D'+%% 0# +?#'$(2" +6$/.-%+ +. '(#$4/ #" < 0# 0(*F/

+4#"%# #$

j#I < J%.1ññI < J

1ñI < J

69 !. */#1(*(#"%# 0# +?#'$(2" '#.+%(?+ +. '(#$4/ #" < 0# 0(*F/ +4#"%# #$

j) I < J%. <1ññI < J

1ñI < J

Proposición 1.5

8#+ 1 -"+ 1-"*(2" 0# -%(.(0+0 0#. 0("#'/ 0/$ ?#*#$ 0(1#'#"*(+6.# #" n#5 Co5 */''#$&/"90(#"%# + -" +4#"%#7

$9 e(*F/ +4#"%# %(#"# -" */#1(*(#"%# 0# M''/Z9D'+%% 0# +?#'$(2" +6$/.-%+ "-.+ $(@ $2./ $( 1I < J% .! ,< 7 !" #$# *+$/ #$ "#-%'+. +. '(#$4/7

69 e(*F/ +4#"%# %(#"# -" */#1(*(#"%# 0# M''/Z9D'+%% 0# +?#'$(2" +6$/.-%+ */"$9%+"%# a &/$(%(?+ $( @ $2./ $( 1I < J%..%.a <

!, 5 $(#"0/ . b f5 0/"0# a b f7*9 e(*F/ +4#"%# %(#"# -" */#1(*(#"%# 0# M''/Z9D'+%% 0# +?#'$(2" +6$/.-%+ */"$9

%+"%# a "#4+%(?+ $( @ $2./ $( 1I < J% .%.a < !, 5 $(#"0/ . b f5 0/"0# a a f7

?)#"'!%$*+>,@

$9 j#I < J%.1ññI < J

1ñI < J%f á 1ññI < J%f á 1ñI < J%, á 1I < J%,< ! .

69 A#+)/$ *+0+ -"+ 0# .+$ 0/$ ()&.(*+*(/"#$7

úJ 8#+ j#I < J%.1ññI < J

1ñI < J% ab f7 e#1("()/$ *I < J%1ñI < J5 &/' ./ <-# $# %(#"# <-#

. *ñI < J *I < J

%a5 / ./ <-# #$ ./ )($)/ ,*,< %.a *7 _#$/.?#)/$ 0(*F+ #*-+*(2" 0(1#'#"*(+.

&/' ?+'(+6.#$ $#&+'+6.#$5 /6%#"(#"0/ <-# *I < J%3%.a < 5 #" 0/"0# 3 #$ -"+ */"$%+"%#&/$(%(?+7 D/' %+"%/5

,1

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%3%.a < ú 1I < J%.

3

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!, 5 $(#"0/ . b f

õJ 8# $-&/"# <-# 1I < J%..%.a < !, 5 $(#"0/ . b f5 0/"0# a b f7

!"%/"*#$ 1ñI < J% .a%.a < ú 1ññI < J%..aU%.a < 7

D/' %+"%/5 j#I < J%.1ññI < J

1ñI < J%ab f7

*9 8# 0#)-#$%'+ 0# )+"#'+ +">./4+ + 697

Algunos comentarios sobre las distintas escalas de medidaL+ #$*+.+ #" <-# $# )(0# .+ -%(.(0+0 #$&#'+0+ 0# A/" B#-)+""9C/'4#"$%#'" #$ *+'0("+.("%#'?+./ I*/)/ .+ 0# %#)&#'+%-'+ #" 4'+0/$ *#"%,4'+0/$J7 8( .+ %'+"$1/')+)/$ )#0(+"%#-"+ +&.(*+*(2" +1," &/$(%(?+5 .+ 1-"*(2" 0# -%(.(0+0 #$&#'+0+ '#$-.%+"%# #$ #<-(?+.#"%#+ .+ &'()#'+ I*/)/ .+ #$*+.+ g+F'#"F#(% #$ #<-(?+.#"%# + .+ *#"%,4'+0+5 */" */''#$&/"90#"*(+ f /Y + TU /g @ Pff /Y + UPU /g5 <-# $# */"$(4-# )#0(+"%# .+ %'+"$1/')+*(2"M I < /J% P5V < /!TUJ7

W(#"# $#"%(0/ 0#*(' k.+ -%(.(0+0 0# 6 #$ )+@/' <-# .+ 0# 0l @ %+)6(=" k.+ 0(1#'#"*(+0# -%(.(0+0 #"%'# 0 @ 3 #$ *("*/ ?#*#$ )+@/' <-# #"%'# 6 @ 3 l5 &#'/ "/ ./ %(#"# 0#*(' k.+

-%(.(0+0 0# 6 #$ 0/6.# <-# .+ 0# 0l / k.+ -%(.(0+0 0# 6 #$ N -"(0+0#$ )+@/' <-# .+ 0# 0l7!. %(&/ 0# #$*+.+ +&.(*+6.# &+'+ '#&'#$#"%+' -"+ )+4"(%-0 0#&#"0# 0# .+$ *+'+*%#',$%(9*+$ 1,$(*+$ @ .24(*+$ 0# 0(*F+ )+4"(%-07 L+ .($%+ $(4-(#"%# )#"*(/"+ .+$ #$*+.+$ )>$ ()9&/'%+"%#$5 $#4-(0+$ 0# +.4H" #3#)&./ 0# )+4"(%-0 +. <-# $# +&.(<-#"E

Escala Ordinal

?.%)G$ I 6 %(#"# (4-+. / )+@/' 0-'#:+ <-# 0 $( @ $2./ $( ? I 6J n ? I 0JJ7B/ %(#"# $#"%(0/ 0#*(' k 6 %(#"# 0/6.# 0-'#:+ <-# 0l7 B/ $# F+ #"*/"%'+0/5 <-# $#&+9

)/$5 .+ )+"#'+ 0# )#0(' .+ 0-'#:+ 0# ./$ )+%#'(+.#$ */" -"+ #$*+.+ )>$ '(*+ / #.+6/'+0+<-# .+ /'0("+.7

Escala Cardinal-Intervalo

B)#2)%$!.%$ I 6 %(#"# (4-+. / )+@/' %#)&#'+%-'+ <-# 0 $( @ $2./ $( ? I 6J n ? I 0JJ7B/ %(#"# $#"%(0/ 0#*(' k 6 %(#"# 0/6.# %#)&#'+%-'+ <-# 0l5 &#'/ $, 0#*(' <-# k.+ 0(1#9

'#"*(+ 0# %#)&#'+%-'+ #"%'# 6 @ 0 #$ 0/6.# <-# #"%'# 3 @ 4l7

Escala Cardinal-Ratio

H)'" " '$&("3 '+, )'2)*+8+*$% &$ .,+($( () #)(+($ I 6 %(#"# (4-+. / )+@/' &#$/ / $+.0/<-# 0 $( @ $2./ $( ? I 6J n ? I 0JJ7

B/ %(#"# $#"%(0/ 0#*(' k# %(#"# -" &#$/ %'#$ -"(0+0#$ )+@/' <-# C5 &#'/ $, 0#*(' <-# #%(#"# -" &#$/ 0/6.# <-# Cl7

Escala Cardinal Absoluta

I$&(" ), ).%"' I 6 %(#"# (4-+. / )+@/' $+.0/ #" #-'/$ <-# 0 $( @ $2./ $( ? I 6J n ? I 0JJ7W(#"#" $#"%(0/ %/0+$ .+$ $#"%#"*(+$ +"%#'(/'#$7

1.4. JUEGOS EN FORMA EXTENSIVA

D+'+ ("%'/0-*(' .+ 1/')+ #;%#"$(?+ 0# '#&'#$#"%+*(2" 0# -" 3-#4/ "/$ $#'?()/$ 0#. $(94-(#"%# 3-#4/ $#"*(../7

Ejemplo 1.11

D#0'/ $-6+$%+ -" 6(..#%# 0# af #-'/$ #"%'# Y+'./$ @ N.+"*+ */" .+$ $(4-(#"%#$ '#4.+$E $# 3-#4+ &/' %-'"/$7 M<-=. + <-(#" .# %/*+ 3-4+' &-#0# &+$+'5 / &-3+' */" Uf #-'/$ )>$ <-##. +"%#'(/' I$-&/"(#"0/ <-# ./$ %(#"#J7 !)&(#:+ N.+"*+ I&+$+"0/ / &-3+"0/ */" Uf

#-'/$J7 8( -" 3-4+0/' 0#*(0# &+$+'5 @+ "/ &-#0# &-3+' #" -"+ 3-4+0+ &/$%#'(/'7 q+"+ #.H.%()/ #" &-3+'5 <-# $# ..#?+ #. 6(..#%#7 8( "("4-"/ F+ &-3+0/ $# ..#?+" Ua #-'/$ *+0+-"/7 M)6/$ 3-4+0/'#$ 0#6#" &+4+' $- H.%()+ &-3+7 M&+'%# 0# .+$ '#4.+$ #$ 0# */"/*(9)(#"%/ */)H" <-# *+0+ 3-4+0/' %(#"# $2./ ^f #-'/$7

8#+"E N.+"*+5 .+ 3-4+0/'+ P5 Y+'./$5 #. 3-4+0/' U7 D/0#)/$ '#&'#$#"%+' .+ $(%-+*(2"0#$*'(%+ #" #. #3#)&./ +"%#'(/' )#0(+"%# >'6/. '#&'#$#"%+0/ #" .+ g(4-'+ P7U7

Jugadora 1

Jugadora 1

Jugador 2

Jugador 2 P

P

P

P

20

40

60

20

25, 25

0, 30

30, 0

20, 10

10,40

Figura 1.2 Subasta de un billete de 50 euros.

!. >'6/. +"%#'(/' %(#"# -" &-"%/ ("(*(+. / '+,:5 0#$0# #. <-# $# #)&(#:+ #. 3-#4/5 #"#$%# *+$/ .+ #.#**(2" 0# N.+"*+ I3-4+0/'+ PJ #"%'# &+$+' IDJ / &-3+' */" Uf #-'/$ IUfJ7!../ 0+ .-4+' + 0/$ '+)+$5 -"+ &+'+ *+0+ &/$(6.# #.#**(2" 0# N.+"*+7

L+ &'()#'+ '+)+5 + .+ <-# $# +**#0# $( N.+"*+ #.(4# D5 + .+ ?#: $# 0(?(0# #" 0/$'+)+$5 0#$0# #. "/0/ <-# */''#$&/"0# + .+ #.#**(2" 0# Y+'./$ #"%'# &+$+' IDJ / &-3+'*/" Uf #-'/$ IUfJ7 8( Y+'./$ 0#*(0# &+$+' IDJ5 $# %#')("+ #. 3-#4/ @ *+0+ 3-4+0/' '#*(6#Ua #-'/$5 )(#"%'+$ <-# $( 0#*(0# &-3+' */" Uf #-'/$5 #. 3-#4/ %#')("+5 ..#?>"0/$# Y+'./$./$ af #-'/$ @ %#"(#"0/ <-# &+4+' $- &-3+5 &/' ./ <-# /6%(#"# -" 6#"#1(*(/ 0# Tf #-'/$5

$(" <-# N.+"*+ %#"4+ <-# &+4+' "( '#*(6(' "+0+ I$(%-+*(2" <-# $# '#*/4# #" #. "/0/ <-#..#?+ +$/*(+0/$ ./$ ?+./'#$ f5 TfJ7

L+ $#4-"0+ '+)+5 + .+ <-# $# +**#0# $( N.+"*+ &-3+ */" Uf #-'/$5 + $- ?#: $# 0(?(0##" 0/$ '+)+$5 0#$0# #. "/0/ <-# */''#$&/"0# + .+ #.#**(2" 0# Y+'./$ #"%'# &+$+' IDJ /&-3+' */" Rf #-'/$ IRfJ5 Uf #-'/$ )>$ <-# .+ &-3+ 0# N.+"*+7 8( #. 3-4+0/' U #.(4# D5 $#+*+6+ #. 3-#4/5 '#*(6(#"0/ N.+"*+ ./$ af #-'/$5 &#'/ %#"(#"0/ <-# &+4+' ./$ Uf 0# $-&-3+5 $(" <-# Y+'./$ '#*(6+ "( &+4-# "+0+7 8( Y+'./$ &-3+ */" Rf #-'/$ ..#4+ #. %-'"/ 0#"-#?/ + N.+"*+ <-# 0#6# #.#4(' #"%'# &+$+' I@ $# +*+6+ #. 3-#4/5 */" &+4/$ .Uf5 Pf'#$&#*%(?+)#"%#5 @+ <-# N.+"*+ 0#6# &+4+' ./$ Uf #-'/$ 0# $- H.%()+ &-3+ @ Y+'./$ ./$Rf #-'/$ 0# $- H.%()+ &-3+5 '#*(6(#"0/ =$%# ./$ af #-'/$J / &-3+' */" ^f5 '#*(6(#"0/N.+"*+ ./$ af #-'/$ @ 0#6(#"0/ &+4+' ^f #-'/$ @ Y+'./$ Rf5 */''#$&/"0(#"%#$ + $- H.%(9)+ &-3+5 */" ./ <-# %+)6(=" $# +*+6+ #. 3-#4/ &-#$ "("4H" 3-4+0/' %(#"# )>$ 0# ^f#-'/$ &+'+ $#4-(' &-3+"0/7

L/$ #.#)#"%/$ <-# 0#1("#" #. >'6/. 0#. 3-#4/ $/"E

L/$ 3-4+0/'#$5 <-# #" #$%# *+$/ $/" .+ 3-4+0/'+ P IN.+"*+J @ #. 3-4+0/' U IY+'./$J7d" */"3-"%/ 0# "/0/$5 ./$ *-+.#$ */''#$&/"0#" + $(%-+*(/"#$ 0# #.#**(2" 0# +.4-"/0# ./$ 3-4+0/'#$ / 0# 1("+. 0#. 3-#4/7d" */"3-"%/ 0# +**(/"#$5 <-# $/" .+$ <-# #".+:+" -" "/0/ */" /%'/5 @ <-# */''#$9&/"0#" + #.#**(/"#$ 0# ./$ 3-4+0/'#$7d"/$ ?#*%/'#$ 0# &+4/$5 *+0+ -"/ 0# ./$ *-+.#$ #$%> +$/*(+0/ + -" "/0/ 0# 1("+. 0# 3-#4/ @ <-# %(#"# 0/$ */)&/"#"%#$5 .+ &'()#'+ 0# .+$ *-+.#$ '#*/4# #. &+4/ I/ .+-%(.(0+0J <-# '#*(6# / <-# /6%(#"# #. 3-4+0/' P5 @ .+ $#4-"0+ 0# .+$ *-+.#$ '#*/4# #.&+4/ I/ .+ -%(.(0+0J <-# '#*(6# / <-# /6%(#"# #. 3-4+0/' U $( #. 3-#4/ %#')("+ #" #$#"/0/7

!. #3#)&./ +"%#'(/' #$ -" 3-#4/ */" )/?()(#"%/$ $-*#$(?/$ 0# ./$ 3-4+0/'#$7 A#+)/$ +*/"%("-+*(2" -" #3#)&./ */" )/?()(#"%/$ $()-.%>"#/$5 #" #. <-# F+6'> <-# ("%'/0-*('-" "-#?/ */"*#&%/ &+'+ &/0#' '#&'#$#"%+' #. 3-#4/ &/' )#0(/ 0# -" 0(+4'+)+7

Ejemplo 1.12 J& 4.)1" () &$' #",)($'

L/$ 3-4+0/'#$ IP @ UJ 0#&/$(%+" 0# )+"#'+ $()-.%>"#+ $#"0+$ )/"#0+$ 0# -" #-'/ $/96'# -"+ )#$+7 8( '#$-.%+" 0/$ *+'+$ / 0/$ *'-*#$5 #. 3-4+0/' P '#*/4# ./$ 0/$ #-'/$5)(#"%'+$ <-# $( F+@ -"+ *+'+ @ -"+ *'-:5 #. 3-4+0/' U $# ..#?+ ./$ 0/$ #-'/$7

L+ '#&'#$#"%+*(2" #" 1/')+ #;%#"$(?+ 0# #$%# 3-#4/ $# '#*/4# #" .+ g(4-'+ P7T7

Jugador 1

Jugador 2

Jugador 2

C

C

C X

X

X

1, 1

1, 1

1, 1

1, 1

Figura 1.3 El juego de las monedas.

!" .+ '#&'#$#"%+*(2" $# /6$#'?+ <-# ./$ 0/$ "/0/$ */''#$&/"0(#"%#$ + -"+ 0#*($(2"0#. 3-4+0/' U #$%>" -"(0/$ )#0(+"%# -" $#4)#"%/ 0# '#*%+ */" %'+:/ 0($*/"%("-/7 8#0(*# <-# #$%/$ 0/$ "/0/$ 1/')+" -" */"3-"%/ 0# ("1/')+*(2" &+'+ #. 3-4+0/' U5 &-#$%/<-# 0(*F/ 3-4+0/' "/ $+6# #" *->. 0# ./$ "/0/$ 0# 0(*F/ */"3-"%/ $# #"*-#"%'+7

!. 3-#4/ +"%#'(/' %+)6(=" $# &-#0# '#&'#$#"%+' 0# .+ $(4-(#"%# 1/')+ #<-(?+.#"%#Ig(4-'+ P7RJ5 #" 0/"0# .+ H"(*+ 0(1#'#"*(+ #$ <-# $# '#&'#$#"%+ +. 3-4+0/' U #" .+ '+,:0#. 3-#4/ @ +. 3-4+0/' P #" #. */"3-"%/ 0# ("1/')+*(2" 1/')+0/ &/' 0/$ "/0/$

Jugador 2

Jugador 1

Jugador 1

C

C

C

X

X

X

1, 1

1, 1

1, 1

1, 1

Figura 1.4 El juego de las monedas.

!" 4#"#'+.5 -" */"3-"%/ 0# ("1/')+*(2" #$ -" */"3-"%/ 0# "/0/$ 0# 0#*($(2" &+'+ #.)($)/ 3-4+0/'7 Y-+"0/ + -" 3-4+0/' .# %/*+ 3-4+' 0#$0# -" */"3-"%/ 0# ("1/')+*(2" #.

3-4+0/' "/ $+6# #" *->. 0# ./$ "/0/$ &#'%#"#*(#"%#$ + 0(*F/ */"3-"%/ $# #"*-#"%'+7L+$ */"0(*(/"#$ <-# 0#6#" *-)&.(' ?+'(/$ "/0/$ &+'+ &#'%#"#*#' +. )($)/ */"3-"%/0# ("1/')+*(2" $/" .+$ $(4-(#"%#$E

L/$ */"3-"%/$ 0# ("1/')+*(2" 0#. 3-4+0/' / à 5 */"%(#"#" $2./ "/0/$ 0# 0#*($(2"0#. 3-4+0/' /7Y+0+ "/0/ 0# 0#*($(2" 0#. 3-4+0/' / #$%> */"%#"(0/ #" -"/ @ $2./ -"/ 0# ./$ */"9 3-"%/$ 0# ("1/')+*(2" 0# #$# 3-4+0/'7L+$ )($)+$ +**(/"#$ 0#6#" #$%+' 0($&/"(6.#$ &+'+ -" 3-4+0/' #" *+0+ -"/ 0# ./$"/0/$ 0# -" */"3-"%/ 0# ("1/')+*(2"7 e# "/ $#' +$,5 0(*F/ 3-4+0/' %#"0',+ -"+&($%+ $/6'# #. "/0/ #" <-# $# #"*-#"%'+5 + &+'%(' 0# .+$ +**(/"#$ <-# #$%>" 0($&/"(9

6.#$ #" #$# "/0/7A#+)/$ + */"%("-+*(2" /%'/ #3#)&./ #" #. *-+. -"/ 0# ./$ )/?()(#"%/$ ./ '#+.(:+ .+"+%-'+.#:+ I/ #. +:+'J7

Ejemplo 1.13

8# %(#"# -"+ 6+'+3+ #$&+\/.+ 0# *+'%+$5 0/"0# .+$ *+'%+$ #$%>" )#:*.+0+$ +.#+%/'(+)#"%#7Y+0+ -"/ 0# ./$ 0/$ 3-4+0/'#$5 N.+"*+ @ Y+'./$5 0#&/$(%+ -" 6(..#%# 0# a #-'/$ #" .+)#$+7 M */"%("-+*(2"5 N.+"*+ %/)+ -"+ *+'%+ 0# .+ 6+'+3+ @ */)&'-#6+ *->. #$7 B+0(#)>$ ?# .+ *+'%+7 !"%/"*#$ N.+"*+ &-#0# +&/$%+'5 &/"(#"0/ a #-'/$ )>$ #" .+ )#$+5 /

'#%('+'$#7 8( $# '#%('+5 #. 0("#'/ <-# F+@ #" .+ )#$+ #$ &+'+ N.+"*+ $( .+ *+'%+ #$*/4(0+ #$-" /'/ / -"+ */&+5 $(#"0/ &+'+ Y+'./$ $( .+ *+'%+ #" *-#$%(2" #$ -"+ #$&+0+ / -" 6+$%/7

8( F+ /&%+0/ &/' +&/$%+'5 Y+'./$ &-#0# '#*/4#' .+ +&-#$%+5 &/"(#"0/ a #-'/$ )>$ #" .+6+"*+5 / &+$+'7 !" #. &'()#' *+$/ $# ..#?+ %/0/ N.+"*+ $( .+ *+'%+ #$*/4(0+ #$ -" /'/ /-"+ */&+5 / %/0/ Y+'./$ $( $# %'+%+ 0# -"+ #$&+0+ / -" 6+$%/7 8( Y+'./$ &+$+ $# ./ ..#?+%/0/ N.+"*+5 *-+.<-(#'+ <-# $#+ .+ *+'%+ #$*/4(0+7

8#+"E N.+"*+5 .+ 3-4+0/'+ P5 Y+'./$ #. 3-4+0/' U7L+ '#&'#$#"%+*(2" 0# #$%# 3-#4/ #" 1/')+ #;%#"$(?+ +&+'#*# #" .+ g(4-'+ P7a7

Jugador 0

Jugadora 1

Jugador 2

Jugador 2

Jugadora 1

O-C

A

A

RA

RA

P

P

E-B

R

R

10, 10

10, 105, 5

5, 5

5, 5

5, 5

1/2

1/2

Figura 1.5 Juego de cartas.

Y-+"0/ F+@ -" )/?()(#"%/ <-# ./ '#+.(:+ .+ "+%-'+.#:+5 $# '#&'#$#"%+ */)/ $( ./'#+.(:+'+ #. 3-4+0/' f7 A#)/$ #" #. 0(+4'+)+ <-# .+ '+,: 0#. 3-#4/5 #" #$%# *+$/5 */''#$9

&/"0# + -"+ ("%#'?#"*(2" 0#. +:+' I3-4+0/' fJ5 <-# 0#%#')("+ $( .+ *+'%+ <-# %/)+ .+ 3-4+0/'+ P #$ 0# +.4-"/ 0# ./$ &+./$ <-# .+ 1+?/'#*#"5 /'/$ / */&+$ Im9YJ / 0# ./$ <-#"/ .+ 1+?/'#*#"5 #$ 0#*('5 #$&+0+$ / 6+$%/$ I!9NJ7 L+ &'/6+6(.(0+0 0# <-# $#+ 0# *+0+ -"+0# .+$ 1/')+$ I1+?/'+6.# / 0#$1+?/'+6.# + N.+"*+J #$ (4-+. + P/ U5 %+. */)/ +&+'#*# #" #.0(+4'+)+7

D/' %+"%/5 *-+"0/ #" -" 3-#4/ F+@ +.4H" )/?()(#"%/ '#+.(:+0/ &/' #. +:+' / &/' .+"+%-'+.#:+5 $# ("%'/0-*# #. 3-4+0/' f @ $# #$&#*(1(*+ .+ &'/6+6(.(0+0 0# *+0+ -"+ 0# .+$'+)+$ <-# $-'4#" 0#. "/0/ */''#$&/"0(#"%# +. 3-4+0/' f7

Elementos de un juegoM */"%("-+*(2" $# 0#1("#" ./$ #.#)#"%/$ <-# *+'+*%#'(:+" + -" 3-#4/ #" 1/')+ #;%#"$(?+7

<= 8#+ #. */"3-"%/ 0# 3-4+0/'#$E 5 % hf5 P5 U5 7775 +i78( "/ F+@ )/?()(#"%/$ 0# +:+' / 0# .+ "+%-'+.#:+5 #"%/"*#$ 5 % hP5 U5 7775 +i7

;= 8#+ ; #. */"3-"%/ 0# "/0/$7 d" "/0/ '#&'#$#"%+ -"+ &/$(6.# $(%-+*(2" 0#. 3-#4/7!"%'# ./$ "/0/$ F+@ -"/ 0# #../$ <-# #$ .+ '+,: 0#. 3-#4/5 &-"%/ 0# */)(#":/ 0#. 3-#4/7 e(*F/ "/0/ $# '#&'#$#"%+ &/' ( I'#1#'#"%# +. /'(4#"J7

M */"%("-+*(2" $# 0#1("# .+ $(4-(#"%# 1-"*(2"E

p E ; r ; < r pI < J

#" 0/"0# pIOJ%O5 @ &+'+ < Ç O5 pI < J #$ #. "/0/ (")#0(+%+)#"%# &'#0#*#$/' 0# < 7 8#+ pUI < J%pIpI < JJ5 pTI < J%pIpIpI < JJJ5 @ +$, $-*#$(?+)#"%#5 &/' ./ <-# (%#9'+"0/ pI < J $# /6%(#"#" %/0/$ ./$ "/0/$ &'#0#*#$/'#$ 0# < 7

8#+ HI < J%p.PI < J5 #. */"3-"%/ 0# "/0/$ <-# $(4-#" (")#0(+%+)#"%# + < 7d" "/0/ #$ %#')("+. $( "/ .# $(4-# "("4H" /%'/ "/0/5 $(#"0/ -" "/0/ 0# 0#*(9

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#. */"3-"%/ 0# "/0/$ %#')("+.#$ 0#. 3-#4/7 4I ; J% h < à ; E HI < J Çi% ; . & I ; J5 #. */"3-"%/ 0# "/0/$ 0# 0#*($(2" 0#.

3-#4/7F= 8#+ 6 #. */"3-"%/ 0# %/0+$ .+$ &/$(6.#$ +**(/"#$7

8# 0#1("# .+ 1-"*(2"E

a E ; . hOi r 6

< r aI < J

<-# F+*# */''#$&/"0#' + *+0+ "/0/ 0($%("%/ 0#. /'(4#" +<-#..+ +**(2" aI < J <-#..#?+ 0#$0# #. "/0/ (")#0(+%/ &'#0#*#$/' pI < J +. "/0/ < 7

8# ?#'(1(*+ <-# $( < ñ5 < ññ à HI < J5 $(#"0/ < ñ Ç < ññ5 #"%/"*#$ aI < ñJ Ç aI < ññJ7 !$0#*('5 +**(/"#$ <-# &+'%#" 0#. )($)/ "/0/ @ */"0-*#" + "/0/$ 0($%("%/$5 0#6#"$#' 0($%("%+$7

D+'+ *-+.<-(#' "/0/ 0# 0#*($(2" < à 4I ; J5 '#&'#$#"%+)/$ #. */"3-"%/ 0# +*9*(/"#$ 0($&/"(6.#$ + &+'%(' 0# < &/'E

6I < J% h# à 6E P < ñ à HI < J */" #%aI < ñJi

K= D+'+ *+0+ 3-4+0/' / $#+ ; / #. */"3-"%/ 0# "/0/$ 0# 0#*($(2" #" ./$ <-# #. 3-4+0/'/ %(#"# <-# #.#4(' -"+ +**(2"7 !" -" "/0/ &+'%(*-.+' 0# 0#*($(2" $2./ )-#?# -"/0# ./$ 3-4+0/'#$7 8# %(#"# <-#

Z/ à 5

; /% 4I ; J

O /5 I à 5 5 */" / Ç I5 $# ?#'(1(*+ <-# ; / ç ; I%

A#)/$5 &/' %+"%/5 <-# .+ 1+)(.(+ h ; /i/ à 5 */"$%(%-@# -"+ &+'%(*(2"5 &/' 3-4+0/9

'#$5 0#. */"3-"%/ 0# "/0/$ 0# 0#*($(2" 4I ; J7L= d"+ 1+)(.(+ 0# */"3-"%/$ 0# ("1/')+*(2" P 5 @ -"+ 1-"*(2"

' E ; r P

< r 'I < J

<-# +$(4"+ + *+0+ "/0/ < -" */"3-"%/ 0# ("1/')+*(2" 'I < J +. <-# &#'%#"#*#7 L/$*/"3-"%/$ 0# ("1/')+*(2" 1/')+" -"+ &+'%(*(2" 0# 4I ; J7

Y/)/ F#)/$ ?($%/ +"%#'(/')#"%#5 %/0/$ ./$ "/0/$ 0# 0#*($(2" <-# &#'%#"#9*#" + -" )($)/ */"3-"%/ 0# ("1/')+*(2" %(#"#" .+$ )($)+$ +**(/"#$ 0($&/"(6.#$5

#$ 0#*('E 6I < J% 6I < ñJ5 $( 'I < J%'I < ñJ

8#+ '%'I < J5 -" */"3-"%/ 0# ("1/')+*(2" &#'%#"#*(#"%# + P 7 D/' %+"%/5 &/0#9)/$ '#&'#$#"%+' &/' 6I'J #. */"3-"%/ 0# +**(/"#$ 0($&/"(6.#$ #" #. */"3-"%/ 0#("1/')+*(2" '7

6I'J% h# à 6E # à 6I < J &+'+ < à 'i

8#+ P / #. */"3-"%/ 0# %/0/$ ./$ */"3-"%/$ 0# ("1/')+*(2" 0#. 3-4+0/' /78#+ P #. */"3-"%/ <-# */"%(#"# + %/0/$ ./$ */"3-"%/$ 0# ("1/')+*(2" */"%#"(9

0/$ #" ./$ P /5 &+'+ / à 5 7 !$ 0#*('5

P %Z/ à 5

P /

M= d"+ 1-"*(2"

o E P f# 6 r nf5 PoI'5 #J r oI'5 #J

<-# +$(4"+ &'/6+6(.(0+0#$ + +**(/"#$ #" */"3-"%/$ 0# ("1/')+*(2" 0/"0# #. )/9?()(#"%/ */''#$&/"0# + .+ "+%-'+.#:+ / +. +:+'7

8# %(#"# <-# ?#'(1(*+' <-#E

oI'5 #J%f5 $( # â 6I'J @ ;# à 6I'J

oI'5 #J% P5 O' à P f

N= d"+ 1-"*(2" 0# &+4/$

) E & I ; J r @+

< r ) I < J% I) PI < J5 7775 ) +I < JJ

#" 0/"0# ) /I < J ("0(*+ #. &+4/ / -%(.(0+0 <-# '#*(6# #. 3-4+0/' / $( $# F+ +.*+":+0/#. "/0/ %#')("+. < 7

D/' %+"%/5 '#*/4(#"0/ %/0/$ ./$ #.#)#"%/$ +"%#'(/'#$ &/0#)/$ 0+' .+ $(4-(#"%# 0#1("(*(2"7

Definición 1.13

d" 3-#4/ #" 1/')+ #;%#"$(?+ A ?(#"# #$&#*(1(*+0/ &/' ./$ $(4-(#"%#$ #.#)#"%/$E

A% h 5 5 I ; 5 pJ5 I 65 aJ5 h ; /i/ à 5 5 h P /i/ à 5 5 I 6I'JJ' à P 5 o5 ) i

Y/)/ (.-$%'+*(2" 0# *+0+ -"/ 0# ./$ #.#)#"%/$ 0#1("(0/$5 */"$(0#'#)/$ 0# "-#?/ #.!3#)&./ P7PT5 &#'/ */"$(0#'+"0/ +F/'+ #. 0(+4'+)+ 0# .+ g(4-'+ P7^5 <-# $# */''#$&/"0##;+*%+)#"%# */" #. 0(+4'+)+ 0# .+ g(4-'+ P7a5 &#'/ -%(.(:+"0/ /%'+ "/%+*(2"7 D+'+ #$%#

3-#4/ I*/" #$%# 0(+4'+)+J ?+)/$ + (' *+.*-.+"0/ *+0+ -"/ 0# ./$ #.#)#"%/$ <-# +&+'#*#"#" .+ e#1("(*(2" P7PT7

Jugador 0

Jugadora 1

Jugadora 1

Jugador 2

Jugador 2

ao

c

e

g

g

h

h

b

x1

x3

x7

x8

x9

x6

x10

x4

x5

x2

d

f

10, 10

10, 105, 5

5, 5

5, 5

5, 5

1/2

1/2


!" #$%# *+$/ $# %(#"# <-#E

<= 5 %hf5 P5 Ui7 !$ #. */"3-"%/ 0# ./$ 3-4+0/'#$5 #"%'# ./$ <-# $# #"*-#"%'+ #. 3-4+90/' "H)#'/ f <-# */''#$&/"0# +. +:+'7

;= ; % hO5 < P5 < U5 < T5 < R5 < a5 < 5 < 5 < V5 < Q5 < Pfi #$ #. */"3-"%/ 0# "/0/$7 M$/*(+0++ #$%# */"3-"%/ $# 0#1("# .+ 1-"*(2" p5 <-# F+*# */''#$&/"0#' + *+0+ "/0/0($%("%/ 0#. /'(4#" $- "/0/ (")#0(+%+)#"%# &'#0#*#$/'5 @ +. /'(4#" #. &'/&(/ /'(4#"7 D/'%+"%/5 #" #$%# *+$/ $# %(#"#E

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*(2" <-# */"0-*# + 0(*F/ "/0/7 D/' %+"%/5 #" #$%# *+$/E

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P f% hhOii

P P% hh < Pi5 h < Uii

P U% hh < T5 < aii

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6Ih < T5 < aiJ% hQ5 'i

M= L+ $(4-(#"%# 1-"*(2" +$(4"+ &'/6+6(.(0+0#$ + *+0+ -"+ 0# .+$ +**(/"#$ 0# +:+'E

oIhOi5 #J%P

U

oIhOi5 CJ%P

U

N= e#1("()/$ +F/'+ .+ 1-"*(2" 0# &+4/$5 <-# F+*# */''#$&/"0#' -" ?#*%/' 6(0()#"9$(/"+. + *+0+ -"/ 0# ./$ "/0/$ %#')("+.#$7

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Pf5 PfJ) I < PfJ% Ia5 .aJ

Ejemplo 1.14

O0#"%(1(*+' *+0+ -"/ 0# ./$ #.#)#"%/$ */"$(0#'+0/$ #" .+ e#1("(*(2" P7PT &+'+ #. 3-#4/'#&'#$#"%+0/ #" .+ g(4-'+ P7`7

Jugador 1

Jugador 2

Jugador 2

Jugador 3

a

o

e

i

i

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i

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h

h

h

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h

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g

c

x1

x3

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x8

x9

x6

x10

x11

x12

x13

x14

x15

x16

x17

x4

x5

x2

d

f

1, 4, 3

1, 3, 2

1, 5, 2

1, 0, 2

2, 1, 2

3, 1, 1

1, 1, 3

2, 1, 1

2, 1, 1

2, 4, 1

Figura 1.7 Juego del Ejemplo 1.14.

!" #$%# *+$/ $# %(#"# <-#E

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D/' %+"%/5 #" #$%# *+$/ #. 3-#4/ A5 '#&'#$#"%+0/ #" .+ g(4-'+ P7` <-#0+ *+'+*%#'(:+90/ &/'E

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1.5. JUEGOS EN FORMA ESTRATÉGICA

D+'+ '#&'#$#"%+' -" 3-#4/ #" 1/')+ #$%'+%=4(*+ "#*#$(%+)/$ &+'%(' 0#. */"*#&%/ 0# #$%'+9%#4(+ 0# -" 3-4+0/'7 d"+ #$%'+%#4(+ #$ -" &.+" */"%("4#"%#5 */)&.#%/ / '#4.+ 0# 0#*($(2"5&+'+ -" 3-4+0/'5 <-# #$&#*(1(*+ *2)/ +*%-+'> #. 3-4+0/' #" *+0+ *('*-"$%+"*(+ &/$(6.# #"<-# .# */''#$&/"0+ )/?#'7 Y/)/ F#)/$ ?($%/ #" #. +&+'%+0/ +"%#'(/'5 #. */"3-"%/ 0#%+.#$ *('*-"$%+"*(+$ $# */''#$&/"0# */" .+ 1+)(.(+ 0# ./$ */"3-"%/$ 0# ("1/')+*(2"

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Definición 1.14

d"+ #$%'+%#4(+ &-'+ &+'+ #. 3-4+0/' / à hP5 7775 +i #$ -"+ 1-"*(2"

H/ E P / r 6

' r H/I'J

*/" H/I'J à 6I'J7

D/' %+"%/5 -"+ #$%'+%#4(+ &-'+ &+'+ #. 3-4+0/' / F+*# */''#$&/"0#'5 + *+0+ */"3-"%/ 0#("1/')+*(2" 0#. 3-4+0/' / -"+ 0# .+$ +**(/"#$ 0($&/"(6.#$ #" 0(*F/ */"3-"%/ 0# ("1/')+9*(2"7

8#+ 9 / #. */"3-"%/ 0# %/0+$ .+$ #$%'+%#4(+$ &-'+$ 0#. 3-4+0/' /7e+0+ -"+ #$%'+%#4(+ &-'+ H/ à 9 / &+'+ *+0+ -"/ 0# ./$ 3-4+0/'#$ P5 U5 7775 + <-#0+

0#%#')("+0/ -" 0#$+''/../ */)&.#%/ 0#. 3-#4/5 ..#4>"0/$# + -" "/0/ %#')("+. I$+.?/ <-##;($%+" )/?()(#"%/$ 0# +:+'5 <-# $# #$%-0(+'>" &/$%#'(/')#"%#J7 8# 0(*# <-#

H% IHP5 HU5 7775 H+J à 9 P#9 U#ñ#9 +%9

#$ -" &#'1(. / -"+ */)6("+*(2" 0# #$%'+%#4(+$ &-'+$78#+ +IHJ #. "/0/ %#')("+. <-# $# +.*+":+ $( ./$ 3-4+0/'#$ 3-#4+" .+ */)6("+*(2" 0#

#$%'+%#4(+$ H5 @ $#+ 1/IHJ% ) /I+IHJJ #. &+4/ <-# '#*(6# #. 3-4+0/' / *-+"0/ ./$ 3-4+0/'#$ 3-#4+" .+ */)6("+*(2" 0# #$%'+%#4(+$ H5 ..#4>"0/$# &/' %+"%/ +. "/0/ %#')("+. +IHJ7

Ejemplo 1.15

Y+.*-.+' #. */"3-"%/ 0# #$%'+%#4(+$ &-'+$ &+'+ *+0+ -"/ 0# ./$ 3-4+0/'#$5 +$, */)/ ./$&+4/$ <-# '#*(6# *+0+ -"/ 0# ./$ 3-4+0/'#$ &+'+ *+0+ */)6("+*(2" 0# #$%'+%#4(+$ &-'+$5#" #. $(4-(#"%# 3-#4/5 *-@+ '#&'#$#"%+*(2" #" 1/')+ #;%#"$(?+ +&+'#*# #" .+ g(4-'+ P7V7

Jugador 2

Jugador 1

Jugador 1

Jugador 1

Jugador 2

Figura 1.8 Juego del Ejemplo 1.15.

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HUP z IHUPIhOiJ5 HUPIh < R5 < aiJJ% I#5 QJ

HTP z IHTPIhOiJ5 HTPIh < R5 < aiJJ% IC5 B J

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Definición 1.15

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Ejemplo 1.16

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Ejemplo 1.17

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Ejemplo 1.18 J& 4.)1" () ("6&) " #+!$(

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Representación en forma estratégica de juegos que contenganmovimientos de azar

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Ejemplo 1.19_#&'#$#"%+' #" 1/')+ #$%'+%=4(*+ #. 3-#4/ 0#. !3#)&./ P7PT5 *-@+ '#&'#$#"%+*(2" #"1/')+ #;%#"$(?+ +&+'#*# #" .+$ g(4-'+$ P7a @ P7^7

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Representación estratégica en juegos con tres jugadores

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Ejemplo 1.20

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_#&'#$#"%+)/$ .+$ #$%'+%#4(+$ &-'+$ 0#. 3-4+0/' U 0# .+ $(4-(#"%# 1/')+E

I, 5 B J5 <-# */"$($%# #" #.#4(' .+ +**(2" , $( #. 3-#4/ ..#4+ +. "/0/ < P@ #" #.#4(' .+ +**(2" B $( #. 3-#4/ ..#4+ +. "/0/ < U7

I, 5 QJ5 <-# */"$($%# #" <-# #. 3-4+0/' U #.(4# , 0#$0# < P @ Q 0#$0# < U7I%5 B J5 <-# */"$($%# #" <-# #. 3-4+0/' U #.(4# % 0#$0# < P @ B 0#$0# < U7

I%5 QJ5 #" <-# #. 3-4+0/' U #.(4# % 0#$0# < P @ Q 0#$0# < U7

D/' %+"%/5

9 U% hI, 5 B J5 I, 5 QJ5 I%5 B J5 I%5 QJi

Y/)/ #. 3-4+0/' T %(#"# -" H"(*/ */"3-"%/ 0# ("1/')+*(2" 1/')+0/ &/' ./$ "/0/$ < T5 < R5 < a5 < @ < 5 .+$ #$%'+%#4(+$ &-'+$ 0#. 3-4+0/' T $# */''#$&/"0#" */" .+$ +**(/"#$0($&/"(6.#$ 0#$0# *+0+ -"/ 0# ./$ "/0/$ <-# 1/')+" $- H"(*/ */"3-"%/ 0# ("1/')+*(2"7D/' %+"%/5

9 T% h'5 /i

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

./$ &+4/$ &+'+ *+0+ */)6("+*(2" 0# #$%'+%#4(+$7 L+ '#&'#$#"%+*(2" <-# $# /6%(#"# #$ .+$(4-(#"%#E

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1.6. JUEGOS COOPERATIVOS

[+ #" #. .(6'/ 0# A/" B#-)+"" @ C/'4#"$%#'" IPQR`J $# 0($%("4-# #"%'# 3-#4/$ "/ *//9&#'+%(?/$ @ 3-#4/$ *//&#'+%(?/$7 L+$ 1/')+$ #;%#"$(?+ @ #$%'+%=4(*+ 0# '#&'#$#"%+*(2" 0#-" 3-#4/ <-# F#)/$ ("%'/0-*(0/ #" +&+'%+0/$ +"%#'(/'#$ $# -%(.(:+" &+'+ #$%-0(+' 3-#4/$"/ *//&#'+%(?/$5 #" ./$ <-# ("%#'#$+" .+$ #$%'+%#4(+$ 0# ./$ 3-4+0/'#$ @ .+$ -%(.(0+0#$ /./$ &+4/$ <-# $# /6%(#"#" &+'+ */)6("+*(/"#$ 0# #$%'+%#4(+$7

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

8#+ 5 % hP5 U5 7775 +i

#. */"3-"%/ 0# 3-4+0/'#$7 m6$='?#$# <-# #. */"3-"%/ 0# 3-4+0/'#$ #$ 1("(%/78#+

7I 5 J

#. */"3-"%/ 0# .+$ &+'%#$ 0# 5 5 <-# #$%> 1/')+0/ &/' *+0+ -"+ 0# .+$ &/$(6.#$ */+.(*(/"#$<-# $# &-#0#" 1/')+' I("*.-@#"0/ .+ */+.(*(2" $(" 3-4+0/'#$ <-# #$ J7

8-&/"4+)/$ <-# .+$ -%(.(0+0#$ 0# ./$ 3-4+0/'#$ $/" %'+"$1#'(6.#$5 ./ *-+. <-(#'# 0#*('<-# .+$ 4+"+"*(+$ / &='0(0+$ <-# $# /6%(#"#" +. +*%-+' */)/ */+.(*(2" &-#0#" '#&+'%('$##"%'# ./$ 3-4+0/'#$ <-# .+ */)&/"#"7

8# ..+)+ 1-"*(2" *+'+*%#',$%(*+ + -"+ 1-"*(2" <-# +$(4"+ + *+0+ */+.(*(2" -" "H)#'/

'#+.5 +$(4"+"0/ +. */"3-"%/ ?+*,/ #. ?+./' *#'/7 !$ 0#*('5v E 7I 5 J r @

9 r vI9 J?#'(1(*+"0/ <-# vIJ% f7

D+'+ -"+ */+.(*(2" 9 5 + vI9 J $# .# ..+)+ ?+./' 0# .+ */+.(*(2" @ #$ #. ?+./' ),"()/ <-#&-#0# /6%#"#' .+ */+.(*(2" $( %/0/$ $-$ )(#)6'/$ $# +$/*(+" @ 3-#4+" #" #<-(&/7 8# %'+%+&/' %+"%/ 0#. ?+./' <-# -"+ */+.(*(2" &-#0# 4+'+"%(:+'$# <-# /6%#"0'> $( '#+.)#"%# 1-"9*(/"+ */)/ %+. */+.(*(2" @ %/)+ $-$ 0#*($(/"#$ 0# )+"#'+ +0#*-+0+7

M */"%("-+*(2" $# 0#1("# 1/')+.)#"%# <-= $# #"%(#"0# &/' '#&'#$#"%+*(2" 0# -" 3-#9

4/ #" 1/')+ */+.(*(/"+. / #" 1/')+ 0# 1-"*(2" *+'+*%#',$%(*+7

Definición 1.16

d" 3-#4/ #" 1/')+ */+.(*(/"+. / #" 1/')+ 0# 1-"*(2" *+'+*%#',$%(*+ */" -%(.(0+0#$%'+"$1#'(6.#$ */"$($%# #"E

d" */"3-"%/ 1("(%/ 0# 3-4+0/'#$ 5 %hP5 U5 7775 +i7d"+ 1-"*(2" *+'+*%#',$%(*+5 <-# +$/*(+ + *+0+ $-6*/"3-"%/ 9 0# 5 I/ */+.(*(2"J-" "H)#'/ '#+. vI9 J I?+./' 0# .+ */+.(*(2"J5 $(#"0/ vIJ%f7

D/' %+"%/5 "% I 5 5 vJ #$ -" 3-#4/ #" 1/')+ */+.(*(/"+. / #" 1/')+ 0# 1-"*(2" *+'+*%#9',$%(*+ */" -%(.(0+0#$ %'+"$1#'(6.#$ $( 5 @ v #$%>" #$&#*(1(*+0/$7

A#+)/$ +.4-"/$ #3#)&./$E

Ejemplo 1.22

8# */"$(0#'+" %'#$ #)&'#$+$ <-# &'/0-*#" #. )($)/ 6(#"7 e+0+$ $-$ %#*"/./4,+$5 .+

#)&'#$+ P &-#0# &'/0-*(' f5 V / P^ -"(0+0#$ 0# (1GE1G +. */$%# -"(%+'(/ 0# U -"(0+0#$)/"#%+'(+$5 .+ #)&'#$+ U &-#0# &'/0-*(' f5 R / PU -"(0+0#$ +. */$%# -"(%+'(/ 0# U -7)7@ .+ #)&'#$+ T &-#0# &'/0-*(' f5 V / PU -"(0+0#$ +. */$%# -"(%+'(/ 0# U -7)7 L+ ("?#'$+0# .+ 1-"*(2" 0# 0#)+"0+ 0#. 6(#" #$ */"/*(0+ &/' .+$ %'#$ #)&'#$+$ @ %(#"# .+ 1/')+$(4-(#"%#E

EI < J%Ta . f5à <

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)#'*+0/ #$ < % < P! < U! < T7!. &+4/ <-# /6%(#"# *+0+ 3-4+0/'+ / ?(#"# 0#%#')("+0/ &/' .+ 1-"*(2" 0# 6#"#1(9

*(/$5 0#. $(4-(#"%# )/0/E

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L+ '#&'#$#"%+*(2" 0#. 3-#4/ #" 1/')+ #$%'+%=4(*+ #$ .+ $(4-(#"%#E

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)+; hf5 Ù5 RVi%Ù

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vIhPiJ% Ù

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vIhUiJ% T^

<-# .+ 3-4+0/'+ U %(#"# +$#4-'+0/ $( 3-#4+ $- #$%'+%#4(+ < U% R / 6(#" < U%PU @

?IhTiJ%RV

<-# .+ 3-4+0/'+ T %(#"# +$#4-'+0/ 3-4+"0/ $- #$%'+%#4(+ < T% V7Y/"$(0#'#)/$ +F/'+ .+ */+.(*(2" 1/')+0+ &/' .+$ 3-4+0/'+$ P @ U7 D+'+ *+0+ */)6("+9

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)(" hPUf5 Q^5 VRi%VR

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Tabla 1.2

)< ); H$1" -.) ') 1$%$,!+G$ &$ *"$&+*+>, V<3 ;W

f f )(" hf5 f5 fi%f

f R )(" hPUf5 Q^5 VRi%VR

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V R )(" hUVV5 UP^5 PVfi%PVf

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P^ f )(" hTT^5 URf5 PQUi%PQU

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)+; hf5 VR5 PVf5 PRR5 PVf5 PVf5 PQU5 PVf5 VRi%PQU

<-# #$ #. &+4/ <-# .+ */+.(*(2" hP5 Ui $# 4+'+"%(:+ + $, )($)+ #.(4(#"0/ */)/ #$%'+%#9

4(+$ < P%P^5 < U%f7 m6$='?#$# <-# <-#0+ $(" #$&#*(1(*+' *2)/ $# '#&+'%# #. ?+./' /69%#"(0/ &/' .+ */+.(*(2" #"%'# .+$ 0/$ 3-4+0/'+$ <-# .+ */)&/"#"7

D/' %+"%/5 $# %(#"# <-#vIhP5 UiJ% PQU

D'/*#0(#"0/ 0# )+"#'+ +">./4+ */" .+$ /%'+$ 0/$ */+.(*(/"#$ 1/')+0+$ &/' 0/$ 3-94+0/'+$5 $# ..#4+ + <-#

vIhP5 TiJ% PQU

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M">./4+)#"%#5vIhU5 TiJ% PRR

<-# #$ #. &+4/ <-# .+ */+.(*(2" hU5 Ti $# 4+'+"%(:+ + $, )($)+ #.(4(#"0/ */)/ #$%'+%#94(+$ < U%PU5 < T% f / 6(#" < U% R5 < T%V5 / 6(#" < U%f5 < T% PU5 / 6(#" < U% R5

< T%PU7

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<-# #$ #. &+4/ <-# .+ */+.(*(2" hP5 U5 Ti $# 4+'+"%(:+ + $, )($)+ #.(4(#"0/ */)/ #$%'+9%#4(+$ *-+.<-(#'+ 0# .+$ $(4-(#"%#$E

< P%V5 < U% PU5 < T%f

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Ejemplo 1.23

8-&/"4+)/$ <-# $# &.+"%#+ .+ "#*#$(0+0 0# +6+$%#*#' 0# #.#*%'(*(0+0 + %'#$ &/6.+*(/9"#$7 D+'+ #../ $# */"$%'-('> -"+ '#0 0# %#"0(0/$ #.=*%'(*/$ <-# */"#*%# 0(*F+$ &/6.+9*(/"#$ */" .+ *#"%'+. #.=*%'(*+7 !" .+ g(4-'+ P7Q $# &'#$#"%+" ./$ */$%#$ 0# %/0/$ ./$

&/$(6.#$ %#"0(0/$ <-# ("%#'*/"#*%+" .+$ &/6.+*(/"#$ IP5 U5 TJ @ .+ *#"%'+. #.=*%'(*+ IfJ7_#&'#$#"%+' #. 3-#4/ #" 1/')+ */+.(*(/"+.7

1

1 2

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3

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Figura 1.9

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L+ 1-"*(2" *+'+*%#',$%(*+ #$ -"+ 1-"*(2" */" 0/)("(/ 0#. */"3-"%/ 0# .+$ &+'%#$ 0#hP5 U5 Ti5 <-# %/)+ ?+./'#$ #" @ @ <-# #$%> 0#1("(0+ 0# .+ $(4-(#"%# 1/')+E


vI9 J f f f f f P P P

EJERCICIOS PROPUESTOS

1.1 IN(#')+" @ g#'">"0#: IPQQTJJ7 d"+ &#'$/"+ #$ #.#4(0+ +.#+%/'(+)#"%# @ $# .#&'#$#"%+" .+$ $(4-(#"%#$ T ./%#',+$E

$9 q+"+' a -7)7 */" &'/6+6(.(0+0 f5a @ "/ 4+"+' "+0+ */" &'/6+6(.(0+0 f5a769 q+"+' Pf -7)7 */" &'/6+6(.(0+0 f5Ua @ "/ 4+"+' "+0+ */" &'/6+6(.(0+0f5`a7

*9 q+"+' Pf/ T -7)7 */" &'/6+6(.(0+0 f5`a @ "/ 4+"+' "+0+ */" &'/6+6(.(0+0f5Ua7

r$%+ &'#1(#'# +J + 6J @ 6J + *J7 8-&/"0'#)/$ <-# #$ -" )+;()(:+0/' 0# .+-%(.(0+0 #$&#'+0+ @ <-# .+ 1-"*(2" 0# -%(.(0+0 #$&#'+0+ +$/*(+0+ + $-$ &'#1#'#"9*(+$ #$ 0# A/" B#-)+""9C/'4#"$%#'"7 W/)+"0/ 1If -7)7J%f @ 1IPf -7)7J%P5@ */"$(0#'+"0/ $-$ &'#1#'#"*(+$5 #"*-#"%'# ./$ .,)(%#$ )>;()/$ @/ / ),"()/$ &+'+1Ia -7)7J @ 1IPf/ T -7)7J7 W#"(#"0/ #" *-#"%+ <-# .+ 1-"*(2" 0# -%(.(0+0 <-# $#

1/')+ */" #$%+$ &'#1#'#"*(+$ #$ */"%("-+ @ 0(1#'#"*(+6.#5 s<-= &/0#)/$ 0#*('+*#'*+ 0# .+ +?#'$(2" +. '(#$4/ 0# .+ &#'$/"+ #" #. ("%#'?+./ nf5 Pfot

1.2 d" ("0(?(0-/ F+ &#"$+0/ '#+.(:+' -"+ ("?#'$(2" #" -" +*%(?/ 1("+"*(#'/ 0# 4'+"?/.+%(.(0+05 <-# &'/&/'*(/"+ -"+ 4+"+"*(+ 6'-%+ 0# f -7)7 I#$ 0#*('5 &='0(0+ 0#.+ *+"%(0+0 ("?#'%(0+J */" &'/6+6(.(0+0 T/ R @ 0# ^ -7)7 */" &'/6+6(.(0+0 P/ R &/'*+0+ -7)7 ("?#'%(0+ IP -7)7 0# '#*-&#'+*(2" 0# .+ ("?#'$(2"! a -7)7 0# '#"0(9)(#"%/ "#%/J7 8(#"0/ $-$ &'#1#'#"*(+$ '#&'#$#"%+6.#$ )#0(+"%# .+ 1-"*(2" 0# -%(9.(0+0 1IWJ% ."IW! QJ @ $- '(<-#:+ +*%-+. Wf b P5 s*->"%/ 0#*(0('> ("?#'%('t

1.3 I]#"0#'$/" @ u-+"0% IPQVaJJ7 d" */"$-)(0/' *-@+ */"0-*%+ $# +0+&%+ + ./$ +;(/9)+$ 0# A/" B#-)+""9C/'4#"$%#'" @ *-@+ '(<-#:+ ("(*(+. #$ 0# Wf%P^f7fff -7)75#$%> $-3#%/ +. '(#$4/ 0# -" ("*#"0(/7 L+ &'/6+6(.(0+0 0# -" 4'+" ("*#"0(/5 */"`f7fff -7)7 #" &='0(0+$5 #$ f5fa @ .+ 0# -" ("*#"0(/ 0#$%'-*%/'5 */" PUf7fff-7)7 #" &='0(0+$5 #$ %+)6(=" f5fa7 8- 1-"*(2" 0# -%(.(0+0 #$ 1IWJ%WP/ U7 sY->.#$ .+ )>;()+ *+"%(0+0 <-# #$%+'> 0($&-#$%/ + &+4+' &/' -"+ &2.(:+ 0# $#4-'/$<-# .# +$#4-'# */"%'+ #. '(#$4/ 0# ("*#"0(/t

1.4 N.+"*+ %(#"# -" '(<-#:+ +*%-+. 0# Wf%U7fff -7)7 @ F+ 0# 0#*(0(' $( ("?#'%('> /

"/ #" -" &'/@#*%/ <-# '#<-(#'# <-# ("?(#'%+ %/0/$ $-$ +F/''/$ IWfJ5 @ <-# 4#"#'+./$ $(4-(#"%#$ '#"0()(#"%/$E .+ &='0(0+ 0#. *+&(%+. ("?#'%(0/ */" -"+ &'/6+6(.(0+00# P/ U5 @ -" '#"0()(#"%/ 6'-%/ 0# ^7fff -7)7 IU7fff!R7fffJ */" &'/6+6(.(0+0P/ U7 8+6(#"0/ <-# $-$ &'#1#'#"*(+$ &-#0#" $#' '#&'#$#"%+0+$ &/' .+ 1-"*(2" 0#-%(.(0+0 1IWJ%WP/ U5 s<-= 0#*($(2" %/)+'>t

8-&/"4+)/$ <-# Y+'./$ */)&+'%# .+$ )($)+$ &'#1#'#"*(+$ <-# N.+"*+ @ &/9$## #. )($)/ "(?#. 0# '(<-#:+5 Wf%U7fff -7)7 8( %-?(#'+" <-# 0#*(0(' #"%'# -"+("?#'$(2" */"3-"%+ Iafv *+0+ -"/5 #$ 0#*('5 2 /% P7fff -7)7J / "/ ..#?+' + *+6/#. &'/@#*%/5 s<-= 0#*($(2" %/)+',+"t

1.5 IY+)&6#.. IPQQaJJ7 d" ("0(?(0-/ I$(" #$*'H&-./$ *,?(*/$J */" -"+ 1-"*(2" 0#-%(.(0+0 $/6'# .+ '(<-#:+ <-# ?(#"# 0+0+ &/' 1IWJ% ." IW!UfJ @ %(#"# -"+ '#"9

+-)#"%+',+ #" UR7fff #-'/$7 !. #"%'+"%# &/%#"*(+. &-#0# #"%'+' / "/ #"%'+' #" .+("0-$%'(+7 8(" */$%#$ 0# */"%'/. 0# */"%+)("+*(2"5 ./$ 6#"#1(*(/$ 0#. )/"/&/.(/$/" PUf7fff #-'/$ @ ./$ 0#. 0-/&/.(/ RV7fff #-'/$7 8( #. #"%'+"%# &/%#"*(+. 0#*(90# "/ #"%'+'5 $-$ 6#"#1(*(/$ $/" *#'/7

$9 8-&/"4+ <-# #. #"%'+"%# %(#"# <-# %/)+' $- 0#*($(2" 0# #"%'+0+ +"%#$ 0#*/"/*#' .+ 0#*($(2" 0#. )/"/&/.($%+7$=<9 _#&'#$#"%# #. 3-#4/ #" 1/')+ #;%#"$(?+7$=;9 _#&'#$#"%# #. 3-#4/ #" 1/')+ #$%'+%=4(*+7

69 8-&/"4+ +F/'+5 &/' #. */"%'+'(/5 <-# #. #"%'+"%# */"/*#5 +"%#$ 0# %/)+' $-0#*($(2"5 .+ 0#*($(2" 0#. )/"/&/.($%+76=<9 _#&'#$#"%# #. 3-#4/ #" 1/')+ #;%#"$(?+76=;9 _#&'#$#"%# #. 3-#4/ #" 1/')+ #$%'+%=4(*+7

1.9 d" #)&'#$+'(/ F+ */"?#"*(0/ + 0/$ ("?#'$/'#$ &+'+ ("?#'%(' #" -" &'/@#*%/ + U5a

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8-&/"4+)/$ <-# +. 1("+. 0# *+0+ +\/ 0# ?(4#"*(+ 0#. &'/@#*%/ ./$ ("?#'$/'#$F+" 0# 0#*(0(' $()-.%>"#+)#"%# $( '#*-&#'+" / "/ $- ("?#'$(2"5 @ <-# $# $-*#0#"./$ $(4-(#"%#$ &+4/$ #" 1-"*(2" 0# %+.#$ 0#*($(/"#$7 8( +. 1("+. 0#. &'()#' +\/5

+)6/$ ("?#'$/'#$ 0#*(0#" +6+"0/"+' #. &'/@#*%/5 *+0+ -"/ '#*(6# PU7fff #-'/$@ #. 3-#4/ $# +*+6+7 8( $2./ -" ("?#'$/' 0#*(0# +6+"0/"+'5 =$%# '#*(6# PV7fff#-'/$ @ #. /%'/ '#*(6# ^7fff #-'/$5 @ #. 3-#4/ $# +*+6+7 g("+.)#"%#5 $( +)6/$("?#'$/'#$ 0#*(0#" )+"%#"#' $- ("?#'$(2" 0-'+"%# #. $#4-"0/ +\/5 #. &'/@#*%/..#4+ + $- 1("+.(:+*(2" @ ./$ ("?#'$/'#$ F+" 0# 0#*(0(' .+ 1/')+ #" <-# $# 0#?-#.9?# .+ ("?#'$(2"E )#0(+"%# 0("#'/ / )#0(+"%# +**(/"#$ */" %/%+. .(<-(0#: #" #.)#'*+0/ 6-'$>%(.7 8( +)6/$ 0#*(0#" #. */6'/ #" 0("#'/5 *+0+ -"/ 0# #../$ '#*(6#UR7fff #-'/$ @ #. 3-#4/ $# +*+6+7 8( $2./ -"/ &'#1(#'# #. 0("#'/5 #$# ("?#'$/''#*(6# Tf7fff #-'/$5 #. /%'/ '#*(6# PV7fff #-'/$ @ #. 3-#4/ $# +*+6+7 [5 &/'H.%()/5 $( "("4-"/ 0#$#+ #. 0("#'/5 .+ #)&'#$+ #"%'#4+ + *+0+ -"/ -" &+<-#%# 0#

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1.10 8# $+6# <-# -" 3-#4/ #" 1/')+ #;%#"$(?+ #$%> */)&.#%+)#"%# #$&#*(1(*+0/ &/'

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Figura 1.10 Juego del Ejercicio Propuesto 1.10.

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Definición 2.1

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Descripción y representación de algunos ejemplos importantes de juegos

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Juegos estáticos con información completa (I) 63

Ejemplo 2.1

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Ejemplo 2.2

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Ejemplo 2.5

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Solución de un juego

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2.2. SOLUCIONES DE UN JUEGO MEDIANTE ARGUMENTOSDE DOMINACIÓN

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Estrategias dominadas y estrategias estrictamente dominadas.Estrategias dominantes

Definición 2.2

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Ejemplo 2.7

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Primer concepto de solución: Uso de Estrategias Dominantes (UED)

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Segundo concepto de solución: Eliminación Iterativa Estricta (EIE)

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Definición 2.4

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Ejemplo 2.14

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Juegos resolubles por dominación

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Definición 2.6

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Ejemplo 2.15

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Ejemplo 2.16

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Ejemplo 2.17

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Comparación entre los anteriores conceptos de solución

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Teorema 2.1

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2.3. APLICACIÓN: EL MECANISMO DE CLARK-GROVESPARA LA ASIGNACIÓN DE UN BIEN PÚBLICO

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Introducción a un problema de asignación de un bien público.Primer intento de resolución

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Ejemplo 2.18

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2.4. SOLUCIONES DE UN JUEGO MEDIANTE ARGUMENTOSDE EQUILIBRIO. EL EQUILIBRIO DE NASH

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El Equilibrio de Nash (EN) en estrategias puras

Definición 2.7

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Cálculo de los EN (en estrategias puras) en los juegos anteriores

Ejemplo 2.20

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Ejemplo 2.21

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Ejemplo 2.22

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Ejemplo 2.23

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Correspondencia de respuesta óptima

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Definición 2.8

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%ñ$à

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Teorema 2.3

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Ejemplo 2.24

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Ejemplo 2.25

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La mayor diferencia Las peticiones de Nash

1/2 1/2

1/2 1/2

R s1 2( ) R s2 1( )

1 1

1 1

3/4

1/4

3/4 3/4

s2 s2

s1 s1

A1

A2

B

Figura 2.1 Correspondencias de respuesta óptima.

Ejemplo 2.26

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1/2

1/2

R q2 1( ) = (1 q1)/2

q1* /3= 1

q2* /3= 1

R q1 2( ) = (1 q2)/2

1

1

q2

q1

q2*

q1*

Figura 2.2 Correspondencia de respuesta óptima del juego básico de Cournot.

Ejemplo 2.27

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Ejemplo 2.28

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Algunos comentarios adicionales sobre el significadodel equilibrio de Nash

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Relación entre el equilibrio de Nash y los anterioresconceptos de solución

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Teorema 2.4

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Teorema 2.5

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S EIE (perfiles que sobreviven a EIE)

S EID (perfiles que sobreviven a EID)

S EN (equilibrios de Nash)

Figura 2.3 Relaciones de inclusión.

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Ejemplo 2.29

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Ejemplo 2.30

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Eficiencia de Pareto. Relación entre los equilibrios de Nashy la eficiencia de Pareto

Definición 2.9Y'0, #+ 2*#3, '% T# J4 8884 # "W &J4 8884 &"V4

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Ejemplo 2.31

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Ejemplo 2.32

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Algunos comentarios sobre cooperación y conflicto

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2.5. APLICACIONES: EL OLIGOPOLIO DE COURNOT

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Duopolio de Cournot. Modelo simplificado

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Tabla 2.1 Resultados de equilibrio, según el número de empresas.

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Equilibrio de Nash

Figura 2.5 Ineficiencia del equilibrio en el duopolio de Cournot.

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Figura 2.6 Cantidad y precio de equilibrio, según el número de empresas.


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2.6. APLICACIONES: EL OLIGOPOLIO DE BERTRAND

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Figura 2.8 Función de demanda y función de beneficios de monopolio.

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Oligopolio de Bertrand. Modelo continuo en productos homogéneos

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Juegos estáticos con información completa (II) 147

2'$ (,0 1*"&-,"#$ 2# *%-+-2'2 ,02-"'+#$= /' >*# F$%'$ ", (#07-%-0)'" &,7('0'0 +,%#0)'$: ;#A'&# (0#&-$, >*# +' #$&'+' 2# 0#(0#$#"%'&-9" 2# +'$ (0#1#0#"&-'$ $#' &,7, 7)"-7, &'02-8"'+8-"%#0B'+,= 2# 7,2, >*# $#' (,$-?+# &'+&*+'0 ('5,$ #$(#0'2,$:

HA,0' ?-#"= $- $*(,"#7,$ C&,7, A'0#7,$ ' ('0%-0 2# 'A,0'D >*# +'$ 5'"'"&-'$ ('0'&,7?-"'&-,"#$ 2# #$%0'%#5-'$ (*0'$ %-#"#" #+ $-5"-1-&'2, 2# 1*"&-,"#$ 2# *%-+-2'2 2# ^,"@#*7'""8_,05#"$%#0"= $) %-#"# $#"%-2, &'+&*+'0 C/ %0'?'4'0 &,"D ('5,$ #$(#0'2,$= >*#('$'" ' 2#(#"2#0 2# +'$ (0,?'?-+-2'2#$ 2# +'$ +,%#0)'$ #$&,5-2'$:

!" '2#+'"%#= / #" &,"$,"'"&-' &," +' ",%'&-9" *%-+-3'2' #" #+ &'()%*+, J ('0' +' *%-+-82'2 2# ^," @#*7'""8_,05#"$%#0"= *$'0#7,$ 7'/.$&*+'$ ('0' 0#1#0-0",$ ' +,$ ('5,$, *%-+-2'2#$ #$(#0'2'$ >*# &'2' 4*5'2,0 ,?%-#"# &*'"2, $# 4*#5' *" (#01-+ 2# #$%0'%#5-'$7-<%'$:

Ejemplo 3.25.!'/ +! ,&" )/1!+&"

5.'&+/% 9

W'0' W0*3

5.'&+/% 8W'0' J= .J .J= J

W0*3 .J= J J= .J

!" #+ 4*#5, 2# +'$ 7,"#2'$ +,$ &,"4*"%,$ 2# #$%0'%#5-'$ (*0'$ $," & J%& O% N!"#"=!#$%P: E,0 %'"%,= *"' #$%0'%#5-' 7-<%' ('0' #+ 4*5'2,0 J ", #$ 76$ >*# *"' 2-$%0-?*&-9"2# (0,?'?-+-2'2 $,?0# #+ &,"4*"%, & J= #$ 2#&-0= C .= J. .D 2,"2# . 0#(0#$#"%' +' (0,?'?-8+-2'2 2# #+#5-0 !"#" / CJ. .D +' (0,?'?-+-2'2 2# #+#5-0 !#$%: [ 2#+ 7-$7, 7,2,= *"'#$%0'%#5-' ('0' #+ 4*5'2,0 O &,"$-$%-06 #" *"' +,%#0)' C2= J. 2D $,?0# #+ &,"4*"%, & O #" +'>*# 2 0#(0#$#"%' +' (0,?'?-+-2'2 2# #+#5-0 !"#" / CJ. 2D +' (0,?'?-+-2'2 2# #+#5-0 !#$%:

E,0 #4#7(+,= +,$ ('5,$ #$(#0'2,$ ('0' &'2' 4*5'2,0 2# +'$ &,7?-"'&-,"#$ 2# #$%0'8%#5-'$ CC .= J. .D= !"#"D / CC .= J. .D= !#$%D $," +,$ $-5*-#"%#$K

3 JCC .= J. .D= !"#"D% . ` $JC!"#"= !"#"D! CJ. .D ` $JC!#$%= !"#"D%

% . `CJD! CJ. .D ` C.JD%O .. J

3 OCC .= J. .D= !"#"D% . ` $OC!"#"= !"#"D! CJ. .D ` $OC!#$%= !"#"D%

% . ` C.JD! CJ. .D`CJD%J. O .

/ '"6+,5'7#"%#=

3 JCC .= J. .D= !#$%D% . ` $JC!"#"= !#$%D! CJ. .D ` $JC!#$%= !#$%D%

. ` C.JD! CJ. .D`CJD% J. O .

3 OCC .= J. .D= !#$%D% . ` $OC!"#"= !#$%D! CJ. .D ` $OC!#$%= !#$%D%

. `CJD! CJ. .D ` C.JD% O .. J

[ +,$ ('5,$ #$(#0'2,$ 2# +' &,7?-"'&-9" 2# #$%0'%#5-'$ CC .= J. .D= C2= J. 2DD $,"K

3 JCC .= J. .D= C2= J. 2DD%2 ` 3 JCC .= J. .D= !"#"D! CJ. 2D ` 3 JCC .= J. .D= !#$%D%

%2 ` aO .. Jb! CJ. 2D ` a J. O .b%J. O .. O2!L .2

3 OCC .= J. .D= C2= J.2DD% .`2`C.JD!CJ. .D` 2`CJD! .`CJ.2D`CJD!CJ. .D`CJ.2D`C.JD%

%.J!O .!O2. L .2

H$)= ('0' +,$ B'+,0#$ &,"&0#%,$ CCJ/ M= O/ MD= !"#"D / CCJ/ M= O/ MD= CL/ c= J/ cDD ,?%#"#87,$ +'$ $-5*-#"%#$ *%-+-2'2#$ #$(#0'2'$K

3 JCCJ/ M= O/ MD= !"#"D% CJ/ MD$JC!"#"= !"#"D! CO/ MD$JC!#$%= !"#"D%

% CJ/ MDCJD! CO/ MDC.JD%.J/ M

3 OCCJ/ M= O/ MD= !"#"D% CJ/ MD$OC!"#"= !"#"D! CO/ MD$OC!#$%= !"#"D%

% CJ/ MDC.JD! CO/ MDCJD% J/ M

3 JCCJ/ M= O/ MD= CL/ c=J/ cDD%

% aCJ/ MDCJD! CO/ MDC.JDbCL/ cD! aCJ/ MDC.JD! CO/ MDCJDbCJ/ cD%.M/ Jc

3 OCCJ/ M= O/ MD= CL/ c=J/ cDD%

% CJ/ MDCL/ cDC.JD! CO/ MDCL/ cDCJD! CJ/ MDCJ/ cDCJD! CO/ MDCJ/ cDC.JD%M/ Jc

Ganancias esperadas en juegos bipersonales

;#' 4 *" 4*#5, &," 2,$ 4*5'2,0#$ &*/,$ &,"4*"%,$ 2# #$%0'%#5-'$ $," & J%N(JJ= (O

J= :::= (-J P

/ & O% N(JO= (O

O= :::= (5OP= / $#' pO% CpJO= pO

O= :::= p5OD *"' #$%0'%#5-' 7-<%' 2#+ 4*5'2,0 O:

;- #+ 4*5'2,0 J 4*#5' ('J / #+ 4*5'2,0 O 4*#5' pO= +'$ 5'"'"&-'$ #$(#0'2'$ $,"K

3 JC('J= pOD%pJO$JC('J= (J

OD!pOO$JC('J= (O

OD!ñ!p5O$JC('J= (5OD%

5

; *%J

p *O$JC('J= ( *OD

3 OC('

J= pOD%pJ

O$OC('

J= (J

OD!pO

O$OC('

J= (O

OD!ñ!p5

O$OC('

J= (5

OD%

5

; *%J p *

O$OC('

J= ( *

OD

;- #+ 4*5'2,0 J 4*#5' pJ% CpJJ= pO

J= :::= p-J D / #+ 4*5'2,0 O 4*#5' pO= +'$ 5'"'"&-'$

#$(#0'2'$ $,"K

3 JCpJ= pOD%pJJ3 JC(J

J= pOD!pOJ3 JC(O

J= pOD!ñ!p-J 3 JC(-J = pOD%

%pJJ

5

; *%J

p *O$JC(J

J= ( *OD!pOJ

5

; *%J

p *O$JC(O

J= ( *OD!ñ!p-J

5

; *%J

p *O$JC(-J = ( *OD%

%

-

;'%J

p'JA

5

; *%J

p *O$JC('J= ( *OD

B%

-

;'%J

5

; *%J

p'Jp

*O$JC('J= ( *OD

3 OCpJ= pOD%pJJ3 OC(J

J= pOD!pOJ3 OC(O

J= pOD!ñ!p-J 3 OC(-J = pOD%

%pJJ

5

; *%J

p *O$OC(J

J= ( *OD!pOJ

5

; *%J

p *O$OC(O

J= ( *OD!ñ!p-J

5

; *%J

p *O$OC(-J = ( *OD%

%

-

;'%J

p'JA

5

; *%J

p *O$OC('J= ( *OD

B%

-

;'%J

5

; *%J

p'Jp

*O$OC('J= ( *OD

Expresión matricial de las ganancias esperadas

;#' 4 *" 4*#5, &," 2,$ 4*5'2,0#$ &*/,$ &,"4*"%,$ 2# #$%0'%#5-'$ (*0'$ $,"& J%N(J

J= (OJ= :::= (-

J P / & O% N(JO= (O

O= :::= (-O P/ $#'" pJ%CpJ

J= pOJ= :::= p-

J D / pO% CpJO= pO

O= :::= p5OD

#$%0'%#5-'$ 7-<%'$ 2# +,$ 4*5'2,0#$ J / O= 0#$(#&%-B'7#"%#:I' 1,07' #$%0'%F5-&' 2# #$%# 4*#5,= #" #$%0'%#5-'$ (*0'$= (*#2# 0#(0#$#"%'0$# &,7, +'

$-5*-#"%# %'?+' ?-7'%0-3K

5.'&+/% 9

(JO (O

O ñ (5O

5.'&+/% 8

(JJ $JC(J

J= (JOD= $OC(J

J= (JOD $JC(J

J= (OOD= $OC(J

J= (OOD ñ $JC(J

J= (5OD= $OC(JJ= (5OD

(OJ $JC(O

J= (JOD= $OC(O

J= (JOD $JC(O

J= (OOD= $OC(O

J= (OOD ñ $JC(O

J= (5OD= $OC(OJ= (5OD

ñ ñ ñ ñ ñ

(-J $JC(-J = (JOD= $OC(-J = (J

OD $JC(-J = (OOD= $OC(-J = (O

OD ñ $JC(-J = (5OD= $OC(-J = (5OD

>*# (,2#7,$ $#('0'0 #" 2,$ 7'%0-&#$K 6J% a$JC('J= ( *ODb'= */ 6O% a$OC('J= ( *ODb'= *= 2,"2# 6J

0#&,5# +'$ 5'"'"&-'$ 2#+ 4*5'2,0 J / 6O +'$ 5'"'"&-'$ 2#+ 4*5'2,0 O= '"%# +'$ 2-$%-"%'$&,7?-"'&-,"#$ 2# #$%0'%#5-'$ (*0'$ 2# +,$ 4*5'2,0#$:

R#"-#"2, #" &*#"%' #$%' $#('0'&-9" 2# +'$ 5'"'"&-'$ 2# +,$ 4*5'2,0#$ #" 2,$ 7'%0-8&#$= 6J / 6O= (,2#7,$ #<(0#$'0 +' 5'"'"&-' #$(#0'2' 2# &'2' 4*5'2,0 &*'"2, 4*#5'" +'$#$%0'%#5-'$ 7-<%'$ pJ / pO 0#$(#&%-B'7#"%#= $-7(+#7#"%# &,7, +,$ $-5*-#"%#$ (0,2*&%,$K

3 JCpJ= pOD%pJ 6Jp1 O 3 OCpJ= pOD%pJ 6Op

1 O

2,"2# (,0 p1 #"%#"2#7,$ #+ B#&%,0 2# #$%0'%#5-'$ 7-<%'$ %0'$(*#$%,:

Ejemplo 3.3

W,"$-2#0#7,$ 2# "*#B, #+ Y*#5, M:J:

5.'&+/% 9

U V

5.'&+/% 8

H M= O J= L

W J= M O= J

X O= O O= Q

^'7,$ ' &'+&*+'0 +' *%-+-2'2 #$(#0'2' 2# &'2' 4*5'2,0 &*'"2, 4*#5'" +'$ $-5*-#"%#$&,7?-"'&-,"#$ 2# #$%0'%#5-'$ 7-<%'$K

&: pJ% CO/ M= J/ d= J/ dD / pO% CJ/ M= O/ MD:I' 7'%0-3 6J #$%6 1,07'2' (,0 #+ (0-7#0 #+#7#"%, 2# +' %'?+' ('0' &'2' #$%0'%#5-'

(*0' 2# &'2' 4*5'2,0 / +' 7'%0-3 6O (,0 #+ $#5*"2, #+#7#"%,K

6J%AM J

J O

O OB = 6O%AO L

M J

O QB/ +' 5'"'"&-' #$(#0'2' 2# &'2' 4*5'2,0 (,0 4*5'0 pJ / pO 0#$(#&%-B'7#"%# #$K

3 JCpJ= pOD%

pJ 6Jp1

O%

CO/ M J/ d J/ dD

A

M J

J OO O

BAJ/ M

O/ M

B%

Cc/ O L/ MD

AJ/ M

O/ M

B%

MJ/ Je

3 OCpJ= pOD% pJ 6Op1 O% CO/ M J/ d J/ dDA

O L

M J

O QBAJ/ M

O/ MB% CJM/ d Jf/ dDAJ/ M

O/ MB%Lf/ Je

;: pJ% CO/ M= J/ d= J/ dD / (O% U: V'2, >*# &*'+>*-#0 #$%0'%#5-' (*0' ", #$ 76$ >*#*"' #$%0'%#5-' 7-<%' &," $,(,0%# *"-%'0-, C2#5#"#0'2'D= (,2#7,$ #<(0#$'0 (O% U &,7,+' #$%0'%#5-' 7-<%' pO >*# '$-5"' (0,?'?-+-2'2 J ' +' #$%0'%#5-' U / (0,?'?-+-2'2 Q ' +'

#$%0'%#5-' V= #$ 2#&-0= pO% CJ= QD: E,0 %'"%,= +' *%-+-2'2 #$(#0'2' ('0' +,$ 4*5'2,0#$ #$K

3 JCpJ= pOD%pJ 6Jp1 O% CO/ M J/ d J/ dDA

M J

J O

O OBAJ

QB% Cc/ O L/ MDAJ

QB%c/ O

3 OCpJ= pOD%pJ 6Op1 O% CO/ M J/ d J/ dDA

O L

M J

O QBAJ

QB% CJM/ d Jf/ dDAJ

QB%JM/ d

3;"!%<&=(>1 678?

H B#&#$ 0#$*+%' &97,2' +' ",%'&-9" &,"$-$%#"%# #" #<(0#$'0 +'$ #$%0'%#5-'$ 7-<%'$ &,7, &,78?-"'&-,"#$ +-"#'+#$ &,"B#<'$ 2# +'$ #$%0'%#5-'$ (*0'$: E,0 #4#7(+,= #" #+ Y*#5, M:J #+ &,"8 4*"%, 2# #$%0'%#5-'$ (*0'$ 2# YJ #$ & J%NH= W= XP= / *"' #$%0'%#5-' 7-<%' $,?0# #$#&,"4*"%, $,(,0%#= (,0 #4#7(+, C .= 2= J. .. 2D= (*#2# $#0 #<(0#$'2' &,7,

.H! 2W! CJ. .. 2DX: !+ ('5, #$(#0'2, 2# YJ $- 2-&A, 4*5'2,0 4*#5' #$' #$%0'%#5-'7-<%' / #+ ,%0, 4*5'2,0 4*#5' &*'+>*-#0 #$%0'%#5-' (*0' , 7-<%' pO (*#2# &'+&*+'0$# 16&-+87#"%# '$)K

3 JC .H! 2W! CJ. .. 2DX= pOD% .3 JCH= pOD!23 JCW= pOD! CJ. .. 2D3 JCX= pOD

Definición ampliada de equilibrio de Nash

I' 2#1-"-&-9" 2#+ #>*-+-?0-, 2# @'$A &*'"2, (#07-%-7,$ +' #<-$%#"&-' 2# #$%0'%#5-'$ 7-<8%'$ ", #$ 76$ >*# *"' #<%#"$-9" 2#+ &,"&#(%, B-$%, ('0' #$%0'%#5-'$ (*0'$:

Definición 3.3

!" #+ 4*#5, 4%N& J= :::= & 5g $J= :::= $5P= 2#&-7,$ >*# #+ (#01-+ 2# #$%0'%#5-'$ 7-<%'$ph% CphJ = :::= ph' = :::= p75 D #$ *" @A.(,(;%(/ +! B&"C C'?0#B-'2, !@D $- ('0' &'2' 4*5'82,0 'K

3 'Cp7J = :::= p7'.J= p7' = p7'!J= :::= ph5 Dn3 'Cp7J = :::= p7'.J= p'= p7'!J= :::= ph5 D ('0' %,2, p' 2#BC& 'D:

!$ 2#&-0= ('0' &'2' 4*5'2,0 '= p7' #$ *"' $,+*&-9" 2#+ (0,?+#7'

7'< 3 'Cp7J = :::= p7'.J= p'= p7'!J= :::= ph5 D #" +' B'0-'?+# p'= 2,"2# p' (#0%#"#&# ' BC& 'D

, 2-&A, 2# ,%0, 7,2,= ('0' &'2' 4*5'2,0 '= p7' 0( #0(.$0(1" 8.1'-" " p7.':RF"5'$# #" &*#"%' >*# *"' #$%0'%#5-' 7-<%' ", #$ 76$ >*# *"' +,%#0)' $,?0# #$%0'%#8

5-'$ (*0'$ / >*# +' 1*"&-9" 2# ('5,$ , 5'"'"&-'$ #$ +-"#'+= ('0' &'2' 4*5'2,0= #" +'$(0,?'?-+-2'2#$ 2# $*$ 2-$%-"%'$ #$%0'%#5-'$ (*0'$: E,0 %'"%,= #+ ('5, #$(#0'2, ('0' *" 4*85'2,0 2# *"' #$%0'%#5-' 7-<%'= $*(,"-#"2, 1-4'$ +'$ #$%0'%#5-'$ 2# +,$ ,%0,$ 4*5'2,0#$=0#$*+%' $#0 *"' &,7?-"'&-9" &,"B#<' 2# +,$ ('5,$ 2# +'$ #$%0'%#5-'$ (*0'$ $,(,0%# 2#2-&A' #$%0'%#5-' 7-<%'= / #" &,"$#&*#"&-'= +' 5'"'"&-' #$(#0'2' 2# *"' #$%0'%#5-' 7-<%'

%-#"# &,7, +)7-%#$ -"1#0-,0 / $*(#0-,0 +'$ 5'"'"&-'$ 7)"-7' / 76<-7' 2# +'$ #$%0'%#5-'$(*0'$ $,(,0%# 2# 2-&A' #$%0'%#5-' 7-<%': !$%, (*#2# B#0$# &+'0'7#"%# #" #+ !4#7(+, M:L:

Ejemplo 3.4

W,"$-2#0#7,$ 2# "*#B, #+ Y*#5, M:JK

5.'&+/% 9

U V

5.'&+/% 8

H M= O J= L

W J= M O= J

X O= O O= Q

W,7, (*#2# B#0$# #" #$%# 4*#5,= 2'2' +' #$%0'%#5-' U 2#+ 4*5'2,0 O= #+ 4*5'2,0 J ",(*#2# #$(#0'0 5'"'0 7#",$ 2# J C5'"'"&-' &*'"2, 4*#5' WD= "- 76$ 2# M C5'"'"&-'&*'"2, 4*#5' HD &," "-"5*"' 2# $*$ #$%0'%#5-'$= /' $#'" (*0'$ , 7-<%'$K

3 JaC .= 2= J. .. 2D= Ub% .$JCH= UD!2$JCW= UD! CJ. .. 2D$JCX= UD%

% . ` M! 2 ` J! CJ. .. 2D ` O

3 JaC .= 2= J. .. 2D= Ub %,7' *" B'+,0 76<-7, 2# M &*'"2, .%J= 2% Q /J. .. 2% Q= / *" B'+,0 7)"-7, 2# J &*'"2, .%Q= 2%J / J. .. 2% Q:

V# +, 2-&A, #" #+ (600'1, '"%#0-,0 / 2# +, B-$%, #" #+ #4#7(+,= $# 2#$(0#"2# -"%*-%-B'8

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

!" #+ R#,0#7' M:J $# 1,07'+-3' +' -2#' '"%#0-,0:

Teorema 3.1

!" #+ 4*#5, 4% N& J= :::= & 5g $J= :::= $5P= +' &,7?-"'&-9" 2# #$%0'%#5-'$ 7-<%'$ph% CphJ = phO = :::= ph5 D #$ *" !A.(,(;%(/ +! B&"C $- / $9+, $- ('0' &'2' 4*5'2,0 ' &,"#$%0'%#5-' 7-<%' p7' % CpJ7' = pO7' = :::= p *7' = :::D %#"#0 p *7' b Q -7(+-&' >*# +' #$%0'%#5-'(*0' ( *' #$ *"' 0#$(*#$%' 9(%-7' ' p7.'% Cp7J = :::= p7'.J= p7'!J= :::= ph5 D:

D!)/"$%&=(>1?

õD ;*(,"5'7,$ >*# #+ (#01-+ ph% CphJ = phO = :::= ph5 D &*7(+#= ('0' &'2' 4*5'2,0 '= >*#

%,2'$ $*$ #$%0'%#5-'$ (*0'$ ( *' $,(,0%# 2# p7' $," 0#$(*#$%' 9(%-7' '

ph.'% CphJ = :::= ph'.J= ph'!J= :::= ph5 D

I+'7#7,$ 97' ' +' 76<-7' *%-+-2'2 >*# (*#2# $#0 '+&'"3'2' (,0 #+ 4*5'2,0 ' $- +,$2#76$ 4*#5'" ph.' = #$ 2#&-0=

97' %7'<N3 'Cp7J = :::= p7'.J= ('= p7'!J= :::= ph5 D K (' à & 'P%

%7'<N3 'Cp7J = :::= p7'.J= p-= ph'!J= :::= ph5 D Kp' à BC& 'DP

!" #$# &'$,=

3 'Cp7J = :::= p7'.J= ( *'= p7'!J= :::= ph5 D%7'<N3 'Cp7J = :::= p7'.J= ('= p7'!J= :::= ph5 D K (' à & 'P%

% 97' = O( *' à ;SECp7' D

/= #" &,"$#&*#"&-'=

3 'Cp7J = :::= p7'.J= p7' = p7'!J= :::= ph5 D% ;p

ij7bQ

p *7' 3 'Cp7J = :::= p7'.J= ( *'= p7'!J= :::= ph5 D%

% ;p

ij7bQ

p *7' 97' % 97' n3 'Cp7J = :::= p7'.J= p'= p7'!J= :::= ph5 D= Op' à BC& 'D

H$) (*#$= p7' #$ 0#$(*#$%' 9(%-7' ' p7.' ('0' %,2, 4*5'2,0 '= (,0 +, >*# ph #$!>*-+-?0-, 2# @'$A:

úD ;#' ph% CphJ = phO = :::= ph5 D *" !@: ;*(,"5'7,$ >*#= 2'2, *" 4*5'2,0 &*'+8>*-#0' '= +' #$%0'%#5-' (*0' ( *' (#0%#"#&# '+ $,(,0%# 2# p7' : V#7,$%0'0#7,$ (,0 0#2*&8&-9" '+ '?$*02, >*# ( *' #$ 0#$(*#$%' 9(%-7' ' ph.'% CphJ = :::= ph'.J= ph'!J= :::= ph5 D: ;- ", +,1*#0'= #<-$%-0)' *"' #$%0'%#5-' (*0' () ' %'+ >*#

3 'Cp7J = :::= p7'.J= () ' = p7'!J= :::= ph5 Db3 'Cp7J = :::= p7'.J= ( *'= p7'!J= :::= ph5 D

HA,0' ?-#"= $- pñ' #$ +' #$%0'%#5-' 7-<%' >*# #$ #" %,2, -5*'+ ' p7' %CpJ7' = pO7' = :::= p *7' = :::D=$'+B, >*# 4*#5' +' #$%0'%#5-' (*0' ( *' &," (0,?'?-+-2'2 &#0, / +' #$%0'%#5-' (*0' () ' &,"(0,?'?-+-2'2 p *7' !p) 7' = 2-&A' #$%0'%#5-' pñ' #$ *"' 0#$(*#$%' #$%0-&%'7#"%# 7#4,0 ' p7.'

>*# p7' : !" #1#&%,=

3 'Cp7J = :::= p7'.J= pñ-= ph'!J= :::= ph5 D% ;:Ç *= :Ç)

p:7' 3 'Cp7J = :::= p7'.J= (:' = p7'!J= :::= ph5 D!

!Cp *7' !p) 7' D3 'Cp7J = :::= p7'.J= () ' = p7'!J= :::= ph5 Db

b3 'Cp7J = :::= p7'.J= p7' = p7'!J= :::= ph5 D%

% ;:Ç *= :Ç)

p:7' 3 'Cp7J = :::= p7'.J= (:' = p7'!J= :::= ph5 D!

!p *7' 3 'Cp7J = :::= p7'.J= ( *'= p7'!J= :::= ph5 D!

!p) 7' 3 'Cp7J = :::= p7'.J= = ph'!J= :::= ph5 D

!" &,"$#&*#"&-'= p7' ", #$ 0#$(*#$%' 9(%-7' ' ph.' = +, >*# &,"%0'2-&# +' A-(9%#$-$/ (0*#?' +' &,"2-&-9" i$,+, $-j:

Z" (#01-+ 2# #$%0'%#5-'$ 7-<%'$ #$ *" #>*-+-?0-, 2# @'$A $- / $9+, $- ('0' &'2' 4*5'82,0= %,2'$ +'$ #$%0'%#5-'$ (*0'$ $,(,0%# 2# $* #$%0'%#5-' 7-<%' $," *"' 0#$(*#$%' 9(%-7' '+' &,7?-"'&-9" 2# #$%0'%#5-'$ 2# #>*-+-?0-, 2#+ 0#$%, 2# +,$ 4*5'2,0#$:

!$%, $-5"-1-&' >*# +'$ #$%0'%#5-'$ 7-<%'$ 2# #>*-+-?0-, '$-5"'" *"' (0,?'?-+-2'2 #$8%0-&%'7#"%# (,$-%-B' $9+, ' '>*#++'$ #$%0'%#5-'$ (*0'$ >*# $," 0#$(*#$%' 9(%-7' ' +'$ #$8%0'%#5-'$ 2#+ 0#$%, 2# 4*5'2,0#$: V# #++, $# 2#2*&# >*# *"' #$%0'%#5-' 7-<%' #$ 0#$(*#$%'9(%-7' ' #$%0'%#5-'$ C(*0'$ , 7-<%'$D 2'2'$= $9+, $- $*$ #$%0'%#5-'$ (*0'$ $,(,0%# +, $,"%'7?-F": !" &,"$#&*#"&-'E +'$ #$%0'%#5-'$ (*0'$ $,(,0%# 2# *"' #$%0'%#5-' 7-<%' 2# #>*-+-8?0-, (0,2*&#" 5'"'"&-'$ -5*'+#$:

!" 0#'+-2'2= #$%' (0,(-#2'2 #$ 5#"#0'+-3'?+# ' &*'+>*-#0 (#01-+ 2# #$%0'%#5-'$ / ", $9+,' '>*#++'$ >*# 1,07'" ('0%# 2# *" !@: !" #1#&%,= $- *" 4*5'2,0 %-#"# *"' #$%0'%#5-' 7-<%'p' >*# #$ 0#$(*#$%' 9(%-7' ' *"' &,7?-"'&-9" 2# #$%0'%#5-'$ 2#+ 0#$%, 2# 4*5'2,0#$= #"8%,"&#$ &*'+>*-#0 #$%0'%#5-' (*0' 2#+ $,(,0%# 2# p' , &*'+>*-#0 #$%0'%#5-' 7-<%' >*# $#

(*#2' 1,07'0 &," '+5*"'$ , %,2'$ +'$ #$%0'%#5-'$ (*0'$ 2# 2-&A, $,(,0%#= $," %'7?-F"0#$(*#$%'$ 9(%-7'$:

Ejemplo 3.5

^,+B'7,$ '+ Y*#5, M:JK

5.'&+/% 9

U V

5.'&+/% 8

H M= O J= L

W J= M O= J

X O= O O= Q

!$%# 4*#5, %-#"# *" ."-&, !@ #" #$%0'%#5-'$ 7-<%'$ &,"$-$%#"%# #" #+ (#01-+ aCJ/ O= Q=J/ OD= CJ/ O= J/ ODb: W,7, $# 2#7*#$%0' ' &,"%-"*'&-9"= &*'+>*-#0 #$%0'%#5-' 2#+ 4*5'2,0 J&*/, $,(,0%# #$%F &,"%#"-2, #" #+ &,"4*"%, NH= XP= -"&+*-2'$ +'$ #$%0'%#5-'$ (*0'$ H /X= #$ 0#$(*#$%' 9(%-7' ' +' #$%0'%#5-' 7-<%' 2#+ 4*5'2,0 O= phO % CJ/ O= J/ OD: V#+ 7-$7,7,2,= &*'+>*-#0 #$%0'%#5-' 2#+ 4*5'2,0 O C(*0' , 7-<%'D #$ 0#$(*#$%' 9(%-7' ' +' #$%0'%#85-' phJ % CJ/ O= Q= J/ OD 2#+ 4*5'2,0 J:

&: V'2' +' #$%0'%#5-' phO % CJ/ O= J/ OD 2#+ 4*5'2,0 O= #+ 4*5'2,0 J ,?%-#"# +'$ 7-$87'$ 5'"'"&-'$ '+ *%-+-3'0 2-$%-"%'$ #$%0'%#5-'$ &," $,(,0%# NH= XPK

3 JCp7J = p7O D%p7J 6Jp71 O % CJ/ O Q J/ OD

A

M J

J O

O O

B AJ/ O

J/ O

B% Cc/ O M/ OD

A

J/ O

J/ O

B%O

3 JCH= phO D% CJ Q QDAM J

J O

O OB AJ/ O

J/ OB% CM JDAJ/ O

J/ OB%O

3 JCX= phO D% CQ Q JDAM J

J O

O OBAJ/ O

J/ OB% CO ODAJ/ O

J/ OB%O

3 JCCJ/ M= Q= O/ MD= p7O D% CJ/ M Q O/ MDAM J

J O

O OB AJ/ O

J/ OB% Cf/ M c/ MDAJ/ O

J/ OB%O

!" 5#"#0'+= 2'2' +' #$%0'%#5-' p7O 2#+ 4*5'2,0 O= &*'+>*-#0 #$%0'%#5-' 2#+ 4*5'2,0 JpJ% C .)

J= Q= J. .) JD >*# %-#"# $,(,0%# NH= XP 5#"#0' *"' 5'"'"&-' 2# O ('0' #+ 4*5'2,0 JK

3 JCpJ= p7O D% C .) J Q J. .)

JD

A

M J

J O

O O

BAJ/ O

J/ O

B% CO! .)

J O. .) JD

A

J/ O

J/ O

B%O

;: V'2' +' #$%0'%#5-' phJ % CJ/ O= Q= J/ OD 2#+ 4*5'2,0 J= #+ 4*5'2,0 O ,?%-#"# +'$7-$7'$ 5'"'"&-'$ '+ *%-+-3'0 2-$%-"%'$ #$%0'%#5-'$K

3 OCp7J = p7O D%p7J 6Op71

O % CJ/ O Q J/ OD

AO L

M JO Q

B AJ/ O

J/ O

B% CO OD

AJ/ O

J/ O

B% O

3 OCp7J = UD% CJ/ O Q J/ ODAO L

M J

O QBAJ

QB% CO ODAJ

QB% O

3 OCp7J = VD% CJ/ O Q J/ OD

A

O L

M J

O Q

BAQ

J

B% CO OD

A

Q

J

B% O

3 OCp7J = CL/ c= J/ cDD% CJ/ O Q J/ ODAO L

M J

O QB AL/ c

J/ cB% CO ODAL/ c

J/ cB%O

!" 5#"#0'+= 2'2' +' #$%0'%#5-' phJ 2#+ 4*5'2,0 J &*'+>*-#0 #$%0'%#5-' 2#+ 4*5'2,0 OpO% C .J

O= J. .JOD= 5#"#0' *"' 5'"'"&-' 2# O ('0' #+ 4*5'2,0 OK

3 OCp7J = pOD% CJ/ O Q J/ ODAO L

M J

O QB A .J

O

J. .JOB% CO ODA

.JO

J. .JOB%O

Ejemplo 3.6

W,"$-2#0#7,$ #+ $-5*-#"%# 4*#5, 2#",7-"'2, 4(!+%&#4&0!,#F(-!%&"KV,$ 4*5'2,0#$ #+-5#" $-7*+%6"#'7#"%# *", 2# +,$ $-5*-#"%#$ ,?4#%,$K (-#20' CkD=

('(#+ CED , %-4#0'$ CRD: @,07'+7#"%#= &'2' 4*5'2,0 A'&# *" 5#$%, >*# 0#(0#$#"%' *"'#+#&&-9" #"%0# (-#20' C(*l, &#00'2,D= ('(#+ C7'", #<%#"2-2'D , %-4#0'$ C2#2,$ '"*+'0 /&,0'39" #<%#"2-2,$D: ;- +,$ 2,$ 4*5'2,0#$ #+-5#" #+ 7-$7, ,?4#%,= A'/ #7('%# / &'2'*", 2# #++,$ ,?%-#"# *" ('5, 2# &#0,: ;- #+-5#" 2-$%-"%, ,?4#%,= #+ >*# 5'"' 0#&-?# *"#*0, 2#+ >*# (-#02#= 2# '&*#02, &," +'$ 0#5+'$ $-5*-#"%#$K

&: E-#20' 5'"' ' %-4#0'$ C(,0>*# *"' (-#20' #$ &'('3 2# 0,7(#0 *"'$ %-4#0'$D:;: R-4#0'$ 5'"' ' ('(#+ C(,0>*# +'$ %-4#0'$ &,0%'" '+ ('(#+D:=: E'(#+ 5'"' ' (-#20' C(,0>*# #+ ('(#+ #"B*#+B# ' +' (-#20'D:

I' 0#(0#$#"%'&-9" 2#+ 4*#5, #" 1,07' #$%0'%F5-&' #$K

5.'&+/% 9

k E R

5.'&+/% 8

k Q= Q .J= J J= .J

E J= .J Q= Q .J= J

R .J= J J= .J Q= Q

!" #$%# 4*#5,= ('0' &*'+>*-#0 4*5'2,0 +' #$%0'%#5-' 7-<%' ph% CJ/ M= J/ M= J/ MD #$0#$(*#$%' 9(%-7' ' $) 7-$7'= #$ 2#&-0= #+ (#01-+ Cph= phD #$ *" #>*-+-?0-, 2# @'$A: ;#(*#2# 2#7,$%0'0 >*# &*'+>*-#0 #$%0'%#5-' p= &*/, $,(,0%# $#' *" $*?&,"4*"%, 2#+ $,(,08%# 2# ph #$ %'7?-F" 0#$(*#$%' 9(%-7' ' ph:

E,0 #4#7(+,= 2'2' +' #$%0'%#5-' 2#+ 4*5'2,0 O phO % CJ/ M= J/ M= J/ MD= #+ 4*5'2,0 J,?%-#"# +,$ $-5*-#"%#$ ('5,$ &," +'$ $-5*-#"%#$ #$%0'%#5-'$K

&: phJ % CJ/ M= J/ M= J/ MDK

3 JCp7J = p7O D%p7J 6Jp71 O % CJ/ M J/ M J/ MDA

Q .J J

J Q .J

.J J QBAJ/ M

J/ M

J/ MB% Q

;: pJ% CJ/ O= Q= J/ ODK

3 JCpJ= p7O D%pJ 6Jp71 O % CJ/ O Q J/ OD

A

Q .J J

J Q .J

.J J Q

B A

J/ M

J/ M

J/ M

B%Q

=: pJ% CJ= Q= QDK

3 JCpJ= p7O D% pJ 6Jp71 O % CJ Q QDA

Q .J J

J Q .J

.J J QB AJ/ M

J/ M

J/ MB%Q

+: pJ% C .JJ= .O

J= J. .JJ. .O

JDK

3 JCpJ= p7O D%pJ 6Jp71

O % C .JJ= .

OJ= J. .

JJ. .

OJD

A Q .J J

J Q .J.J J QBA

J/ M

J/ MJ/ MB

%

% C .JJ!O .O

J. J= J. O .JJ. .O

J= .JJ. .O

JDAJ/ M

J/ M

J/ MB% Q

S?$F0B#$# >*# #" 2D A#7,$ &,"$-2#0'2, *"' #$%0'%#5-' 7-<%' &*'+>*-#0' ('0' #+ 4*5'2,0J: V# A#&A,= +' #$%0'%#5-' &,"$-2#0'2' #" 'D #$ +' ('0%-&*+'0-3'&-9" 2# +' &,"$-2#0'2' #"2D= ('0' .J

J%J/ M= .OJ%J/ M: !" ?D %#"#7,$ +' ('0%-&*+'0-3'&-9" 2# 2D ('0' .J

J%J/ O= .O

J%Q: !" &D %#"#7,$ +' ('0%-&*+'0-3'&-9" 2# 2D ('0' .JJ% J= .O

J%Q:

87 ;#'" C .= J. .D / C2= J. 2D #$%0'%#5-'$ 7-<%'$ 5#"F0-&'$ ('0' +,$ 4*5'2,0#$ J /O 0#$(#&%-B'7#"%#:

97 T-4'2' +' #$%0'%#5-' 7-<%' 5#"F0-&' C2= J. 2D 2#+ 4*5'2,0 O= ('0' #+ 4*5'2,0 J&'+&*+'7,$K

3 JC!"#"= C2= J. 2DD%2CJD! CJ. 2DC.JD%O2. J

3 JC!#$%= C2= J. 2DD% 2C.JD! CJ. 2DCJD% J. O2

67 ;# %-#"# >*#

3 JC!"#"= C2= J. 2DDb3 JC!#$%= C2= J. 2DD á O2. Jb J. O2 á 2b J/ O

3 JC!"#"= C2= J. 2DDa3 JC!#$%= C2= J. 2DD á O2. Ja J. O2 á 2a J/ O

3 JC!"#"= C2= J. 2DD%3 JC!#$%= C2= J. 2DD á O2. J% J. O2 á 2%J/ O

E,0 %'"%,= +' 0#$(*#$%' 9(%-7' #$ !"#" $- 2b J/ O= !#$% $- 2a J/ O= / &*'+>*-#0' $-2%J/ O:

;JC2D%E .% QC!#$%D $- 2a J/ O

.%JC!"#"D $- 2b J/ O

. à aQ= Jb C&*'+>*-#0 #$%0'%#5-'D $- 2% J/ O

m061-&'7#"%# +' &,00#$(,"2#"&-' 2# 0#$(*#$%' 9(%-7' 2#+ 4*5'2,0 J ' &*'+>*-#0 #$8%0'%#5-' 7-<%' 2#+ 4*5'2,0 O #$%6 #" +' T-5*0' M:J:

1 (cara)

q

0 (cruz) 1/2

1/2

1 (cara)

R q1( )

p

Figura 3.1 Correspondencia de respuesta óptima del jugador 1.

J7 T-4'2' +' #$%0'%#5-' 7-<%' 5#"F0-&' C .= J. .D 2#+ 4*5'2,0 J= ('0' #+ 4*5'2,0 O&'+&*+'7,$K

3 OCC .= J. .D= !"#"D% .C.JD! CJ. .DCJD%J. O .

3 OCC .= J. .D= !#$%D% .CJD! CJ. .DC.JD% O .. J

K7 ;# %-#"# >*#

3 OCC .= J. .D= !"#"Db3 OCC .= J. .D= !#$%D á J. O .b O .. J á .a J/ O

3 OCC .= J. .D= !"#"Da3 OCC .= J. .D= !#$%D á J. O .a O .. J á .b J/ O

3 OCC .= J. .D= !"#"D%

3 OCC .= J. .D= !#$%D á J. O .%

O .. J á .%

J/ OE,0 %'"%,= +' 0#$(*#$%' 9(%-7' #$ !"#" $- .a J/ O= !#$% $- (b J/ O= / &*'+>*-#0' $-

.%J/ O:

;OC .D%E 2%JC!"#"D $- .a J/ O

2% QC!#$%D $- .b J/ O

2 à aQ= Jb C&*'+>*-#0 #$%0'%#5-'D $- .% J/ O

m061-&'7#"%# +' &,00#$(,"2#"&-' 2# 0#$(*#$%' 9(%-7' 2#+ 4*5'2,0 O ' &*'+>*-#0 #$%0'%#85-' 7-<%' 2#+ 4*5'2,0 J #$%6 #" +' T-5*0' M:O:

1 (cara)

q

0 (cruz) 1/2

1/2

1 (cara)

R p2( )

p

Figura 3.2 Correspondencia de respuesta óptima del jugador 2.

L7 !" #+ (+'", .. 2 $# 0#(0#$#"%'" 5061-&'7#"%# +'$ &,00#$(,"2#"&-'$ ;JC2D / ;OC .D= ,?%#"-F"2,$# +,$ #>*-+-?0-,$ 2# @'$A #" #$%0'%#5-'$ 7-<%'$ #" +,$ (*"%,$ #" +,$>*# '7?'$ $# &,0%'" CT-5*0' M:MD:

R#"-#"2, #" &*#"%' >*# #+ 4*5'2,0 O #$ -"2-1#0#"%# #"%0# $*$ #$%0'%#5-'$ (*0'$ /

7-<%'$ C+# 5#"#0'" +' 7-$7' *%-+-2'2 #$(#0'2'D &*'"2, #+ 4*5'2,0 J 4*#5' $* #$%0'%#5-'7-<%' C .= J. .D% CJ/ O= J/ OD= / >*# #+ 4*5'2,0 J #$ -"2-1#0#"%# #"%0# $*$ #$%0'%#5-'$(*0'$ / 7-<%'$ &*'"2, #+ 4*5'2,0 O 4*#5' $* #$%0'%#5-' 7-<%' C2= J. 2D% CJ/ O= J/ OD=(,2#7,$ 2#&-0 >*# #+ !@ #" #$%0'%#5-'$ 7-<%'$ #$ '>*#+ #" #+ >*# &'2' 4*5'2,0 4*#5'2-&A' #$%0'%#5-' 7-<%'= >*# &,00#$(,"2# '+ (*"%, #" >*# $# &,0%'" +'$ &,00#$(,"2#"&-'$2# 0#$(*#$%' 9(%-7':

E,0 %'"%,=

2@B% MH8N9E 8N9:E H8N9E 8N9:O

!$%# #$ #+ !@ #" #$%0'%#5-'$ 7-<%'$ 2#+ 4*#5, /' >*# "-"5." 4*5'2,0 %-#"# -"&#"%-B, '

2#$B-'0$# *"-+'%#0'+7#"%# C$-= (,0 #4#7(+,= #+ 4*5'2,0 J $# 2#$B)' *"-+'%#0'+7#"%# 2#

I' &,00#$(,"2#"&-' 2# 0#$(*#$%' 9(%-7' 2#+ 4*5'2,0 J #$K

;JC2D%

E

.% J C!'50D $- 2b O/ M

.%Q C>?1@,/D $- 2a O/ M

. à aQ= Jb C&*'+>*-#0 #$%0'%#5-'D $- 2%

O/ M

;#' C .= J. .D *"' #$%0'%#5-' 7-<%' 2#+ 4*5'2,0 J: I' *%-+-2'2 >*# +' 4*5'2,0' O,?%-#"# &," &'2' *"' 2# $*$ #$%0'%#5-'$ (*0'$ #$K

3 OCC .= J. .D= !'50D% . ` O! CJ. .D ` Q%O .

3 OCC .= J. .D= >?1@,/D% . ` Q! CJ. .D ` J% J. .

;# (0,2*&# +' -5*'+2'2 O .%J. . $- / $9+, $- .% J/ M: W*'"2, .%J/ M +' 4*5'2,0'O ,?%-#"# +' 7-$7' 5'"'"&-' 2# $*$ 2,$ #$%0'%#5-'$ (*0'$= / (,0 %'"%, 2# &*'+>*-#0' 2#$*$ #$%0'%#5-'$ 7-<%'$: !" &,"$#&*#"&-'= &*'+>*-#0 #$%0'%#5-' (*0' , 7-<%' #$ 0#$(*#$%'9(%-7' 2# +' 4*5'2,0' O ' C .= J. .D% CJ/ M= O/ MD 2#+ 4*5'2,0 J:

;# %-#"# >*#

;OC .D%E 2%JC!'50D $- .b J/ M

2%QC>?1@,/D $- .a J/ M

2 à aQ= Jb C&*'+>*-#0 #$%0'%#5-'D $- .% J/ M

m061-&'7#"%#= +,$ #>*-+-?0-,$ 2# @'$A $*05#" 2# +' -"%#0$#&&-9" 2# +'$ &,00#$(,"8

2#"&-'$ 2# 0#$(*#$%' 9(%-7' CT-5*0' M:LD:

1 (cine)

q

0 (fútbol) p*5 1/3

q* 2/35

1 (cine)

R p2( )

p

R q1( )

Figura 3.4 Correspondencias de respuesta óptima en la Batalla de los Sexos.

H$) (*#$= #+ &,"4*"%, 2# (#01-+#$ 2# #$%0'%#5-'$ >*# 1,07'" *" !@ #$K

2@B% MHP(1!E P(1!:E HQR$;/,E QR$;/,: I SH8N6E 9N6:E H9N6E 8N6:TO

Otro procedimiento de cálculo de los EN en estrategias mixtas para juegos 2 #2

!" +,$ 4*#5,$ 1-"-%,$ +' -"%0,2*&&-9" 2# #$%0'%#5-'$ 7-<%'$ $*(,"#= (,0 *"' ('0%#= +' &,"8B#0$-9" 2# +,$ &,"4*"%,$ 2# #$%0'%#5-'$ 2# +,$ 4*5'2,0#$ #" &,"4*"%,$ &,"%-"*,$ / &,"B#8<,$= / (,0 ,%0' ('0%#= +' &,"B#0$-9" 2# +'$ 5'"'"&-'$ 2# +,$ 4*5'2,0#$ #" 1*"&-,"#$&,"%-"*'$ / +-"#'+#$ #" +'$ (0,?'?-+-2'2#$: !$%' &'0'&%#0)$%-&' (#07-%# #" #+ &'$, 2# +,$ 4*#5,$ O# O *%-+-3'0 #+ &6+&*+, 2-1#0#"&-'+ ('0' ,?%#"#0 +'$ 0#$(*#$%'$ 9(%-7'$ 2# *" 4*5'2,0 ' +'$ #$%0'%#5-'$ 7-<%'$ 2#+ ,%0,= &,7, -+*$%0'0#7,$ #" #+ $-5*-#"%# #4#7(+,:

Ejemplo 3.9

W,"$-2#0#7,$ +' $-5*-#"%# B#0$-9" 2#+ 4*#5, G'+&9"8E'+,7'= 2,"2# A %O / ! %LK

5.'&+/% 9

E'+,7' G'+&9"

5.'&+/% 8E'+,7' J= J Q= O

G'+&9" O= Q .M= .M

W,7, (,2#7,$ ,?$#0B'0= #$%# 4*#5, (0#$#"%' O !@ #" #$%0'%#5-'$ (*0'$ &,00#$(,"82-#"%#$ ' +,$ (#01-+#$ CB"/,-"= C"/+85D / C C"/+85= B"/,-"D: ^#'7,$ $- #<-$%# '+5."

!@ #" #$%0'%#5-'$ 7-<%'$:;- #+ 4*5'2,0 J 4*#5' +' #$%0'%#5-' (*0' B"/,-" &," (0,?'?-+-2'2 . / C"/+85 &,"(0,?'?-+-2'2 CJ. .D= / 2#+ 7-$7, 7,2, #+ 4*5'2,0 O 4*#5' B"/,-" &," (0,?'?-+-2'2 2

/ C"/+85 &," (0,?'?-+-2'2 CJ. 2D= +' *%-+-2'2 #$(#0'2' 2# '7?,$ 4*5'2,0#$ #$K

3 JaC .= J. .D= C2= J.2Db% .3 JCB"/,-"= C2= J.2DD!CJ. .D3 JC C"/+85= C2= J.2DD%

% .a2CJD! CJ. 2DCQDb! CJ. .DaO2! C.MDCJ. 2Db%

%M .!c2. L .2. M

3 OaC .= J. .D= C2= J.2Db%

.3 OCB"/,-"= C2= J.2DD!

CJ. .D3 OC C"/+85= C2= J.2DD%

% .a2CJD! CJ. 2DCODb! CJ. .Da2CQD! C.MDCJ. 2Db%

%c .!M2. L .2. M

!+ 4*5'2,0 J 2#&-2-06 #+ B'+,0 2# . 2# %'+ 7,2, >*# 7'<-7-&# $*$ 5'"'"&-'$ #$(#0'82'$ 2'2' +' #$%0'%#5-' 7-<%' 2#+ 4*5'2,0 OK

7'< .

3 JaC .= J. .D= C2= J. 2Db%M .! c2. L .2. M

$'?-#"2, >*# Qm .m J= / Qm 2m J:

;- #<-$%-#0'" $,+*&-,"#$ -"%#0-,0#$= $# -2#"%-1-&'0)'" 2#0-B'"2, +' *%-+-2'2 #$(#0'2'2# YJ &," 0#$(#&%, ' . # -5*'+'"2, ' Q: V# #$# 7,2, ,?%#"20)'7,$K

L3 J

L .%Q% M. L2

;- 2b M/ L= #"%,"&#$L3 J

L .a Q= +, >*# $-5"-1-&' >*# .7% Q= / #+ 4*5'2,0 J $#+#&8

&-,"' C"/+85:

;- 2a M/ L= #"%,"&#$L3 J

L .b Q= +, >*# $-5"-1-&' >*# .7% J= / #+ 4*5'2,0 J $#+#&8

&-,"' B"/,-":

;- 2%

M/ L= #"%,"&#$

L3 J

L .%

Q= +, >*# $-5"-1-&' >*# #+ 4*5'2,0 J #$ -"2-1#0#"%##"%0# $*$ #$%0'%#5-'$ (*0'$ B"/,-" / C"/+85= '$) &,7, &," &*'+>*-#0 2-$%0-?*&-9"2# (0,?'?-+-2'2 $,?0# 2-&A'$ #$%0'%#5-'$ (*0'$:

k#'+-3'"2, #+ 7-$7, (0,&#$, 2# &6+&*+, ('0' #+ 4*5'2,0 OK

7'<2

3 OaC .= J. .D= C2= J. 2Db%c .! M2. L .2. M

$'?-#"2, >*# Qm .m J= / Qm 2m J:;- #<-$%-#0'" $,+*&-,"#$ -"%#0-,0#$= $# -2#"%-1-&'0)'" 2#0-B'"2, +' *%-+-2'2 #$(#0'2'

2# YO &," 0#$(#&%, ' 2 # -5*'+'"2, ' Q: V# #$# 7,2, ,?%#"20)'7,$K

L3 O

L2%Q% M. L .

;- .b M/ L= #"%,"&#$L3 O

L2a Q= +, >*# $-5"-1-&' >*# 27% Q= / #+ 4*5'2,0 O $#+#&8

&-,"' C"/+85:;- .a M/ L= #"%,"&#$

L3 O

L2b Q= +, >*# $-5"-1-&' >*# 27% J= / #+ 4*5'2,0 O $#+#&8

&-,"' B"/,-":

;- .%M/ L= #"%,"&#$L3 O

L2% Q= +, >*# $-5"-1-&' >*# #+ 4*5'2,0 O #$ -"2-1#0#"%#

#"%0# $*$ #$%0'%#5-'$ (*0'$ B"/,-" / C"/+85= '$) &,7, &," &*'+>*-#0 2-$%0-?*&-9"2# (0,?'?-+-2'2 $,?0# 2-&A'$ #$%0'%#5-'$ (*0'$:

E,0 %'"%,= $# %-#"#" +'$ 1*"&-,"#$ ;JC2D / ;OC .D >*# $# 0#(0#$#"%'" 5061-&'7#"%# #" +'

T-5*0' M:c:

1 (Paloma)

q

0 (Halcón) p*5 3/4

q* 3/45

1 (Paloma)

R p2( )

p

R q1( )

Figura 3.5 Correspondencias de respuesta óptima en el juego Halcón-Paloma.

H +' B-$%' 2# +' 0#(0#$#"%'&-9" 5061-&'= $# ,?%-#"#" +,$ $-5*-#"%#$ #>*-+-?0-,$ 2# @'$AK

2@BNCB"/,-"= C"/+85D= C C"/+85= B"/,-"D=

aC .% M/ L= J. .%J/ LD= C2% M/ L= J. 2%J/ LDbP

Análisis general de los equilibrios de Nash en estrategias mixtas

en juegos 2 #2

X'$6"2,",$ #" +' '(+-&'&-9" 2#+ 7F%,2, 2# -5*'+'&-9" 2# +'$ 5'"'"&-'$ #$(#0'2'$= (,2#87,$ 0#'+-3'0 *" '"6+-$-$ 5#"#0'+-3'2, 2# +,$ #>*-+-?0-,$ 2# @'$A #" #$%0'%#5-'$ 7-<%'$('0' %,2, 4*#5, &," 2,$ 4*5'2,0#$ / 2,$ #$%0'%#5-'$ (*0'$ ('0' &'2' 4*5'2,0: W,"$-2#0#87,$ #+ $-5*-#"%# 4*#5,K

5.'&+/% 9

U V

5.'&+/% 8H 'HU= ?HU 'HV= ?HV

X 'XU= ?XU 'XV= ?XV

;#' C .= J. .D *"' #$%0'%#5-' 7-<%' 5#"F0-&' ('0' #+ 4*5'2,0 J= / C2= J. 2D *"' #$%0'8%#5-' 7-<%' 5#"F0-&' ('0' #+ 4*5'2,0 O: H(+-&'"2, #+ 7F%,2, 2# -5*'+'&-9"= / &,"$-2#0'"82, 1-4' +' #$%0'%#5-' C .= J. .D 2#+ 4*5'2,0 J= +'$ *%-+-2'2#$ #$(#0'2'$ 2#+ 4*5'2,0 O&*'"2, 4*#5' +' #$%0'%#5-'$ (*0'$ U / V $,"= 0#$(#&%-B'7#"%#K

3 OCC .= J. .D= UD%

. ` ?HU!

CJ. .D ` ?XU%

?XU!

.C?HU. ?XUD3 OCC .= J. .D= VD% . ` ?HV! CJ. .D ` ?XV%?XV! .C?HV. ?XVD

U5*'+'"2, '7?'$= / 2#$(#4'"2, #+ ('067#%0, . ,?%#"#7,$K

3 OCC .= J. .D= UD%$OCC .= J. .D= VDg ?XU! .C?HU. ?XUD% ?XV! .C?HV. ?XVDg

.% C?XV. ?XUD/ C?HU!?XV. ?XU. ?HVD

V#+ 7-$7, 7,2,= &,"$-2#0'"2, 2'2' +' #$%0'%#5-' C2= J. 2D 2#+ 4*5'2,0 O= +'$ *%-+-82'2#$ #$(#0'2'$ 2#+ 4*5'2,0 J &*'"2, 4*#5' +' #$%0'%#5-'$ (*0'$ H / X $,"= 0#$(#&%-B'87#"%#K

3 JCH= C2= J. 2DD%2 ` 'HU! CJ. 2D ` 'HV% 'HV! 2C'HU. 'HVD

3 JCX= C2= J. 2DD% 2 ` 'XU! CJ. 2D ` 'XV% 'XV! 2C'XU. 'XVD

U5*'+'"2, / 2#$(#4'"2,= 2# 7'"#0' '"6+,5'= 2#2*&-7,$K

2% C'XV. 'HVD/ C'HU! 'XV. 'XU. 'HVD

k#$*7-#"2,K

.%C?XV.?XUD/ C?HU!?XV.?XU.?HVD / CJ. .D%C?HU.?HVD/ C?HU!?XV.?XU.?HVD

2%C'XV.'HVD/ C'HU!'XV.'XU.'HVD / CJ.2D%C'HU.'XUD/ C'HU!'XV.'XU. 'HVD

;- +,$ B'+,0#$ '$) ,?%#"-2,$ 2# . / 2 (#0%#"#&#" '+ -"%#0B'+, &#00'2, aQ= Jb= 2-0#7,$>*# #+ (#01-+ aC .= J. .D= C2= J. 2Db #$ *" #>*-+-?0-, 2# @'$A #" #$%0'%#5-'$ 7-<%'$: W,7,(*#2# '(0#&-'0$#= +' #$%0'%#5-' 7-<%' 2# #>*-+-?0-, ('0' &'2' *", 2# +,$ 4*5'2,0#$ #" %,2, 4*#5, O#O 2#(#"2# #<&+*$-B'7#"%# 2# +'$ 5'"'"&-'$ 2#+ ,%0, 4*5'2,0 #" &'2' (#01-+ 2##$%0'%#5-'$ (*0'$= / ", 2# +'$ 5'"'"&-'$ (0,(-'$:

;-" #7?'05,= #++, ", $-5"-1-&' >*# +'$ 5'"'"&-'$ 2# *" 4*5'2,0 ", '1#&%#" ' $* #$%0'8%#5-' 2# #>*-+-?0-,: E,0 #4#7(+,= &*'"2, *" 4*5'2,0 (,$## *"' #$%0'%#5-' #$%0-&%'7#"%#2,7-"'"%#= F$%# %-#"# >*# $#+#&&-,"'0 #" #>*-+-?0-, #$' #$%0'%#5-'= -"2#(#"2-#"%#7#"%# 2#+'$ 5'"'"&-'$ 2#+ 0#$%, 2# 4*5'2,0#$= (*#$ &," #++' $-#7(0# #$%'06 7'<-7-3'"2, $*$ 5'8"'"&-'$:

Cálculo sistemático de los EN en estrategias mixtasde un juego bipersonal finito

E'0' 4*#5,$ ?-(#0$,"'+#$ 1-"-%,$ #<-$%# *" (0,&#2-7-#"%, 2# &6+&*+, 5#"#0'+ 2# +,$ !@ #"

#$%0'%#5-'$ 7-<%'$: V-&A, (0,&#2-7-#"%, &,"$%' 2# +'$ $-5*-#"%#$ #%'('$K

87 !+-7-"'0 %,2'$ +'$ #$%0'%#5-'$ (*0'$ >*# ", $,?0#B-B'" '+ (0,&#$, 2# #+-7-"'&-9"-%#0'%-B' 2# #$%0'%#5-'$ #$%0-&%'7#"%# 2,7-"'2'$:

97 ;- #+ 4*#5, 0#$*+%'"%# #$ O#O '(+-&'0 &*'+>*-#0' 2# +,$ (0,&#2-7-#"%,$ B6+-2,$=/' '"'+-3'2,$= ('0' 2-&A,$ 4*#5,$:

67 ;- #" #+ 4*#5, 0#$*+%'"%# '+ 7#",$ '+5*", 2# +,$ 4*5'2,0#$ %-#"# *" &,"4*"%, 2##$%0'%#5-'$ (*0'$ $*(#0-,0 ' O C/ #+ ,%0, 4*5'2,0 &,7, 7)"-7, ODK

H"'+-3'7,$ +' #<-$%#"&-' 2# !@ ('0' +'$ 2-$%-"%'$ &,7?-"'&-,"#$ (,$-?+#$ 2# $,8(,0%#$ 2# #$%0'%#5-'$ (*0'$ 2# &'2' 4*5'2,0= #$ 2#&-0= ?*$&'7,$ +,$ !@ 2# %,2,$ +,$

4*#5,$ >*# $# (*#2#" 1,07'0 &," +,$ $*?&,"4*"%,$ 2# #$%0'%#5-'$ (*0'$ 2# &'2' 4*5'2,0:

I,$ !@ #"&,"%0'2,$ ('0' &'2' &,7?-"'&-9" 2# $,(,0%#$ $# &,"1-07'06" &,7, !@2#+ 4*#5, &,7(+#%,= $- $," B'+-2'2,$ '+ &,"$-2#0'0 +'$ #$%0'%#5-'$ >*# ", 1-5*0'" #"+,$ $,(,0%#$ '&%*'+7#"%# *%-+-3'2,$: !$%, $-5"-1-&' >*# +'$ #$%0'%#5-'$ 2# +,$ (#01-+#$(0,(*#$%,$ &,7, !@ 2#?#06" 7'"%#"#0$# &,7, 0#$(*#$%'$ 9(%-7'$ 2# *" 4*5'2,0&," 0#$(#&%, '+ 0#$%, 2# #$%0'%#5-'$ 2#+ (#01-+= &*'"2, '7(+-'7,$ #+ &,"4*"%, $,(,08%# 2# 2-&A, 4*5'2,0 C#$ 2#&-0= &*'"2, (#07-%-7,$ >*# (*#2'" #"%0'0 #" 4*#5, %,2'$+'$ #$%0'%#5-'$ 2-$(,"-?+#$ ('0' #+ 4*5'2,0D:

;- #" #+ 4*#5, 0#2*&-2, 0#$*+%'"%# '+ 7#",$ *", 2# +,$ 4*5'2,0#$ %-#"# *" &,"4*"%, 2##$%0'%#5-'$ (*0'$ *"-%'0-,= %,2,$ +,$ (#01-+#$ 2# #$%0'%#5-'$ (*0'$ , 7-<%'$ 2#+ 4*#5, 0#2*8&-2, $," !@$:

U+*$%0#7,$ 2-&A, (0,&#2-7-#"%, &," #+ !4#7(+, M:JQ:

Ejemplo 3.10

;# &,"$-2#0' #+ $-5*-#"%# 4*#5,K

5.!'/ 679

5.'&+/% 9

U W V

5.'&+/% 8

H M= O L= M J= L

W J= M f= Q O= J

X O= O e= .c O= Q

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

5.'&+/% 9

U V

5.'&+/% 8

H M= O J= L

W J= M O= J

X O= O O= Q

E'0' &'+&*+'0 +,$ !@ 2#+ 4*#5, $#5*-7,$ #+ $-5*-#"%# (0,&#$,K

&: W'+&*+'7,$ (0-7#0, +,$ !@ #" #$%0'%#5-'$ (*0'$ C'7?,$ $,(,0%#$ *"-%'0-,$DK

!" #$%# 4*#5, ", #<-$%#":

;: W'+&*+'7,$ 'A,0' +,$ !@ >*# %#"5'" $,(,0%# *"-%'0-, #" +' #$%0'%#5-' 2# *"$9+, 4*5'2,0K

;8: ;,(,0%# *"-%'0-, #" +' #$%0'%#5-' 2#+ 4*5'2,0 JK!" "*#$%0, 4*#5,= #+ $,(,0%# ", (*#2# $#0 H= (*#$ +' ."-&' 0#$(*#$%' 9(%-7' #$ V= "-

W= (*#$ +' ."-&' 0#$(*#$%' 9(%-7' #$ U= "- X= (*#$ +' ."-&' 0#$(*#$%' 9(%-7' #$ U:;9: ;,(,0%# *"-%'0-, #" +' #$%0'%#5-' 2#+ 4*5'2,0 OK!" "*#$%0, 4*#5,= #+ $,(,0%# ", (*#2# $#0 U C."-&' 0#$(*#$%' 9(%-7' HD= '*">*# $)

(*#2# $#0 V= (*#$ %-#"# &,7, 0#$(*#$%'$ 9(%-7'$ ' W / X:^#'7,$ $- &," *" $,(,0%# *"-%'0-, NVP 2#+ 4*5'2,0 O / &," $,(,0%# NW= XP 2#+

4*5'2,0 J C#$ 2#&-0= &," $,(,0%# (0,2*&%, NW= XP#VD #<-$%# '+5." !@K[' A#7,$ -"2-&'2, >*# 2'2' +' #$%0'%#5-' V 2#+ 4*5'2,0 O= +'$ #$%0'%#5-'$ W / X $,"

0#$(*#$%'$ 9(%-7'$ (,0 ('0%# 2#+ 4*5'2,0 J /= (,0 %'"%,= &*'+>*-#0 #$%0'%#5-' 7-<%' >*#$# (*#2' 1,07'0 &," #++'$ %'7?-F" +, #$:

V'2' *"' #$%0'%#5-' 7-<%' CQ= .= J. .D 2#+ 4*5'2,0 J $,?0# #+ $,(,0%# NW= XP= B#'8

7,$ ('0' >*F B'+,0#$ #$ V 0#$(*#$%' 9(%-7'K

m'"'"&-' #$(#0'2' ('0' #+ 4*5'2,0 O $- #+-5# UK . ` M! CJ. .D ` O% .! O

m'"'"&-' #$(#0'2' ('0' #+ 4*5'2,0 O $- #+-5# VK . ` J! CJ. .D ` Q% .

n*#006 7#3&+'0 W / X $- .! O% .= +, &*'+ ", ,&*00# "*"&'= (*#$ $-#7(0# (0#1#0-06#$%0-&%'7#"%# U:

4/% $&1$/E 1/ C&I 1(1'R1 @B =/1 !"$! "/0/%$!7W,"&+*-7,$ >*# #" &*'+>*-#0 !@= #+ $,(,0%# ('0' #+ 4*5'2,0 J 2#?# $#0 NH= WP= ,

?-#" NH= XP= , ?-#" NW= XP= , ?-#" NH= W= XP= / #+ $,(,0%# ('0' #+ 4*5'2,0 O 2#?# $#0

"#&#$'0-'7#"%# NU= VP:

=: W'+&*+'7,$ 'A,0' +,$ !@ >*# %#"5'" $,(,0%# ", *"-%'0-, #" +' #$%0'%#5-' 2#'7?,$ 4*5'2,0#$K

=8: ;,(,0%# NH= WP# NU= VP= #" &*/, &'$, #+ (,$-?+# !@ $#0)' aC .= J. .= QD=C2= J. 2Db:

V'2' +' #$%0'%#5-' C2= J. 2D 2#+ 4*5'2,0 OK

I' 5'"'"&-' #$(#0'2' ('0' #+ 4*5'2,0 J $- #+-5# H #$K 2 ` M! CJ. 2D ` J%O2!JI' 5'"'"&-' #$(#0'2' ('0' #+ 4*5'2,0 J $- #+-5# W #$K 2 ` J! CJ. 2D ` O%.2!O!+ 4*5'2,0 J >*#006 7#3&+'0 H / W $- O2! J%.2!O= #$ 2#&-0= $- 2% J/ M

V'2' +' #$%0'%#5-' C .= J. .= QD 2#+ 4*5'2,0 JK

I' 5'"'"&-' #$(#0'2' ('0' #+ 4*5'2,0 O $- #+-5# U #$K . ` O! CJ. .D ` M%. .!MI' 5'"'"&-' #$(#0'2' ('0' #+ 4*5'2,0 O $- #+-5# VK . ` L! CJ. .D ` J%M .! J!+ 4*5'2,0 O >*#006 7#3&+'0 U / V $- . .!M% M .!J= #$ 2#&-0= $- .%J/ O:

!+ (#01-+ aCJ/ O= J/ O= QD= CJ/ M= O/ MDb #$ (,0 %'"%, *" (,$-?+# &'"2-2'%, ' !@: E'0' >*#$#' #1#&%-B'7#"%# *" !@ A'&# 1'+%' >*# CJ/ O= J/ O= QD $#' 0#$(*#$%' 9(%-7' ' CJ/ M= O/ MD=#$ 2#&-0= >*# #+ 4*5'2,0 J ", %#"5' -"&#"%-B,$ ('0' *%-+-3'0 ,%0' #$%0'%#5-' C/' $#' X ,>*# +' -"&+*/'D= 2'2' +' #$%0'%#5-' CJ/ M= O/ MD 2# YO: ^#'7,$ $- #$ '$)K

3 JaCJ/ O= J/ O= QD= CJ/ M= O/ MDb%J/ O ` J/ M ` M!J/ O ` O/ M ` J!J/ O ` J/ M ` J!J/ O ` O/ M ` O%c/ M

;-" #7?'05,= +' 5'"'"&-' 2# YJ $- 4*#5' X #$K

3 Ja D= CJ/ M= O/ MDb% O/ M!O ` O/ M% Ob c/ M

I, >*# $-5"-1-&' >*# CJ/ O= J/ O= QD ('0' #+ 4*5'2,0 J ", #$ 0#$(*#$%' 9(%-7' '

CJ/ M= O/ MD 2#+ 4*5'2,0 O: I*#5, #+ (#01-+ -"2-&'2, ", #$ *" !@: E,0 %'"%,= 1/ C&I 1(1#'R1 @B =/1 !"$! "/0/%$!7

=9: ;,(,0%# NH= XP# NU= VP= #" &*/, &'$, #+ (,$-?+# !@ $#0)' aC .= Q= J. .D=C2= J. 2Db

V'2' +' #$%0'%#5-' C2= J. 2D 2#+ 4*5'2,0 OK

I' 5'"'"&-' #$(#0'2' ('0' #+ 4*5'2,0 J $- #+-5# H #$K 2 ` M! CJ. 2D ` J%O2!JI' 5'"'"&-' #$(#0'2' ('0' #+ 4*5'2,0 J $- #+-5# X #$K 2 ` O! CJ. 2D ` O%O!+ 4*5'2,0 J >*#006 7#3&+'0 H / X $- O2! J%O= #$ 2#&-0= $- 2%J/ O:

V'2' +' #$%0'%#5-' C .= Q= J. .D 2#+ 4*5'2,0 JK

I' 5'"'"&-' #$(#0'2' ('0' #+ 4*5'2,0 O $- #+-5# U #$K . ` O! CJ. .D ` O% OI' 5'"'"&-' #$(#0'2' ('0' #+ 4*5'2,0 O $- #+-5# V #$K . ` L! CJ. .D ` Q%L .

!+ 4*5'2,0 O >*#006 7#3&+'0 U / V $- O%L .= #$ 2#&-0= $- .% J/ O:

V# "*#B,= *" (#01-+= aCJ/ O= Q= J/ OD= CJ/ O= J/ ODb #$ *" (,$-?+# &'"2-2'%, ' !@: E'0'>*# $#' #1#&%-B'7#"%# *" !@ A'&# 1'+%' >*# #+ 4*5'2,0 J ", %#"5' -"&#"%-B,$ ' *%-+-3'0,%0' #$%0'%#5-' C#" #$%# &'$, W , '+5*"' >*# +' -"&+*/'D= 2'2' +' #$%0'%#5-' CJ/ O= J/ OD 2#+ 4*5'2,0 OK

3 JaCJ/ O= Q= J/ OD= CJ/ O= J/ ODb%J/ O ` J/ O ` M!J/ O ` J/ O ` J!J/ O ` J/ O ` O!J/ O ` J/ O ` O%O

I' 5'"'"&-' 2#+ 4*5'2,0 J $- 4*#5' W #$K

3 JaW= CJ/ O= J/ ODb% J/ O!O ` J/ O%M/ Oa O

I, >*# $-5"-1-&' >*# CJ/ O= Q= J/ OD ('0' #+ 4*5'2,0 J $) #$ 0#$(*#$%' 9(%-7' 'CJ/ O= J/ OD 2#+ 4*5'2,0 O: I*#5, #+ (#01-+ -"2-&'2, SH8N9E UE 8N9:E H8N9E 8N9:T "V !" .1 @B :

=6: ;,(,0%# NW= XP# NU= VP= #" &*/, &'$, #+ (,$-?+# !@ $#0)' aCQ= .= J. .D=C2= J. 2Db

V'2' +' #$%0'%#5-' C2= J. 2D 2#+ 4*5'2,0 OK

I' 5'"'"&-' #$(#0'2' ('0' #+ 4*5'2,0 J $- #+-5# W #$K 2 ` J! CJ. 2D ` O%.2!O

I' 5'"'"&-' #$(#0'2' ('0' #+ 4*5'2,0 J $- #+-5# X #$K 2 ` O! CJ. 2D ` O%O!+ 4*5'2,0 J >*#006 7#3&+'0 $- .2!O%O= #$ 2#&-0= $- 2% Q:E#0, $- 2%Q $-5"-1-&' >*# #+ 4*5'2,0 O #$%6 *$'"2, *" $,(,0%# *"-%'0-,= +, &*'+

#<&+*/#= &,7, /' A#7,$ B-$%, #" #+ '('0%'2, ?O= >*# aCQ= .= J. .D= C2= J. 2Db $#' *"!@: E,0 %'"%,= 1/ C&I 1(1'R1 @B =/1 !"$! "/0/%$!7

=J: ;,(,0%# NH= W= XP#NU= VP= #" &*/, &'$, #+ (,$-?+# !@ $#0)' aC .= 1 = J. ..1 D=C2= J. 2Db

V'2' +' #$%0'%#5-' C2= J. 2D 2#+ 4*5'2,0 OK

I' 5'"'"&-' #$(#0'2' ('0' #+ 4*5'2,0 J $- #+-5# H #$K 2 ` M! CJ. 2D ` J%O2!JI' 5'"'"&-' #$(#0'2' ('0' #+ 4*5'2,0 J $- #+-5# W #$K 2 ` J! CJ. 2D ` O%.2!O

I' 5'"'"&-' #$(#0'2' ('0' #+ 4*5'2,0 J $- #+-5# X #$K 2 ` O! CJ. 2D ` O%O

!+ 4*5'2,0 J >*#006 7#3&+'0 H= W / X $- O2! J%.2!O% O C+, &*'+ ", ,&*00#"*"&'D:

E,0 %'"%,= 1/ C&I 1(1'R1 @B =/1 !"$! "/0/%$!7

!" &,"&+*$-9"= #+ ."-&, !@ 2#+ 4*#5, 0#2*&-2, #$ aCJ/ O= Q= J/ OD= CJ/ O= J/ ODb: !"

&*'"%, '+ 4*#5, ,0-5-"'+= &,"$-2#0'"2, +' #$%0'%#5-' W 2#+ 4*5'2,0 O (,2#7,$ &,"&+*-0K2@B% MSH8N9E UE 8N9:E H8N9E UE 8N9:TO:

Teoremas de existencia del equilibrio de Nash

H"%#$ 2# #"*"&-'0 / 2# (,2#0 2#7,$%0'0= $->*-#0' ('0&-'+7#"%#= +,$ %#,0#7'$ >*# '$#5*80'" +' #<-$%#"&-' 2# #>*-+-?0-, 2# @'$A ('0' '+5*"'$ &+'$#$ '7(+-'$ # -7(,0%'"%#$ 2# 4*#5,$= $#06 "#&#$'0-, #$%'?+#&#0 '+5*"' %#07-",+,5)' "*#B' / 0#('$'0 '+5*",$ &,"&#(%,$&,",&-2,$= 0#1#0-2,$ ' &,"4*"%,$ #" #$('&-,$ #*&+-2#,$ # $= / ' 1*"&-,"#$ / &,00#$(,"2#"8&-'$ #"%0# 2-&A,$ &,"4*"%,$:

Repaso terminológico

V'2, *" &,"4*"%, HÑ # $= 2#&-7,$ >*# #$K

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P!%%&+/ $- $* &,7(+#7#"%'0-, # $.H #$ '?-#0%,: !$ 2#&-0= 2'2, &*'+>*-#0 (*"%, % â H=#<-$%# *"' ?,+' '?-#0%' XC %= # D 2# &#"%0, % / 0'2-, # "-"5*", 2# &*/,$ (*"%,$ (#0%#"#&# ' H:

W=/$&+/ $- #<-$%# *" # b Q %'+ >*# 2'2,$ 2,$ (*"%,$ &*'+#$>*-#0' %& ' à H= +' 2-$%'"&-'#"%0# % # ' #$ 7#",0 >*# # :

P/)0&=$/ $- #$ &#00'2, / '&,%'2,:P/1<!*/ $- 2'2,$ 2,$ (*"%,$ &*'+#$>*-#0' %& ' à H= &*'+>*-#0 &,7?-"'&-9" &,"B#<' 2# %

# ' (#0%#"#&# ' H= #$ 2#&-0= ('0' %,2, j à aQ= Jb= j %! CJ. jD ' à H:

I,$ #4#7(+,$ 76$ $#"&-++,$ # -7(,0%'"%#$ 2# &,"4*"%,$ &,7('&%,$ / &,"B#<,$ #" # $

$," +,$ -"%#0B'+,$ &#00'2,$ U= +,$ $)7(+-&#$ 2# 2-7#"$-9" &*'+>*-#0' B= +,$ (0,2*&%,$&'0%#$-'",$ 2# -"%#0B'+,$ &#00'2,$ / +,$ (0,2*&%,$ &'0%#$-'",$ 2# $)7(+-&#$:

V'2' *"' 1*"&-9" E K HrX= &," 2,7-"-, HÑ # $ / 0#&,00-2, #" XÑ # (= 2#&-7,$ >*# #$K

P/1$(1.& #" H $- ('0' %,2, % à H= +' $*&#$-9" N F ) P >*# %#"5' (,0 +)7-%# % -7(+-&' >*# +'$*&#$-9" 2# -765#"#$ N E C F ) DP %#"206 (,0 +)7-%# E C F D:

I+'7'7,$ =/%%!"0/1+!1=(& 2# H #" X= / +' 2#",%'7,$ GK HrrX= ' %,2' 0#5+'>*# '$-5"' ' &'2' (*"%, 2# H *" $*?&,"4*"%, 2# X: !$ 2#&-0= GK HrrX #$ *"' &,00#$8(,"2#"&-' 2# H #" X $- ('0' %,2, % à H= GC %DÑX: !" #+ &'$, ('0%-&*+'0 #" >*# GC %D #$*"-%'0-, ('0' %,2, %= G #$ *"' 1*"&-9":

V'2' *"' &,00#$(,"2#"&-' GK HrrX= 2,"2# HÑ # $= XÑ # ( / X #$ &,7('&%,KV#&-7,$ >*# #$ C!)(=/1$(1.& ".0!%(/%)!1$! #" H $- #+ 50'1, 2# G= >*# #$ #+ &,"8

4*"%, JG% NC %& 'D à # $! ( K % à H= ' à GC %DPÑH#X= #$ &#00'2, #" H#X: R'7?-F"

(*#2# 2#&-0$# >*# G #$ A#7-&,"%-"*' $*(#0-,07#"%# $- $-#7(0# >*# +'$ $*&#$-,"#$ N F ) P=N H) P= &," F ) à H # H) à GC % $D %#"5'" (,0 +)7-%# % à H # ' à X= 0#$(#&%-B'7#"%#= $# B#0-1-&'

>*# ' à GC %D: S?$F0B#$# >*# +' A#7-&,"%-"*-2'2 $*(#0-,0 #$ *"' 5#"#0'+-3'&-9" 2# +'&,"%-"*-2'2: V# A#&A,= #" #+ &'$, ('0%-&*+'0 #" >*# GC F D 1*#$# *"-%'0-, ('0' %,2, F 2# H=>*# #+ 50'1, 2# G $#' &#00'2, -7(+-&' >*# G #$ *"' 1*"&-9" &,"%-"*' #" H:

;- H%X= 2#&-7,$ >*# % à H #$ *" 0.1$/ X(-/ 2# +' &,00#$(,"2#"&-' ) $- % à GC %D: I,'"%#0-,0 B#"20)' ' $-5"-1-&'0 %% GC %D $- G 1*#$# *"' 1*"&-9":

V'2' *"' 1*"&-9" 0#'+ E K HrX= &," 2,7-"-, HÑ # $ &,"B#<, / 0#&,00-2, #"XÑ #= 2#&-7,$ >*# #$K

P>1=&<& $- 2'2,$ 2,$ (*"%,$ &*'+#$>*-#0' %& ' à H= +' -7'5#" 2# &*'+>*-#0 &,7?-"'&-9"&,"B#<' 2# % # ' #$ -5*'+ , 7'/,0 >*# +' &,00#$(,"2-#"%# &,7?-"'&-9" &,"B#<' 2# +'$-765#"#$: !$ 2#&-0= E Cj %! CJ. jD 'Dn j E C %D! CJ. jD E C 'D= Oj à aQ= Jb= O %& ' à H:P.&"(=>1=&<& $- E Cj %! CJ. jD 'Dn7-"N E C %*& E C 'DP= Oj à aQ= Jb= O %& ' à H: Z" 7,2,#>*-B'+#"%# 2# 2#1-"-0 +' >*'$-&,"&'B-2'2 #$ >*# #+ &,"4*"%, 2# &,"%,0", $*(#0-,0La% N % à H K E C %Dn aP $#' &,"B#<, ('0' %,2, a à #:

!" +' T-5*0' M:d $# -+*$%0' +' &,"&'B-2'2 / &*'$-&,"&'B-2'2 ('0' 1*"&-,"#$ 0#'+#$ 2#B'0-'?+# 0#'+= &," 2,7-"-, #" #+ -"%#0B'+, aQ= Jb:

f

f

f

f

1 1 1

(a) (b) (c)

Cóncava ycuasicóncava

Cuasicóncava pero nocóncava (sí convexa)

Ni cóncava nicuasi cóncava

Figura 3.6 Ilustración de la concavidad y cuasiconcavidad.

H &,"%-"*'&-9" #"*"&-'0#7,$ $-" 2#7,$%0'&-9" #+ %#,0#7' 2#+ (*"%, 1-4, 2# o'p*%'8"-= / 2#7,$%0'0#7,$= ?'$6"2,",$ #" F+= 2,$ %#,0#7'$ 2# #<-$%#"&-' 2# #>*-+-?0-,$ 2#@'$A: !+ (0-7#0, #$%'?+#&# +' #<-$%#"&-' 2# #>*-+-?0-,$ #" #$%0'%#5-'$ (*0'$ ('0' '+5*",$%-(,$ 2# 4*#5,$ -"1-"-%,$= 7-#"%0'$ >*# #+ $#5*"2,= >*# #$ *" &,0,+'0-, 2#+ '"%#0-,0= #$%'8

?+#&# +' #<-$%#"&-' 2# #>*-+-?0-,$ #" #$%0'%#5-'$ 7-<%'$ ('0' %,2,$ +,$ 4*#5,$ 1-"-%,$:

Teorema 3.2 F!/%!)& +!, 0.1$/ X(-/ +! Y&Z.$&1(

;#' GK HrrH *"' &,00#$(,"2#"&-' &," 2,7-"-, #" 6Ñ # $= 2,"2# H #$ ", B'&),=&,7('&%, / &,"B#<,: ;- G #$ A#7-&,"%-"*' $*(#0-,07#"%#= / '2#76$ GC %DÑH #$ &,"8B#<, / ", B'&), ('0' %,2, % 2# H= #"%,"&#$ #<-$%# *" (*"%, 1-4, %+ 2# G= #$ 2#&-0= *"(*"%, %+ 2# H %'+ >*# %+à GC %+D:

!" +' T-5*0' M:f= $# -+*$%0' #$%# 0#$*+%'2, 7#2-'"%# %0#$ $-%*'&-,"#$: !" +' (0-7#0' $#

B#0-1-&'" +'$ A-(9%#$-$ 2#+ %#,0#7'= #" +' $#5*"2' 1'++' >*# GC %QD $#' &,"B#<,= / #" +'%#0&#0' 1'++' >*# G $#' A#7-&,"%-"*' $*(#0-,07#"%# #" %Q:

1

1

1

1 1

1

(a) (b) (c)

Tres puntos fijos Ningún punto fijo Ningún punto fijo

C

g

g g

g

g

g g

B A

x0 x0 x0 y0 z 0

Figura 3.7 Ilustración del teorema de Kakutani.

V'7,$ ' &,"%-"*'&-9" #+ #"*"&-'2, / 2#7,$%0'&-9" 2# +,$ 2,$ %#,0#7'$ 2# #<-$%#"&-'

76$ -7(,0%'"%#$:

Teorema 3.3

;#' #+ 4*#5, 4% N& J= :::= & 5g $J= :::= $5P= %'+ >*#= ('0' %,2, 4*5'2,0 '= $# &*7(+#K

&: & ' #$ *" $*?&,"4*"%, ", B'&),= &,7('&%, / &,"B#<, 2# *" #$('&-, # $:;: $' #$ &,"%-"*' #" %,2, $* 2,7-"-, & %& J# & O#ñ#& 5= / #$ &*'$-&9"&'B'

#" +' B'0-'?+# (':

!"%,"&#$= #<-$%# '+ 7#",$ *" !@ #" #$%0'%#5-'$ (*0'$ 2# 4:

D!)/"$%&=(>1?

V#1-"'7,$= ('0' &'2' 4*5'2,0 '= $* &,00#$(,"2#"&-' 2# 0#$(*#$%' 9(%-7' ;'= >*# '$-58"' ' &'2' (#01-+ 2# #$%0'%#5-'$ (*0'$ ,% C(J= :::= ('.J= ('= ('!J= :::= (5D% C('= ,.! * #+&,"4*"%, ;'C ,D 2# +'$ #$%0'%#5-'$ (*0'$ 2# ' >*# $," 0#$(*#$%' 9(%-7' ' ,.!: !$ 2#&-0=

;'C ,D% N F ' à & ' K F ' #$ $,+*&-9" 2#+ (0,?+#7' 7'< H

i à &

i

$'C H'= ,.!DP

V#1-"'7,$ 'A,0' +' &,00#$(,"2#"&-' 5+,?'+ 2# 0#$(*#$%' 9(%-7' ; &,7, '>*#++'>*# '$-5"' ' &'2' (#01-+ , #+ (0,2*&%, &'0%#$-'", 2# +,$ &,"4*"%,$ 2# 0#$(*#$%' 9(%-7'

-"2-B-2*'+#$: H$) (*#$= ; #$ *"' &,00#$(,"2#"&-' 2# " #" " %'+ >*#

;C ,D% ;JC ,D#ñ# ;5C ,D% N1 % C1 J= :::= 1 5D à " K 1 ' à ;'C ,D= O'%J= O= :::= 5P

^#'7,$ >*# ; #$ A#7-&,"%-"*' $*(#0-,07#"%#: " #$ ", B'&),= &,7('&%, / &,"B#8<, (,0 $#0 " #+ (0,2*&%, &'0%#$-'", 2# +,$ & ': E'0' &'2' '%J= O= :::= 5= +' 1*"&-9" $' #$&,"%-"*' #" & = +, >*# -7(+-&' >*# #$ &,"%-"*' #" & ': E,0 A-(9%#$-$ & ' #$ ", B'&), /&,7('&%,: !+ %#,0#7' 2# q#-#0$%0'$$ '$#5*0' >*# #+ (0,?+#7' 7'<

Hi à &

i

$'C H'= ,.!D %-#"#

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, à & / - à & = 0#$(#&%-B'7#"%#= #++, -7(+-&' >*# - à ;C ,*: E#0,

+-7) rä

() % ( à & á E'0' &'2' 4*5'2,0 '= +-7) rä

() ' % (' à & '

+-7) rä

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1 ) ' % 1 ' à & '

1 ) à ;C() D ú E'0' &'2' 4*5'2,0 '= 1 ) ' à ;'C() D ú

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(#0, (,0 $#0 $' &,"%-"*' #" & ' $# B#0-1-&' >*#

$'C1 '= (.'Dn $'C %'= (.'D= O %' à & ' ú 1 ' à ;'C(D= O'= / #" &,"$#&*#"&-' 1 à ;C(D

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H$) (*#$= +' &,00#$(,"2#"&-' ;K "rr" &*7(+# +'$ A-(9%#$-$ 2#+ %#,0#7' 2# o'8p*%'"- CA#7-&,"%-"*-2'2 $*(#0-,0 #" *" &,"4*"%, ", B'&),= &,7('&%, / &,"B#<,= /'2#76$ -765#"#$ ", B'&)'$ / &,"B#<'$D= +, >*# ",$ (#07-%# '1-07'0 >*# ; %-#"# '+7#",$ *" (*"%, 1-4, ,+:

HA,0' ?-#"= >*# ,+%C(7J = :::= (7' = :::= (75 D $#' *" (*"%, 1-4, 2# ; $-5"-1-&' >*# ,+ à ;C ,+D=+, >*# -7(+-&' >*# (7' à ;'C ,+D ('0' %,2, '= #$ 2#&-0= >*# (7' #$ 0#$(*#$%' 9(%-7' ' ,+.!

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!+ $-5*-#"%# %#,0#7'= ,?0' 2# @'$A #" #+ 'l, JrcQ= #$ *" $#"&-++, &,0,+'0-, 2#+ '"%#0-,0:

Teorema 3.4

!" %,2, 4*#5, 1-"-%, 4% N& J= :::= & 5g $J= :::= $5P #<-$%# '+ 7#",$ *" !@ #" #$%0'%#5-'$7-<%'$:

D!)/"$%&=(>1?

V'2, #+ 4*#5, 4= $#' & '% N(J' = (O

' = :::= ( *'= :::= (-i' P / & %& J#ñ# & '#ñ# & 5: E,820)'7,$ ++'7'0 BC4D '+ 4*#5, &*/,$ 4*5'2,0#$ $," +,$ 7-$7,$ >*# +,$ 2# 4= &*/,$&,"4*"%,$ 2# #$%0'%#5-'$ (*0'$ ('0' &'2' 4*5'2,0 ' $," +,$ $)7(+-&#$ BC&

'D &,"$%-%*-2,$

(,0 +'$ 2-$%0-?*&-,"#$ 2# (0,?'?-+-2'2 $,?0# & '= / &*/,$ ('5,$ $," +,$ ('5,$ #$(#0'2,$&'+&*+'2,$ 2#+ 7,2, *$*'+K

3 'CpJ= :::= p'= :::= p5D% ;(à &

p *CJDJ ñp *C'D

' ñ p *C5D5 $'C( *CJD

J = :::= ( *C'D' = :::= ( *C5D5 D

2,"2# $' #$%6 2#1-"-2' $,?0# B%BC& JD#ñ#BC& 'D#ñ#BC& 5D= / 2,"2# p *C'D' #$ +'

(0,?'?-+-2'2 &," >*# #+ 4*5'2,0 ' 4*#5' $* #$%0'%#5-' (*0' ( *C'D' #" $* #$%0'%#5-' 7-<%' p':E*#$ ?-#"= #+ 4*#5, BC4D% NBC& JD= :::= BC& 5Dg 3 J= :::= 3 5P &*7(+# +'$ A-(9%#$-$ 2#+

R#,0#7' M:M= /' >*# ('0' %,2, 4*5'2,0 '=

&: BC& 'D #$ *" $*?&,"4*"%, ", B'&),= &,7('&%, / &,"B#<, 2#+ #$('&-, ;-i:

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!" &,"&+*$-9"= #<-$%# *" #>*-+-?0-, 2# @'$A #" #$%0'%#5-'$ (*0'$ 2#+ 4*#5, BC4D=

>*# ", #$ ,%0' &,$' >*# *" #>*-+-?0-, 2# @'$A #" #$%0'%#5-'$ 7-<%'$ 2#+ 4*#5, 4:

Juegos simétricos

Z" &'$, #$(#&-'+ 2# 4*#5,$= >*# 7#0#&# +' (#"' #$%*2-'0 (,0 $* -"%#0F$ #" #&,",7)'= #$ #+2# +,$ 4*#5,$ $-7F%0-&,$= >*# 2#1-"-0#7,$ ' &,"%-"*'&-9"= / &*/' -"%#0(0#%'&-9" -"%*-%-B'#$ >*# %,2,$ +,$ 4*5'2,0#$ $," -5*'+7#"%# %0'%'2,$:

Definición 3.4

V'2, #+ 4*#5, 4% N& J= :::= & 5g $J= :::= $5P=

&: V#&-7,$ >*# 4 #$ "()[$%(=/ $- &*7(+# +'$ 2,$ &,"2-&-,"#$K

87 I,$ &,"4*"%,$ 2# #$%0'%#5-'$ (*0'$ $," -2F"%-&,$K & J%& O%ñ% & 5:97 E'0' &'2' ('0 2# 4*5'2,0#$= ' / *= #+ -"%#0&'7?-, 2# #$%0'%#5-'$= 2'2'$ +'$ #$8

%0'%#5-'$ 2#+ 0#$%, 2# 4*5'2,0#$= 5#"#0' *" -"%#0&'7?-, #" 5'"'"&-'$ ('0' #++,$ 2,$ /", '+%#0' +'$ 5'"'"&-'$ 2# +,$ 2#76$= #$ 2#&-0K

$'C('= ( *= (.'. *D%$ *C( *= ('= (.'. *D O'= *= O('= ( *

$) C('= ( *= (.'. *D% $) C( *= (' (.'. *D O '= *= O('= ( *= O) â N'= *P

2,"2# C( *= ('= (.'. *D #$ #+ (#01-+ 2# #$%0'%#5-'$ (*0'$ #" #+ >*# #+ 4*5'2,0 ' 4*#5' ( * / #+ 4*5'2,0 * 4*#5' (':

C!" #+ &'$, 2# 4*#5,$ &," 2,$ 4*5'2,0#$= +' &,"2-&-9" O $# 0#2*&# ' $'C('=( *D%$ *C( *= ('D= , +, >*# #$ +, 7-$7,= $*$ 2,$ 7'%0-&#$ 2# ('5,$ $," %0'"$(*#$%'$ *"' 2#+' ,%0'K 6 *% 61

' :D;: ;*(*#$%, >*# 4 #$ $-7F%0-&,= 2#&-7,$ >*# *" (#01-+ ,% C(J= :::= ('= :::= (5D #$

$-7F%0-&, $- %,2,$ +,$ 4*5'2,0#$ 4*#5'" +' 7-$7' #$%0'%#5-'= #$ 2#&-0= ('% ( * O'= *:

E'0' #$%# %-(, 2# 4*#5,$ B'7,$ ' #"*"&-'0= $-" 2#7,$%0'0+,= *" %#,0#7' 2# #<-$%#"&-'76$ (0#&-$, >*# +,$ '"%#0-,0#$= / >*# #$ *" &,0,+'0-, 2# F$%,$K

Teorema 3.5

V'2, #+ 4*#5, 4% N& J= :::= & 5g $J= :::= $5P $-7F%0-&,= 2,"2# & '% 6= O ' à NJ= O= :::= 5P:

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;: ;- 4 #$ 1-"-%,= %-#"# '+ 7#",$ *" !@ $-7F%0-&, #" #$%0'%#5-'$ 7-<%'$:

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Ejemplo 3.11

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k E R

5.'&+/% 8

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J Q .J

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Ejemplo 3.12a) !" #+ ,+-5,(,+-, 2# W,*0",% B-$%, #" #+ &'()%*+, '"%#0-,0= $#0)' '(+-&'?+# #+ R#,0#87' 2# #<-$%#"&-' M:M= /' >*# #+ #$('&-, 2# #$%0'%#5-'$ 2# &*'+>*-#0 4*5'2,0= #+ -"%#0B'+,&#00'2, aQ= "b= #$ &,7('&%, / &,"B#<,= / +'$ 1*"&-,"#$ 2# 5'"'"&-'$= n'C2J= 2O= :::= 25D%2'C". +. @2J. @2O.ñ. @25D= $," &,"%-"*'$ / &*'$-&9"&'B'$ #" 2': E,0 %'"%, A'2# %#"#0 '+5." !@ #" #$%0'%#5-'$ (*0'$: H2#76$= +# #$ '(+-&'?+# #+ R#,0#7' M:c= (,0 $#0$-7F%0-&,= +, >*# -7(+-&' >*# A' 2# %#"#0 '+5." !@ $-7F%0-&,= &,7, #$ #+ &'$, 2#+#>*-+-?0-,K

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+: !" #+ (0,?+#7' 2# +,$ #4-2,$ CH('0%'2, O:fD #+ 4*#5, #$ $-7F%0-&,: !" F+= &'2'#$('&-, 2# #$%0'%#5-'$ #$ & '% aQ= äD &,"B#<,= (#0, ", &,7('&%,: @, +# #$ '(+-&'?+#=(,0 %'"%,= #+ R#,0#7' 2# #<-$%#"&-' M:M: ;-" #7?'05,= $) %-#"# !@= >*# '2#76$ #$ $-7F8%0-&,:

Ejemplo 3.13

V# +,$ ,%0,$ 4*#5,$ 2#$&0-%,$ #" +' ;#&&-9" O:J 2#+ &'()%*+, '"%#0-,0= $," $-7F%0-&,$ +,$$-5*-#"%#$K

&: !+ 4*#5, 1-"-%, 2-+#7' 2#+ (0-$-,"#0, C!4#7(+, O:JD:;* ."-&, !@ #$ $-7F%0-&,:;: !+ 4*#5, 1-"-%, G'+&9"8E'+,7' C!4#7(+, O:LD:;- ! aA / O= #$ 2#+ %-(, 2-+#7' 2#+ (0-$-,"#0,:;- ! %A / O= %-#"# *" !@ $-7F%0-&, #" #$%0'%#5-'$ (*0'$= >*# #$ CG'+&9"= G'+&9"D:;- ! bA / O= ", %-#"# "-"5." !@ $-7F%0-&, #" #$%0'%#5-'$ (*0'$= (#0, $) %-#"# *" !@

$-7F%0-&, #" #$%0'%#5-'$ 7-<%'$:

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@, $," $-7F%0-&,$ +,$ $-5*-#"%#$K

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&,7-%F= %-#"# B,%, 2# &'+-2'2 / (,0 %'"%, 2-$%-"%, %0'%,D:

3.2. JUEGOS BIPERSONALES DE SUMA CERO

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Definición 3.5

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$JC(J= (OD!$OC(J= (OD% Q= O(J à & J / O(O à & O

!" +,$ 4*#5,$ ?-(#0$,"'+#$ 2# $*7' &#0,= >*# '2#76$ $#'" 1-"-%,$= (*#$%, >*# +,$('5,$ 2#+ $#5*"2, 4*5'2,0 $," +,$ ,(*#$%,$ 2# +,$ ('5,$ 2#+ (0-7#0,= +' 7'%0-3 2# ('5,$2#+ (0-7#0, ?'$%' ('0' 2#1-"-0 #+ 4*#5, / ('0' 0#'+-3'0 $* '"6+-$-$: E,0 #$%' 0'39"= ' +,$ 4*#5,$ ?-(#0$,"'+#$ 1-"-%,$ 2# $*7' &#0, $# +#$ ++'7' ' B#&#$ 4*#5,$ 7'%0-&-'+#$: W,7#"83'0#7,$ (,0 %'"%, #$%' $#&&-9" #$%*2-'"2, '+5*",$ &,"&#(%,$ 2#1-"-?+#$ #" *"' 7'%0-3:

Valores maximín y minimax de una matriz

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J= (OJ= :::= (-J P / & O% N(J

O= (OO= :::= (5OP: E,0 %'"%,= +'$ 1-+'$ 2# 6J &,00#$(,"2#" ' +'$

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R#"-#"2, #" &*#"%' >*#

7'<(i1à &

1

$JC('J= ( *OD / 7-"(j2à &

2

$JC('J= ( *OD

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Definición 3.6

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7%7'<(i1à &

1

N7-"(j2à &

2

$JC('J= ( *ODP

;: I+'7'7,$ <&,/% )(1()&* 2# 6J= / +, 2#",%'7,$ 7= '+ 7)"-7, 2# +,$ 76<-8

7,$ 2# %,2'$ +'$ &,+*7"'$ 2# 6J: !$ 2#&-0=

7%7-"(j2à &

2

N7'<(i1à &

1

$JC('J= ( *ODP

=: V#&-7,$ >*# #+ %F07-", $JC('0J = ( *0O D= >*# ,&*(' +' 1-+' 'Q / +' &,+*7"' *Q 2# 6J=#$ *" 0.1$/ +! "(,,& 2# 6J $- #$ #+ 76<-7, 2# $* &,+*7"' / #+ 7)"-7, 2# $* 1-+': !$2#&-0=

$JC('0J = ( *0O D%7'<(i1à &

1

$JC('J= ( *0O D%7-"(j2à &

2

$JC('0J = ( *OD

I,$ #4#7(+,$ $-5*-#"%#$ -+*$%0'" #$%,$ &,"&#(%,$:

Ejemplo 3.14

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6J%A Q .J

7 J7

J7 Q .J7

.J7

J7 Q B$# A'" $#l'+'2, 7#2-'"%# *" '$%#0-$&, $*?)"2-&# +,$ 7)"-7,$ 2# &'2' 1-+'= / 7#2-'"%#*" '$%#0-$&, $*(#0)"2-&# +,$ 76<-7,$ 2# &'2' &,+*7"':

s R,2,$ +,$ 7)"-7,$ 2# 1-+' B'+#" .J= +*#5, #+ B'+,0 7'<-7)" 2# 6J #$

7% 7'<(i1à N(1

1= (21

= (31

PN 7-"(j2à N(1

2= (22

= (32

P$JC('J= ( *ODP% 7'<

(i1à N(1

1= (21

= (31

PN.J= .J= .JP%.J

s R,2,$ +,$ 76<-7,$ 2# &,+*7"' B'+#" J= / (,0 %'"%, #+ B'+,0 7-"-7'< 2# 6J #$

7% 7-"(j2à N(1

2= (22

= (32

PN 7'<(i1à N(1

1= (21

= (31

P$JC('J= ( *ODP% 7-"

(j2à N(1

2= (22

= (32

PNJ= J= JP%J

s @-"5." %F07-", #$%6 2,?+#7#"%# $#l'+'2,= / (,0 %'"%, 6J ", %-#"# (*"%, 2# $-++':

Ejemplo 3.15

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6J%

AM7 J

7J7

O7

O7

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s I,$ 7)"-7,$ 2# 1-+' $," J= J / O= 7-#"%0'$ >*# +,$ 76<-7,$ 2# &,+*7"' $," M /O: E,0 %'"%,= #+ B'+,0 7'<-7)" 2# 6J #$

7% 7'<(i1à N(1

1= (21

= (31

PN 7-"(j2à N(1

2= (22

P$

JC('

J= ( *

ODP% 7'<

(i1à N(1

1= (21

= (31

PNJ= J= OP%O

/ #+ B'+,0 7-"-7'< 2# 6J #$

7% 7-"(j2àN(1

2= (22

PN 7'<(i1à N(1

1= (21

= (31

P$JC('J= ( *ODP% 7-"

(j2à N(1

2= (22

PNM= OP% O

s !+ ."-&, %F07-", >*# #$%6 2,?+#7#"%# $#l'+'2,= >*# ,&*(' +' %#0&#0' 1-+' / $#85*"2' &,+*7"'= / &*/, B'+,0 #$ O= #$ #+ (*"%, 2# $-++' 2# 6J: S?$F0B#$# >*# 2-&A, B'+,0&,-"&-2# &," #+ B'+,0 7'<-7)" / &," #+ B'+,0 7-"-7'< 2# 6J: @, $# %0'%' 2# *"' &'$*'8

+-2'2= $-", 2# *"' &,"2-&-9" "#&#$'0-' / $*1-&-#"%# ('0' +' #<-$%#"&-' 2# (*"%, 2# $-++'2# *"' 7'%0-3:

Estrategias puras maximín y minimax de un juego. Niveles de seguridad

;*(,"5'7,$ >*# 6J $#' #1#&%-B'7#"%# +' 7'%0-3 2# ('5,$ C$JC('J= ( *ODD'%J= :::= -g *%J= :::= 5 ('0'#+ 4*5'2,0 J= 2#+ 4*#5, 4%N& J= & Og $J= $OP= 2,"2# & J%N(J

J= (OJ= :::= (-

J P / & O%N(JO= (O

O= :::= (5OP:I'$ 1-+'$ 2# 6J &,00#$(,"2#"= (,0 %'"%,= ' +'$ #$%0'%#5-'$ (*0'$ 2# YJ 7-#"%0'$ >*# $*$

&,+*7"'$ &,00#$(,"2#" ' +'$ #$%0'%#5-'$ (*0'$ 2# YO: ;#' 7%

$JC(

(

J= (

1

OD #+ B'+,0 7'<-7)"2# 6J: H +' #$%0'%#5-' (*0' ((J 2# YJ #" &*/' 1-+' $# '+&'"3' 2-&A, B'+,0 $# +' ++'7' #$%0'%#8

5-' (*0' 7'<-7)": !+ B'+,0 7'<-7)" 2# 6J (*#2# -"%#0(0#%'0$# &,7, #+ "-B#+ 2# $#5*0-82'2 2#+ 4*5'2,0 J= #$ 2#&-0= #+ B'+,0 >*# (*#2# #$%'0 $#5*0, YJ 2# &,"$#5*-0 '+ 4*5'0 *"'#$%0'%#5-' (*0' #" 2-&A, 4*#5,= (,0 76$ >*# YO (*2-#0' #7(#l'0$# #" -7(#2-0+,: !$ 2#&-0= 4*5'"2, $* #$%0'%#5-' $# '$#5*0' +' &,"$#&*&-9" 2# *" ('5, '+ 7#",$ %'" 50'"2# &,7, 7:

E*#$%, >*# #+ 4*#5, 4 2#%#07-"' 2,$ 7'%0-&#$ 2# ('5,$=

6J% C$JC('J= ( *ODD'%J= :::= -g *%J= :::= 5

('0' #+ 4*5'2,0 J /

6O% C$OC('J= ( *ODD'%J= :::= -g *%J= :::= 5

('0' #+ 4*5'2,0 O= (,20)'7,$ -5*'+7#"%# 2#1-"-0= ' ('0%-0 2# 6O= +'$ #$%0'%#5-'$ (*0'$ 7'8<-7)" / #+ "-B#+ 2# $#5*0-2'2 &,00#$(,"2-#"%# ('0' #+ 4*5'2,0 O: I' ."-&' 2-1#0#"&-' #$>*# 'A,0' +'$ #$%0'%#5-'$ (*0'$ 2# YO -2#"%-1-&'" &,+*7"'$ 2# 6O / ", 1-+'$: H +,$ B'+,0#$7'<-7)" 2# YJ / YO +,$ 2#",7-"'0#7,$= 0#$(#&%-B'7#"%#= vJ / vO:

Ejemplo 3.16W,"$-2#0#7,$ 2# "*#B, #+ Y*#5, M:J= 2,"2# '('0#&#" +'$ 7'%0-&#$ 2# ('5,$ 6J ('0' #+ 4*5'2,0 J / 6O ('0' #+ 4*5'2,0 O:

5.'&+/% 9

U V

5.'&+/% 8

H M= O J= L

W J= M O= J

X O= O O= Q

!+ 4*5'2,0 J %-#"# *"' ."-&' #$%0'%#5-' (*0' 7'<-7)"= X= >*# +# (0,(,0&-,"' *""-B#+ 2# $#5*0-2'2 2# O= /' >*#

vJ% $JCX= UD% 7'<(i1à N(1

1= (21

= (31

PN 7-"(j2à N(1

2= (22

P$JC('J= ( *ODP% 7'<

(i1à NH= W=XP

NJ= J= OP% O

E,0 $* ('0%#= #+ 4*5'2,0 O %-#"# %'7?-F" *"' ."-&' #$%0'%#5-' (*0' 7'<-7)"= U= >*# +#(0,(,0&-,"' *" "-B#+ 2# $#5*0-2'2 2# O= /' >*#

vO%$OCX= UD% 7'<(j2àNU= VP

N 7-"(i1à NH= W=XP

$OC('J= ( *ODP% 7'<(j2à NU= VP

NO= QP%O

Ejemplo 3.17

a) !" #+ 4*#5, +' ?'%'++' 2# +,$ $#<,$

5.'&+/%& 9

W-"# T.%?,+

5.'&+/% 8W-"# J= O Q= Q

T.%?,+ Q= Q O= J

+'$ 2,$ #$%0'%#5-'$ (*0'$ 2#+ 4*5'2,0 J $," 7'<-7)"= / '7?'$ +# (0,(,0&-,"'" *" "-B#+

2# $#5*0-2'2 2# Q: I, 7-$7, ,&*00# &," +' 4*5'2,0' O:

;: !" #+ 4*#5, 2# +'$ 7,"#2'$

5.'&+/% 9

W'0' W0*3

5.'&+/% 8W'0' J= .J .J= J

W0*3 .J= J J= .J

+'$ 2,$ #$%0'%#5-'$ (*0'$ 2#+ 4*5'2,0 J $," 7'<-7)"= '7?'$ +# (0,(,0&-,"'" *" "-B#+ 2#$#5*0-2'2 2# .J= / +, 7-$7, ,&*00# &," #+ 4*5'2,0 O:

H*">*# +,$ &,"&#(%,$ 2# #$%0'%#5-'$ (*0'$ 7'<-7)" / B'+,0#$ 2# $#5*0-2'2 $," B6+-82,$ ('0' &*'+>*-#0 4*#5,= $*$ (0,(-#2'2#$ $," -"%#0#$'"%#$ $9+, #" #+ &'$, 2# +,$ 4*#5,$2# $*7' &#0,: E,0 ,%0' ('0%#= #$%'$ (0,(-#2'2#$ $," #$(#&-'+7#"%# -"%#0#$'"%#$ &*'"2,-"%0,2*&-7,$ #" #+ '"6+-$-$ +'$ #$%0'%#5-'$ 7-<%'$: E,0 %'"%,= #" #+ 0#$%, 2# #$%' $#&&-9"",$ $-%*'0#7,$ #" #+ &,"%#<%, 2# 4*#5,$ 1-"-%,$ 2# $*7' &#0, &," #$%0'%#5-'$ 7-<%'$:

Estrategias mixtas maximín y valor de un juego de suma cero

H2'(%#7,$ +'$ 2#1-"-&-,"#$ '"%#0-,0#$ '+ &,"%#<%, 2# +,$ 4*#5,$ 1-"-%,$ 2# $*7' &#0, &,"#$%0'%#5-'$ 7-<%'$:

Definición 3.7

;#' #+ 4*#5, ?-(#0$,"'+ 1-"-%, 2# $*7' &#0, 4% N& J= & Og $J= $OP= 2,"2#

& J% N(JJ= (O

J= :::= (-J P / & O% N(JO= (O

O= :::= (5OP

&*/'$ 7'%0-&#$ 2# ('5,$ $,"

6J% C$JC('J= ( *ODD'%J= :::= -g *%J= :::= 5 / 6O% C$OC('J= ( *OD'%J= :::= -g *%J= :::= 5%. 6J

;#' pJ *"' #$%0'%#5-' 7-<%' 5#"F0-&' 2# YJ= 0#(0#$#"%'?+# 7#2-'"%# *" B#&%,0 1-+'&," - &,7(,"#"%#$ (,$-%-B'$ , "*+'$ >*# $*7'" +' *"-2'2: H"6+,5'7#"%#= $#' pO *"'#$%0'%#5-' 7-<%' 5#"F0-&' 2# YO= 0#(0#$#"%'?+# 7#2-'"%# *" B#&%,0 1-+' &," 5 &,7(,8"#"%#$ (,$-%-B'$ , "*+'$ >*# $*7'" +' *"-2'2:

&: I+'7'7,$ <&,/% )&*()V1 2#+ 4*#5, 4= / +, 2#",%'7,$ vJ= '+ ('5, 76<-7,>*# (*#2# '$#5*0'0$# #+ 4*5'2,0 J $*(,"-#"2, >*# YO 0#$(,"2# ' +' #$%0'%#5-' pJ 2# YJ&," *"' #$%0'%#5-' pO >*# +# (#07-%# 7'<-7-3'0 $* ('5,= / (,0 %'"%, 7-"-7-3'0 #+ 2#YJ: !$ 2#&-0=

vJ% 7'<p1àBC&

1D N 7-"p

2àBC&

2D pJ 6Jp

1 OP

;: I+'7'7,$ <&,/% )(1()&* 2#+ 4*#5, 4 '+ ".7#0,

vO% 7-"p2àBC&

2D

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1 OP

;- #+ B'+,0 7'<-7)" $# '+&'"3' #" +'$ #$%0'%#5-'$ 7-<%'$ &,"&0#%'$ phJ 2# YJ / phO2# YO= 2#&-7,$ >*# phJ #$ *"' !"$%&$!'(& )&*()V1 2# YJ: H"6+,5'7#"%#= $- #+ B'+,07-"-7'< $# '+&'"3' #" +'$ #$%0'%#5-'$ 7-<%'$ &,"&0#%'$ pñhO 2# YO / pñhJ 2# YJ= 2#&-7,$>*# pñhO #$ *"' !"$%&$!'(& )(1()&* 2# YO:

=: I+'7'7,$ <&,/% +!, -.!'/ 4= / +, 2#",%'7,$ v= '+ B'+,0 7'<-7)" / '+ B'+,07-"-7'<= $- #$ >*# '7?,$ &,-"&-2#":

!+ B'+,0 7'<-7)" vJ #$ #+ B'+,0 2# $#5*0-2'2 2#+ 4*5'2,0 YJ= #$ 2#&-0= #+ ('5, 76$'+%, >*# YJ (*#2# '$#5*0'0$# $*(,"-#"2, *" i&,7(,0%'7-#"%, 2# (#,0 &'$,j (,0 ('0%# 2#YO C>*# #$ +' $*(,$-&-9" '2#&*'2'= /' >*# #" +,$ 4*#5,$ 2# $*7' &#0,= YO (0,2*&# #+ ('5,

76$ ?'4, (,$-?+# ('0' YJ &," '>*#++'$ 4*5'2'$ >*# $," 9(%-7'$ ('0' F+D: H2#76$= #+ +#7'$-5*-#"%# ",$ (#07-%-06 0#+'&-,"'0 #+ B'+,0 7-"-7'< &," #+ B'+,0 2# $#5*0-2'2 2#+ 4*5'82,0 O:

Lema 3.1

V'2, #+ 4*#5, ?-(#0$,"'+ 1-"-%, 2# $*7' &#0, 4% N& J= & Og $J= $OP= 2,"2#

& J% N(JJ= (O

J= :::= (-J P / & O% N(JO= (O

O= :::= (5OP

/ &*/'$ 7'%0-&#$ 2# ('5,$ $,"

6J% C$JC('J= ( *ODD'%J= :::= -g *%J= :::= 5 / 6O% C$OC('J= ( *OD'%J= :::= -g *%J= :::= 5%. 6J

#+ B'+,0 7-"-7'< vO #$ #+ B'+,0 2# $#5*0-2'2 2#+ 4*5'2,0 O &'7?-'2, 2# $-5",: !$2#&-0=

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D!)/"$%&=(>1?

!$ ?-#" $'?-2, >*#= 2'2' *"' 1*"&-9" 0#'+ &*'+>*-#0' E C %D= #+ ,(*#$%, 2#+ 76<-7, 2# E C %D #$ -5*'+ '+ 7)"-7, 2#+ ,(*#$%, 2# E C %D= #$ 2#&-0= .7'< E C %D%7-" C. E C %DD= /'"6+,5'7#"%#= .7-" E C %D%7'< C. E C %DD:

H2#76$ $'?#7,$ >*# +' $,+*&-9" 2# (0,?+#7' 2# 7'<-7-3'0 *"' 1*"&-9" $# '+&'"83' #" +,$ 7-$7,$ (*"%,$ F7 2,"2# $# '+&'"3' +' $,+*&-9" 2#+ &,00#$(,"2-#"%# (0,?+#87' 2# 7-"-7-3'0 $* ,(*#$%': X'$6"2,",$ #" #++,= %#"#7,$K

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Ejemplo 3.19

!" #+ 4*#5, 2# +'$ 7,"#2'$

5.'&+/% 9

W'0' W0*3

5.'&+/% 8W'0' J= .J .J= J

W0*3 .J= J J= .J

%'"%, +' #$%0'%#5-' (*0' W'0' &,7, +' #$%0'%#5-' (*0' W0*3 2#+ 4*5'2,0 J $," 7'<-7)"#" #+ &,"%#<%, 2# #$%0'%#5-'$ (*0'$= / '7?'$ +# (0,(,0&-,"'" #" 2-&A, &,"%#<%, *" "-B#+2# $#5*0-2'2 2# .J: ;-" #7?'05,= #" #+ &,"%#<%, 2# #$%0'%#5-'$ 7-<%'$= #+ B'+,0 7'<-87)" #$K

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Qm am JN 7-"

Qm bm JCa J. aDA

J .J.J JB A

bJ. bBP%

% 7'<Qm am J

N 7-"Qmbm J

J. Oa! OCOa. JDbP

!+ 7)"-7, 2# J. Oa! OCOa. JDb= ('0' a 1-4, / b #"%0# Q / J= #$ E CaD%J. Oa $-ab J/ O C7)"-7, >*# $# '+&'"3' A'&-#"2, b%QD= #$ E CaD%Q $- a%J/ O C7)"-7, >*#$# '+&'"3' ('0' &*'+>*-#0 bD /= (,0 .+%-7,= #$ E CaD%Oa. J $- aa J/ O C7)"-7, >*# $#'+&'"3' A'&-#"2, b% JD: E,0 ,%0' ('0%#= / %#"-#"2, #" &*#"%' +, '"%#0-,0= #+ 76<-7, 2# E CaD &*'"2, a B'0)' #"%0# Q / J #$ -5*'+ ' Q= / 2-&A, 76<-7, $# '+&'"3' A'&-#"2,

a%J/ O: E,0 %'"%,= #+ B'+,0 7'<-7)" vJ #$ -5*'+ ' Q= / $# '+&'"3' A'&-#"2, a%J/ O / b-5*'+ ' &*'+>*-#0 B'+,0= #$ 2#&-0= vJ%Q #$ #+ B'+,0 2# $#5*0-2'2 2# YJ= / F$%# +, '+&'"3' 4*5'"2, $* #$%0'%#5-' 7-<%' pJ% CJ/ O= J/ OD= -"2#(#"2-#"%#7#"%# 2# >*F #$%0'%#5-' (*0', 7-<%' 4*#5*# YO:

H"6+,5'7#"%#= #+ B'+,0 7-"-7'< #$K

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b!OCO

b. JD

a= ('0'

b 1-4, /

a #"%0# Q / J= #$ E C

bD% O

b. J $-

bb J/ O C76<-8

7, >*# $# '+&'"3' A'&-#"2, a% JD= #$ E CbD%Q $- b% J/ O C76<-7, >*# $# '+&'"3'('0' &*'+>*-#0 aD /= (,0 .+%-7,= #$ E CbD% J. Ob $- ba J/ O C76<-7, >*# $# '+&'"3'A'&-#"2, a%QD: H$)= #+ 7)"-7, 2# E CbD &*'"2, b B'0)' #"%0# Q / J #$ -5*'+ ' Q= >*# $#'+&'"3' A'&-#"2, b% J/ O: E,0 %'"%,= #+ B'+,0 7-"-7'< vO #$ -5*'+ ' Q C/ (,0 %'"%, #+B'+,0 2# $#5*0-2'2 2# YO #$ Q%.vOD= / $# '+&'"3' A'&-#"2, b% J/ O / a -5*'+ ' &*'+8>*-#0 B'+,0: !" 2#1-"-%-B'= #+ B'+,0 2# $#5*0-2'2 2# YO #$ .vO%Q / F$%# +, '+&'"3' 4*5'"2, $* #$%0'%#5-' 7-<%' pO% CJ/ O= J/ OD= -"2#(#"2-#"%#7#"%# 2# >*F #$%0'%#5-' (*0', 7-<%' 4*#5*# YJ:

!+ 0#$*+%'2, $-5*-#"%#= #+ 76$ -7(,0%'"%# ('0' #$%# %-(, 2# 4*#5,$= 1*# ,?%#"-2, (,0^," @#*7'"" #" JrOe / 0#&-?# #+ ",7?0# 2# %#,0#7' 2#+ 7-"-7'<:

Teorema 3.6

V'2, *" 4*#5, ?-(#0$,"'+ 1-"-%, 2# $*7' &#0, &*'+>*-#0' 4% N& J= & Og $J= $OP= 2-&A, 4*#5, %-#"# *" B'+,0: !$ 2#&-0= #<-$%# *" v à ; %'+ >*# vJ% vO% v= $-#"2, vJ / vO +,$B'+,0#$ 7'<-7)" / 7-"-7'<:

D!)/"$%&=(>1?

X'$'0#7,$ "*#$%0' 2#7,$%0'&-9" #" +' #<-$%#"&-' 2# #>*-+-?0-,$ 2# @'$A: E,0 $#0 4 *" 4*#5, 1-"-%,= #<-$%# '+5." !@ #" 4: ;#' CphJ = phO D *", 2# %'+#$ #>*-+-?0-,$: E,0 2#1-"-8&-9" 2# !@= $# &*7(+#" +'$ '1-07'&-,"#$ $-5*-#"%#$K

3 JCp7J = p7O D%p7J 6Jp71 O % 7'<

p1àBC&

1D

NpJ 6Jp71 O P aM:Jb

3 OCp7J = p7O D%p7J 6Op71 O % 7'<

p2àBC&

2D

Np7J 6Op1 OP aM:Ob

V# aM:Ob $# 2#2*&#

3 JCp7J = p7O D%p7J 6Jp71 O % 7-"

p2àBC&

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Np7J 6Jp1 OP aM:Mb

C2#?-2, ' >*# p7J 6Jp71 O %.p7J 6Op71

O %. 7'<p2àBC&

2D

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p2àBC&

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Np7J 6Jp1 OPD:

HA,0'= 2# aM:Jb $# 2#2*&#

3 JCp7J = p7O D%p7J 6Jp71 O % 7'<

p1àBC&

1D

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p1àBC&

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N 7-"p2àBC&

2DpJ 6Jp

1 OP%

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2Dp7J 6Jp1 O%3 JCp7J = p7O D aM:Lb

/ 2# aM:Mb $# 2#2*&#

3 JCp7J = p7O D%p7J 6Jp71 O % 7-"

p2àBC&

2D

Np7J 6Jp1 OPm 7-"

p2àBC&

2D

N 7'<p1àBC&

1DpJ 6Jp

1 OP%

% vOm 7'<p1àBC&

1DpJ 6Jp71

O %3 JCp7J = p7O D aM:cb

E,0 .+%-7,= 2# aM:Lb / aM:cb $# 2#2*&#

3 JCp7J = p7O D% vJ% vO

!" +,$ !4#7(+,$ M:Je / M:Jr A#7,$ &,7(0,?'2, >*# #+ Y*#5, M:M / #+ 4*#5, 2# +'$7,"#2'$ $'%-$1'&#" #+ R#,0#7' M:d: !+ B'+,0 2# '7?,$ 4*#5,$ #$ Q:

Relación entre los equilibrios de Nash y las estrategias maximín

!" +,$ 4*#5,$ ?-(#0$,"'+#$ 1-"-%,$ 2# $*7' &#0,= #<-$%# *"' #$%0#&A)$-7' 0#+'&-9" #"%0#+,$ #>*-+-?0-,$ 2# @'$A / +'$ #$%0'%#5-'$ 7'<-7)"= $#5." +' &*'+ 1,07'" ('0%# 2# +,$

#>*-+-?0-,$ +'$ #$%0'%#5-'$ 7'<-7)" / $9+, #++'$: !+ R#,0#7' M:f (0#&-$' / &,7(+#%' 2-&A'0#+'&-9":

Teorema 3.7

;#' #+ 4*#5, ?-(#0$,"'+ 1-"-%, 2# $*7' &#0, 4% N& J= & Og $J= $OP= 2,"2#

& J% N(JJ= (O

J= :::= (-J P / & O% N(JO= (O

O= :::= (5OP

/ &*/'$ 7'%0-&#$ 2# ('5,$ $,"

6J% C$JC('J= ( *ODD'%J= :::= -g *%J= :::= 5 / 6O% C$OC('J= ( *ODD'%J= :::= -g *%J= :::= 5%. 6J

;#' pJ *"' #$%0'%#5-' 7-<%' 5#"F0-&' 2# YJ / pO *"' #$%0'%#5-' 7-<%' 5#"F0-&' 2# YO:

&: ;- CphJ = phO D #$ *" #>*-+-?0-, 2# @'$A= phJ / phO $," #$%0'%#5-'$ 7'<-7)" 2# YJ /YO= / #+ ('5, >*# YJ 0#&-?# #" 2-&A, #>*-+-?0-, &,-"&-2# &," #+ B'+,0 2#+ 4*#5,:

;: ;- phJ / phO $," #$%0'%#5-'$ 7'<-7)" 2# YJ / YO= #+ (#01-+ CphJ = phO D #$ *" #>*-+-8?0-, 2# @'$A:

D!)/"$%&=(>1?

&: !" +' 2#7,$%0'&-9" 2#+ R#,0#7' M:d= A#7,$ B-$%, >*# $- CphJ = phO D #$ *" !@ $#&*7(+#K

3 JCp7J = p7O D%p7J 6Jp71 O % 7'<

p1àBC&

1D

N 7-"p2àBC&

2DpJ 6Jp

1 OP% vJ aM:db

3 JCp7J = p7O D%p7J 6Jp71 O % 7-"

p2àBC&

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1 OP% vO aM:fb

I' -5*'+2'2 aM:db $-5"-1-&' >*# phJ #$ *"' #$%0'%#5-' 7'<-7)" 2# YJ= / 2# aM:fb $#

2#2*&# +, $-5*-#"%# '+ '(+-&'0 #+ I#7' M:JK

3 OCp7J = p7O D%.3 JCp7J = p7O D%.vO%. 7-"p2àBC&

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;: ;- phJ / phO $," #$%0'%#5-'$ 7'<-7)" 2# YJ / YO= $# &*7(+#

3 JCp7J = p7O D%p7J 6Jp71

O % 7'<p1àBC&

1D N 7-"p

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6Op

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p2àBC&

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1DpJ 6Jp

1 OP%

% 7'<p1 àBC& 1D

NpJ 6Jp71 O PnpJ 6Jp71

O %3 JCpJ= p71 O D= OpJ à BC& JD

3 OCp7J = p7O D% p7J 6Op71 O % 7-"

p1àBC&

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1 O%3 OCp7J = pOD= OpO à BC& OD

/ #$%'$ 2,$ .+%-7'$ 2#$-5*'+2'2#$ ",$ 2-&#" >*# CphJ = phO D #$ *" !@:

Juegos bipersonales de suma constanteS?%#"#7,$ *"' 5#"#0'+-3'&-9" 2# +,$ 4*#5,$ 2# $*7' &#0, $-= #" +*5'0 2# #<-5-0 >*# +'$*7' 2# +'$ 5'"'"&-'$ 2# +,$ 2,$ 4*5'2,0#$ '"%# &'2' (#01-+ 2# #$%0'%#5-'$ $#' "*+'= #<-5-87,$ >*# $#' *"' &,"$%'"%#:

Definición 3.8

;#' 4% N& J= & Og $J= $OP *" 4*#5, ?-(#0$,"'+: V#&-7,$ >*# 4 #$ *" 4*#5, +! ".)&=/1"$&1$! $- ('0' %,2, ('0 2# #$%0'%#5-'$ (*0'$ 2# '7?,$ 4*5'2,0#$= +' $*7' 2# +,$('5,$ >*# ' '7?,$ &,00#$(,"2#" #$ $-#7(0# +' 7-$7': !$ 2#&-0=

$JC(J= (OD!$OC(J= (OD% += O(J à & J / O(O à & O

;- ?-#" +,$ 4*#5,$ 2# $*7' &,"$%'"%# %-#"#" &,7, &'$, ('0%-&*+'0 +,$ 4*#5,$ 2# $*7'&#0,= *"' $-7(+# %0'"$1,07'&-9" 2# +'$ 5'"'"&-'$ 2# +,$ 4*5'2,0#$ ",$ (#07-%# &,"$-2#0'0+,$ 4*#5,$ 2# $*7' &,"$%'"%# &,7, $- $# %0'%'$# 2# 4*#5,$ 2# $*7' &#0,: ^#'7,$ '+5*",$#4#7(+,$ >*# &,"B-#0%#" &*'+>*-#0 4*#5, 2# $*7' &,"$%'"%# #" *" 4*#5, 2# $*7' &#0,

R0'"$1,07'&-9" 2# +'$ 5'"'"&-'$ 2# YJK

$ñJC(J= (OD%

$JC(J= (OD. += O(J à & J / O(O à & O

R0'"$1,07'&-9" 2# +'$ 5'"'"&-'$ 2# YOK

$ñOC(J= (OD%$OC(J= (OD. += O(J à & J / O(O à & O

R0'"$1,07'&-9" 2# +'$ 5'"'"&-'$ 2# YJ / YOK

$ñJC(J= (OD%$JC(J= (OD. +J $ñOC(J= (OD%$OC(J= (OD. +O

O+J= +O %'+ >*# +J! +O% += O(J à & J / O(O à & O

;-#7(0# >*# *" 4*#5, '27-%' #$%# %-(, 2# %0'"$1,07'&-,"#$= (,20#7,$ '"'+-3'0 *" 4*#5,2# $*7' &,"$%'"%# ' %0'BF$ 2# '+5*", 2# +,$ 4*#5,$ 2# $*7' &#0, '$,&-'2,$: !$%, ",$

&,"2*&# ' +' $-5*-#"%# (0#5*"%'K \&*6"2, (,2#7,$ '27-%-0 #$%# %-(, 2# %0'"$1,07'&-,"#$/ (,0 %'"%, A'&#0 *$, 2#+ %#,0#7' 2#+ 7-"-7'<] I' 0#$(*#$%' ",$ ++#B' 2# "*#B, ' +'-"%#0(0#%'&-9" 2# +'$ 5'"'"&-'$ &,7, *%-+-2'2#$ 2# ^," @#*7'""8_,05#"$%#0": V-&A'-"%#0(0#%'&-9" #$ &,"$-$%#"%# &," *"'$ (0#1#0#"&-'$ >*# '27-%#" *"' #$&'+' &'02-"'+8-"%#0B'+,= / +' #<-5-7,$ $-#7(0# >*# >*#0#7,$ -"&+*-0 ' +'$ #$%0'%#5-'$ 7-<%'$ #" #+ '"68+-$-$:

H$) (*#$= +,$ 0#$*+%'2,$ '+&'"3'2,$ ('0' 4*#5,$ 2# $*7' &#0, $," '(+-&'?+#$ '+ '"68+-$-$ 2# &*'+>*-#0 4*#5, 2# $*7' &,"$%'"%# &*/'$ 5'"'"&-'$ '27-%'" %0'"$1,07'&-,"#$'1-"#$ (,$-%-B'$:

W,7, &,"&+*$-9" 2# #$%' $#&&-9"= 7#0#&# +' (#"' ,?$#0B'0 >*# #" #$%# %-(, 2# 4*#5,$#<-$%# *"' (0#$&0-(&-9" &+'0' $,?0# &97, 4*5'0 C4*5'0 '7?,$ 4*5'2,0#$ &*'+>*-#0 #$%0'%#85-' 7'<-7)"= , +, >*# #$ +, 7-$7,= 4*5'0 &*'+>*-#0 #>*-+-?0-, 2# @'$AD /' >*#= #" B-0%*22#+ %#,0#7' '"%#0-,0= +,$ ('5,$ ' &'2' 4*5'2,0 ", 2#(#"2#" 2# +' #$%0'%#5-' 7'<-7)"#+#5-2': !" #1#&%,= $- 4*#5'" 2# #$%# 7,2,= #+ 4*5'2,0 J 0#&-?-06 #+ ('5, v 2#+ 4*#5,=7-#"%0'$ >*# #+ 4*5'2,0 O 0#&-?-06 #+ ('5, .v:

3.3. ESTRATEGIAS RACIONALIZABLES

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

$,+*&-,"#$ 2# 2,7-"'&-9" C*$'"2, /' #+ &,"&#(%, 2# 2,7-"'&-9" 0#B-$'2,D / +'$ $,+*&-,8"#$ 2# #>*-+-?0-, C*$'"2, #+ #>*-+-?0-, 2# @'$A #" #$%0'%#5-'$ 7-<%'$D: !" %#0&#0 +*5'0=2#1-"-0#7,$ #+ &,"&#(%, 2# #$%0'%#5-' >*# "*"&' #$ 0#$(*#$%' 9(%-7'= &,"&#(%, >*# #$+-5#0'7#"%# 76$ 5#"#0'+ >*# #+ 2# #$%0'%#5-' #$%0-&%'7#"%# 2,7-"'2'= / ' ('0%-0 2# F+&,"$%0*-0#7,$ *" "*#B, &,"&#(%, 2# $,+*&-9"= #+ 2# +'$ #$%0'%#5-'$ 0'&-,"'+-3'?+#$= / +,&,7('0'0#7,$ &," +,$ '"%#0-,0#$ &,"&#(%,$ 2# $,+*&-9":

Revisión de estrategias dominadas

V'2, #+ 4*#5, $-5*-#"%#K

5.!'/ 67J

5.'&+/% 9

H X W

5.'&+/% 8

t O= d d= Q J= J

[ J= M O= f L= L

u O= M c= J M= J

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3 OCt= pOD%Mb3 OCt= WD%J= 3 OC[= pOD%cb3 OC[= WD%L

/

3 OCu= pOD%Ob3 OCu= WD%J

!" 2#1-"-%-B'= W #$%6 #$%0-&%'7#"%# 2,7-"'2' (,0 +' #$%0'%#5-' 7-<%' pO= ' (#$'0 2#>*# ", +, #$%6 (,0 "-"5*"' 2# $*$ #$%0'%#5-'$ (*0'$ $,(,0%#= H , X: E,0 ,%0' ('0%#= #$16&-+ ,?$#0B'0 >*# +' #$%0'%#5-' 7-<%' pJ% CO/ M= J/ M= QD= #" +' >*# #+ 4*5'2,0 J 4*#5' t&," (0,?'?-+-2'2 O/ M # [ &," (0,?'?-+-2'2 J/ M= +# (0,2*&# *",$ ('5,$ #$%0-&%'7#"%#-"1#0-,0#$ ' +,$ 2# u= A'5' +, >*# A'5' #+ 4*5'2,0 O: !1#&%-B'7#"%#=

3 JCpJ= HD%c/ Ma3 JCu= HD% O= 3 JCpJ= XD%JL/ Ma3 JCu= XD%c

/

3 JCpJ= WD%Oa3 JCu= WD%M

!" 2#1-"-%-B'= +' #$%0'%#5-' 7-<%' pJ #$%6 #$%0-&%'7#"%# 2,7-"'2' (,0 u= ' (#$'0 2#>*# ", +, #$%6 "-"5*"' 2# $*$ #$%0'%#5-'$ (*0'$ $,(,0%#= t , [:

I' $-%*'&-9" '"%#0-,0 $*5-#0# +' "#&#$-2'2 2# '7(+-'0 +' 2#1-"-&-9" 2# #$%0'%#5-' 2,87-"'2'= / #$ +, >*# A'&#7,$ ' &,"%-"*'&-9":

Definición 3.9

!" #+ 4*#5, 4% N& J= :::= & 5g $J= :::= $5P= $#'" pñ' / pññ' 2,$ #$%0'%#5-'$ C(*0'$ , 7-<%'$D2#+ 4*5'2,0 ':

&: V#&-7,$ >*# pñ' #$%6 +/)(1&+&E , %'7?-F" +[;(,)!1$! +/)(1&+&E (,0 pññ'&*'"2, +' 2#$-5*'+2'2

3 'C(J= :::= ('.J= pñ'= ('!J= :::= (5Dm3 'C(J= :::= ('.J= pññ' = ('!J= :::= (5D

$# &*7(+# ('0' %,2' &,7?-"'&-9" 2# #$%0'%#5-'$ (*0'$ (.'= / ('0' '+5*"' $# &*7(+# 2#

7,2, #$%0-&%,:;: V#&-7,$ >*# pñ' #$%6 !"$%(=$&)!1$! +/)(1&+& (,0 pññ' &*'"2, +' 2#$-5*'+2'2

3 'C(J= :::= ('.J= pñ'= ('!J= :::= (5Da $'C(J= :::= ('.J= pññ' = ('!J= :::= (5D

$# &*7(+# ('0' %,2' &,7?-"'&-9" 2# #$%0'%#5-'$ (*0'$ (.' 2# +,$ ,%0,$ 4*5'2,0#$:=: V#&-7,$ >*# pñ' #$ 1/ +/)(1&+& $- ", #<-$%# "-"5*"' ,%0' #$%0'%#5-' 2#+ 4*5'8

2,0 '= "- (*0' "- 7-<%'= >*# +' 2,7-"#= / 2#&-7,$ >*# pñ' #$ 1/ +/)(1&+& !"$%(=$&#)!1$! $- ", #<-$%# "-"5*"' ,%0' >*# +' 2,7-"# #$%0-&%'7#"%#:

!$%# "*#B, &,"&#(%, 2# 2,7-"'&-9" ++#B' '$,&-'2,$ %'7?-F" 2,$ (0,&#$,$ 2# #+-7-8

"'&-9" -%#0'%-B'= >*# (,20)'7,$ ++'7'0 @,()(1&=(>1 \$!%&$(<& +! ,&" @"$%&$!'(&" @"$%(=#$&)!1$! D/)(1&+&" 0/% .1& !"$%&$!'(& 0.%& / )(*$& / @,()(1&=(>1 \$!%&$(<& +! ,&"

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

Ejemplo 3.20

&: !" #+ 2-+#7' 2#+ (0-$-,"#0,= #$ 16&-+ ,?$#0B'0 >*# %'"%, +' !U!8E_ &,7, !UV8E_&,"2*&#" ' CW,"1#$'0= W,"1#$'0D= >*# #$ #+ ."-&, (#01-+ $*(#0B-B-#"%#:

;: !" +' ?'%'++' 2# +,$ $#<,$= &*'+>*-#0 (#01-+ CpJ= pOD (,$-?+# 2#+ 4*#5, #$ *"(#01-+ $*(#0B-B-#"%#= %'"%, 2# +' !U!8E_ &,7, 2# +' !UV8E_:

Relación entre EN y Eliminación Iterativa Estricta (EIE-PM)

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Teorema 3.8

V'2, #+ 4*#5, 4% N& J= :::= & 5g $J= :::= $5P / +' &,7?-"'&-9" 2# #$%0'%#5-'$

(7% C(7J = (7O = :::= (7' = :::= (75 DK

&: ;- 4 #$ 1-"-%, / (7 #$ *" !@ #" #$%0'%#5-'$ (*0'$= +'$ #$%0'%#5-'$ >*# +, &,"$%-8%*/#" $,?0#B-B#" '+ (0,&#$, 2# !+-7-"'&-9" U%#0'%-B' !$%0-&%' C!U!8E_D:

;: ;- 4 #$ 1-"-%, / (7 #$%6 &,"$%-%*-2' (,0 +'$ ."-&'$ #$%0'%#5-'$ >*# $,?0#B-B#" '+(0,&#$, 2# !+-7-"'&-9" U%#0'%-B' !$%0-&%' C!U!8E_D= (7 #$ #+ ."-&, !@ 2#+ 4*#5,:

=: ;- +' #$%0'%#5-' (' 0#$*+%' #+-7-"'2' #" #+ (0,&#$, 2# !+-7-"'&-9" U%#0'%-B' !$8%0-&%' C!U!8E_D= / #" ('0%-&*+'0= $- (' #$ #$%0-&%'7#"%# 2,7-"'2'= ", (#0%#"#&# '+ $,8(,0%# 2# "-"5*"' #$%0'%#5-' 7-<%' >*# 1,07# ('0%# 2# *" #>*-+-?0-, 2# @'$A:

D!)/"$%&=(>1?

E'0' +,$ '('0%'2,$ ' / ?= $-5*#" $-#"2, B6+-2'$= #" %,2,$ $*$ 2#%'++#$= +'$ 2#7,$%0'&-,8"#$ 2# +,$ #"*"&-'2,$ '"6+,5,$ CR#,0#7' O:c= '('0%'2,$ ' / ?D 2#+ &'()%*+, '"%#0-,0=A#&A'$ ('0' #+ (0,&#$, 2# #+-7-"'&-9" !U!:

I,1"J E,20)' (#"$'0$# >*# #" +'$ 2#1-"-&-,"#$ '"%#0-,0#$ 2# 2,7-"'&-9" 2F?-+ / 2# 2,87-"'&-9" #$%0-&%'= 2#?#0)' A'?#0$# $*$%-%*-2, (K' C&,7?-"'&-9" 2# #$%0'%#5-'$ (*0'$ 2# +,$,%0,$D (,0 p.' C&,7?-"'&-9" 2# #$%0'%#5-'$ 7-<%'$ 2# +,$ ,%0,$D= (#0, #++, ", #$ "#&#$'8

0-,= (*#$ #$ 16&-+ B#0 >*# $- +'$ 2#$-5*'+2'2#$ $# &*7(+#" ('0' %,2, (.' %'7?-F" $# &*78(+#" ('0' %,2, p.' :

!" &*'"%, '+ '('0%'2, &= $- (' (#0%#"#&-#0' ' ;SECp7' D= 1,07'"2, p7' ('0%# 2# *" !@ph% CphJ = :::= p7' = :::= p75 D= 2# #++, $# 2#2*&-0)' >*# (' #$ 0#$(*#$%' 9(%-7' ' p7.'= +, >*#+# -7(#2-0)' $#0 #+-7-"'2' #" #+ (0,&#$, !U!8E_:

V# #$%# %#,0#7' $# 2#2*&# >*# (,2#7,$ '(+-&'0 #+ (0,&#$, 2# #+-7-"'&-9" -%#0'%-B'#$%0-&%' '"%#$ 2# &'+&*+'0 *" !@ #" #$%0'%#5-'$ 7-<%'$= (*#$ +'$ #$%0'%#5-'$ (*0'$ #+-7-"'82'$ ", B'" ' 1,07'0 ('0%# 2# "-"5." $,(,0%# 2# #$%0'%#5-'$ 7-<%'$ 2# #>*-+-?0-,: !+ $-85*-#"%# #4#7(+, -+*$%0' #$%# A#&A,:

Ejemplo 3.21

5.!'/ 67K

59

WJ WO WM WL Wc

58

TJ c= c c= Q f= Q M= J Q= M

TO O= f O= e e= f Q= c L= f

TM J= Q J= M d= r J= M J= JQ

TL O= O J= M J= M L= L JM= O

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59

WJ WL

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TL O= O L= L

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2@B% MHQ8E P8:E HQJE PJ:E SH8N6:Q8! H9N6:QJE H8NJ:P8! H6NJ:PJTO

Estrategias racionalizables

!+ &,"&#(%, 2# #$%0'%#5-' 0'&-,"'+-3'?+#= >*# %'7?-F" (,20)'7,$ ++'7'0 4*$%-1-&'?+#= $#(0,(,"# #<(+,0'0 *" &'7-", -"%#07#2-, #"%0# +,$ &,"&#(%,$ 2# $,+*&-9" ?'$'2,$ #" +'2,7-"'&-9" #$%0-&%' / #" +' 2,7-"'&-9" 2F?-+= #$ 2#&-0= #"%0# +,$ ?'$'2,$ #" +' -2#' 2#

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

&," +' A-(9%#$-$ 2# >*# %'+ 0'&-,"'+-2'2 #$ 2# 2,7-"-, (.?+-&, ",$ (#07-%-06"= 2# 7,2,'"6+,5, ' &,7, $# A-3, #" #+ &'()%*+, '"%#0-,0= &,"$%0*-0 (0,&#$,$ 2# #+-7-"'&-9" 2##$%0'%#5-'$ -"$'%-$1'&%,0-'$= 2# 7,2, >*# &,"$-2#0'0#7,$ '&#(%'?+#$ +'$ #$%0'%#5-'$ >*#$,?0#B-B'" ' 2-&A, (0,&#$,= / &,"$-2#0'0#7,$ *" "*#B, &,"&#(%, 2# $,+*&-9"= >*# &,"8$-$%# #" -2#"%-1-&'0 &,7, $,+*&-,"#$ 2#+ 4*#5, ' %,2,$ +,$ (#01-+#$ >*# #$%6" &,"$%-%*-2,$(,0 %'+#$ #$%0'%#5-'$ $*(#0B-B-#"%#$:

I,$ &,"&#(%,$ 2# &,"4#%*0' , &0##"&-' / 2# #$%0'%#5-' "*"&' 9(%-7' (#07-%#" (0#&-$'0+'$ -2#'$ '"%#0-,0#$:

Definición 3.10!" #+ 4*#5, 4% N& J= :::= & 5g $J= :::= $5PK

&: I+'7'7,$ =/1-!$.%& 2#+ 4*5'2,0 ' $,?0# &97, 4*5'06" +,$ 2#76$= ' %,2'2-$%0-?*&-9" 2# (0,?'?-+-2'2 p.'% CpJ= :::= p'.J= p'!J= :::= p5D $,?0# #+ #$('&-,& .'%& J#ñ#& '.J# & '!J#ñ# & 5 2# +'$ #$%0'%#5-'$ (*0'$ 2# +,$ 2#76$ 4*5'82,0#$:

;: V#&-7,$ >*# +' #$%0'%#5-' p! !" %!"0.!"$& >0$()& +!, -.'&+/% ! & ". =/1-!#$.%& p.! $- +' 2#$-5*'+2'2

$'Cp'= p.'Dn $'Cpñ'= p.'D

$# &*7(+# ('0' %,2' #$%0'%#5-' 7-<%' pñ' 2# 2-&A, 4*5'2,0:=: V#&-7,$ >*# +' #$%0'%#5-' p' 2#+ 4*5'2,0 ' #$ 1.1=& >0$()& $- ", #<-$%# "-"8

5*"' &,"4#%*0' p.' 2# 2-&A, 4*5'2,0 ' +' &*'+ p' $#' 0#$(*#$%' 9(%-7':

!+ "*#B, &,"&#(%, 2# #$%0'%#5-' "*"&' 9(%-7' #$%6 #$%0#&A'7#"%# 0#+'&-,"'2, &," #+2# #$%0'%#5-' #$%0-&%'7#"%# 2,7-"'2': V# A#&A,= #$ 16&-+ B#0 >*#= ('0' %,2, 4*5'2,0 '=%,2' #$%0'%#5-' #$%0-&%'7#"%# 2,7-"'2' C(,0 '+5*"' #$%0'%#5-' (*0' , 7-<%'D #$ *"' #$%0'8%#5-' "*"&' 9(%-7'= (*#$ '>*#++' >*# +' 2,7-"' (0,2*&# ' 2-&A, 4*5'2,0 ('5,$ #$%0-&%'8

7#"%# 7'/,0#$= -"2#(#"2-#"%#7#"%# 2# &97, 4*#5*#" +,$ 2#76$: _6$ '2#+'"%#&,"1-07'0#7,$ >*# +' -7(+-&'&-9" #" $#"%-2, &,"%0'0-, ", $-#7(0# #$ &-#0%':

Ejemplo 3.22

!" #+ 4*#5, G'+&9"8E'+,7'K

5.'&+/% 9

E'+,7' G'+&9"

5.'&+/% 8E'+,7' A / O= A / O Q= A

G'+&9" A = Q A / OK! = A / OK!

C2,"2# A b Q= ! b QD

;- ! aA / O= +' #$%0'%#5-' E'+,7' 2# &*'+>*-#0 4*5'2,0 #$%6 #$%0-&%'7#"%# 2,7-"'2'(,0 G'+&9"= / (,0 %'"%, ", #$ 0#$(*#$%' 9(%-7' ('0' "-"5*"' &,"4#%*0' 2# 2-&A, 4*5'2,0$,?0# &97, '&%*'06 #+ ,%0, 4*5'2,0: ;-" #7?'05,= $- ! %A / O= +' #$%0'%#5-' E'+,7' $9+,#$%6 2F?-+7#"%# 2,7-"'2' (,0 G'+&9"= / #" #$%# &'$, $) #$ 0#$(*#$%' 9(%-7' ' +' &,"4#8%*0' $#5." +' &*'+ #+ ,%0, 4*5'06 G'+&9" &," &#0%#3':

H$) (*#$= #+ A#&A, 2# $#0 2F?-+7#"%# 2,7-"'2' ", -7(+-&' #+ $#0 #$%0'%#5-' "*"&'9(%-7':

Definición 3.11

!" #+ 4*#5, 4% N& J= :::= & 5g $J= :::= $5P= 2#&-7,$ >*# +' #$%0'%#5-' (*0' (' 2#+ 4*5'2,0 '

#$ %&=(/1&,(^&;,! , -."$(X(=&;,! $- $,?0#B-B# '+ (0,&#$, 2# #+-7-"'&-9" -%#0'%-B' 2##$%0'%#5-'$ "*"&' 9(%-7'$:

I' 0'39" 2# >*# ' +'$ #$%0'%#5-'$ 0'&-,"'+-3'?+#$ $# +#$ ++'7# %'7?-F" 4*$%-1-&'?+#$ #$>*# $* *$, (,0 *" 4*5'2,0 (*#2# $#0 4*$%-1-&'2, (,0 F$%# 7#2-'"%# *"' &'2#"' 2# 0'3,"'87-#"%,$= ?'$'2,$ #" #+ A#&A, 2# >*# #$ 2# 2,7-"-, (.?+-&, >*# '7?,$ 4*5'2,0#$ $," 0'8&-,"'+#$ C/ #" ('0%-&*+'0 7'<-7-3'2,0#$ 2# $*$ 5'"'"&-'$D / &,",&#" +,$ 2#%'++#$ 2#+ 4*#5,:

Ejemplo 3.23

&: V'2, #+ Y*#5, M:LK

5.'&+/% 9H X W

5.'&+/% 8

t O= d d= Q J= J

[ J= M O= f L= L

u O= M c= J M= J

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#%'(' #+-7-"'7,$ +' #$%0'%#5-' [ 2#+ 4*5'2,0 J (,0>*# #$%6 2,7-"'2' #$%0-&%'7#"%# (,0u: !" +' %#0&#0' / .+%-7' #%'(' #+-7-"'7,$ +' #$%0'%#5-' X 2#+ 4*5'2,0 O (,0>*#= #"

'*$#"&-' 2# +' #$%0'%#5-' [ 2# YJ= #$%6 2,7-"'2' #$%0-&%'7#"%# (,0 H: ;,?0#B-B#" (,0%'"%, '+ (0,&#$, 2# #+-7-"'&-9" -%#0'%-B' 2# #$%0'%#5-'$ "*"&' 9(%-7'$ C>*# #" #$%# &'$,&,-"&-2# &," #+ (0,&#$, 2# #+-7-"'&-9" -%#0'%-B' #$%0-&%' !U!8E_D +'$ #$%0'%#5-'$ t / u2#+ 4*5'2,0 J / +' #$%0'%#5-' H 2#+ 4*5'2,0 O: !" &,"&+*$-9"K

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n*# YJ 4*#5*# t #$%6 4*$%-1-&'2, (,0K

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5.'&+/% 9

H X W

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t J= d d= Q J= J

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Teorema 3.9

V'2, #+ 4*#5, 4% N& J= :::= & 5g $J= :::= $5P / *"' #$%0'%#5-' (*0' , 7-<%' &*'+>*-#0' p'

2#+ 4*5'2,0 'K

&: ;- 5%O= p' #$ *"' #$%0'%#5-' #$%0-&%'7#"%# 2,7-"'2' $- / $9+, $- #$ *"' #$%0'8%#5-' "*"&' 9(%-7':

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=: ;- $9+, '&#(%'7,$ &,7, &,"4#%*0'$ 2#+ 4*5'2,0 - '>*#++'$ 2-$%0-?*&-,"#$ 2#(0,?'?-+-2'2 p.'% CpJ= :::= p'.J= p'!J= ::: = p5D $,?0# & .' #" +'$ >*# +'$ B'0-'?+#$ '+#'8%,0-'$ p * $," -"2#(#"2-#"%#$ #"%0# $)= #"%,"&#$= ('0' #+ &'$, 2# M , 76$ 4*5'2,0#$= $#0*"' #$%0'%#5-' "*"&' 9(%-7' ", -7(+-&' $#0 *"' #$%0'%#5-' #$%0-&%'7#"%# 2,7-"'2':

D!)/"$%&=(>1 +! H=:?

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eCJ/ OD%L= / 0#$(,"2-#"2, &," _L $#0)' MCJ/ OD!MCJ/ OD% M:

Ejemplo 3.24

V'2, #+ 4*#5, -"%0,2*&-2, #" #+ !4#7(+, M:OJ=

59

WJ WO WM WL Wc

58

TJ c= c c= Q f= Q M= J Q= M

TO O= f O= e e= f Q= c L= f

TM J= Q J= M d= r J= M J= JQ

TL O= O J= M J= M L= L JM= O

#+ (0,&#$, 2# #+-7-"'&-9" -%#0'%-B' 2# #$%0'%#5-'$ #$%0-&%'7#"%# 2,7-"'2'$= 0#'+-3'2, #"2-&A, #4#7(+,= #$%'?+#&# >*# +'$ ."-&'$ $*(#0B-B-#"%#$ $," +'$ #$%0'%#5-'$ TJ / TL 2# YJ=/ +'$ #$%0'%#5-'$ WJ / WL 2# YO: E*#$%, >*# $9+, A'/ 2,$ 4*5'2,0#$= #+ '('0%'2, C'D 2#+R#,0#7' M:r (#07-%# '1-07'0 >*# +'$ #$%0'%#5-'$ #$%0-&%'7#"%# 2,7-"'2'$ &,-"&-2#" &,"+'$ #$%0'%#5-'$ "*"&' 9(%-7'$= / #" &,"$#&*#"&-' +,$ &,"4*"%,$ 2# #$%0'%#5-'$ 0'&-,"'+-83'?+#$ $," L J% NTJ= TLP / L O%NWJ= WLP:

W#00'7,$ +' $#&&-9" &," #+ $-5*-#"%# %#,0#7'= >*# %'7(,&, 2#7,$%0'0#7,$: !$%# %#,80#7' (#07-%# 2#1-"-0 +'$ #$%0'%#5-'$ 0'&-,"'+-3'?+#$ 2# 7'"#0' &,7('&%'= / #>*-B'+#"%# '+' '"%#0-,0= 2# 7,2, >*# +' &'2#"' 2# 4*$%-1-&'&-,"#$ >*# A#7,$ *%-+-3'2, '"%#0-,07#"%#>*#2' &+'0'7#"%# 1*"2'7#"%'2':

Teorema 3.10

V'2, #+ 4*#5, 4% N& J= :::= & 5g $J= :::= $5P=

&: I,$ &,"4*"%,$ L J= :::= L '= :::= L 5 $," +,$ &,"4*"%,$ 2# #$%0'%#5-'$ (*0'$ 0'&-,"'8+-3'?+#$ 2# +,$ 4*5'2,0#$ $- / $9+, $- ('0' &'2' 4*5'2,0 ' #+ &,"4*"%, L ' #$ #+ 7'/,0$*?&,"4*"%, 2# & ' >*# &*7(+# +, $-5*-#"%#K

R,2' #$%0'%#5-' (*0' %' 2# L ' #$ *"' 0#$(*#$%' 9(%-7' 2#+ 4*5'2,0 ' ' '+5*"' &,"4#8%*0' p.' &*/, $,(,0%# #$ *" $*?&,"4*"%, 2# L .'% L J#ñ# L '.J# L '!J# L 5: C!$2#&-0= #" p.' +' (0,?'?-+-2'2 2# >*# '+5." 4*5'2,0 *Ç ' 4*#5*# '+5*"' #$%0'%#5-' (*0'#<%#0"' ' L * #$ "*+':D

;: !" #+ &'$, #" >*# 5%O= +' #$%0'%#5-' (*0' %' 2#+ 4*5'2,0 ' (#0%#"#&# ' L ' $- /$9+, $- (#0%#"#&# '+ $,(,0%# 2# *"' #$%0'%#5-' p' >*# 1,07' ('0%# 2# *" !@:

U+*$%0#7,$ #$%# %#,0#7' &," *" #4#7(+,:

&,7, (0#2-&&-9" 2#+ &,7(,0%'7-#"%, 2# 4*5'2,0#$ 0'&-,"'+#$: !+ $-5*-#"%# #4#7(+,-+*$%0'06 +' -2#':

Ejemplo 3.26

5.!'/ 67b

5.'&+/% 9

H X W V

5.'&+/% 8

H f= f O= O J= J Q= Q

X O= O O= O J= M/ O M= Q

W J= J M/ O= J J= J Q= Q

V Q= Q Q= M Q= Q Q= Q

!" #$%# 4*#5,= +,$ %0#$ (#01-+#$ $,7?0#'2,$ CH= HD= CX= XD / CW= WD $," #>*-+-?0-,$ 2#@'$A= +,$ ."-&,$ #" #$%0'%#5-'$ (*0'$= (#0, ?'$%' *" '"6+-$-$ $,7#0, ('0' &,"&+*-0 >*#$*$ 7F0-%,$ $," 2-$%-"%,$ / >*#= #" &,"$#&*#"&-' ", $," -5*'+7#"%# $'%-$1'&%,0-,$ / $*B'+,0 (0#2-&%-B, #$ 2-1#0#"%#: !1#&%-B'7#"%#= '"'+-3'"2, #+ %#0&#0, 2# #++,$= >*# #$ #+(#01-+ CW= WD= ,?$#0B'7,$ >*# +' #$%0'%#5-' W 2# YJ #$%6 2F?-+7#"%# 2,7-"'2' (,0 H /(,0 X= / +, 7-$7, +# ,&*00# ' +' #$%0'%#5-' W 2# YO: n*# &*'+>*-#0' 2# #++,$ 4*#5*# W$9+, $# 4*$%-1-&' $- #$%6 $#5*0, 2# >*# #+ ,%0, B' ' 4*5'0 W= / '*" #" #$# &'$, W ", #$

#$%0-&%'7#"%# 7#4,0 >*# H , X= $-", -5*'+ 2# 7'+': I, 7-$7, ,&*00# ('0' YO &," $*#$%0'%#5-' (*0' W: k#$*7-#"2,= ", A'/ "'2' (#,0 >*# '7?,$ 4*5'2,0#$ (*#2'" A'&#0= $-$# #<&#(%.' 4*5'0 V= >*# 4*5'0 $* #$%0'%#5-' W:

;- '"'+-3'7,$ #+ $#5*"2, 2# +,$ #>*-+-?0-,$= #+ CX= XD= ,?$#0B'7,$ *" 7F0-%, 2# #$#(#01-+= >*# &,7('0%# &," #+ (0-7#0, CH= HD: W,"$-$%# #" >*# +' #$%0'%#5-' 2# &'2' 4*5'82,0 #" #$# (#01-+ #$ ", 2,7-"'2':

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

!" #$%' $#&&-9" ",$ -"%#0#$'0#7,$ (,0 #+ $#5*"2, 2# +,$ '$(#&%,$ 0#&-F" 2#$&0-%,$:E'0' #++,= #<(+,0'0#7,$ &,"&#(%,$ 2# $,+*&-9" >*# ' +'$ &,"2-&-,"#$ 2#+ #>*-+-?0-, 2#@'$A 'l'2'" ,%0'$ &,"2-&-,"#$ %#90-&'$ 2#$#'?+#$: I'$ $,+*&-,"#$ 2# *" 4*#5, #" #$#&'$, /' ", $#0)'" %,2,$ +,$ #>*-+-?0-,$ 2# @'$A 2#+ 4*#5, $-", $9+, '>*#++,$ >*# &*7(+#"

+'$ &,"2-&-,"#$ %#90-&'$ '2-&-,"'+#$ >*# A#7,$ -7(*#$%,: E,0 #$,= ' 2-&A,$ &,"&#(%,$ 2#$,+*&-9" $# +#$ ++'7' 0#1-"'7-#"%,$ 2#+ #>*-+-?0-, 2# @'$A:

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l,:

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

Equilibrio admisible

!+ #>*-+-?0-, '27-$-?+# %'7?-F" 0#&-?# +,$ ",7?0#$ 2# !A.(,(;%(/ 1/ +/)(1&+/ / 2#!A.(,(;%(/ !1 !"$%&$!'(&" 1/ +/)(1&+&"7

Definición 3.12

V'2, #+ 4*#5, 4% N& J= :::= & 5g $J= :::= $5P= 2#&-7,$ >*# #+ (#01-+ 2# #$%0'%#5-'$ C(*0'$ ,7-<%'$D ph% CphJ = :::= p75 D #$ *" !A.(,(;%(/ &+)("(;,! $- #$ *" #>*-+-?0-, 2# @'$A /'2#76$ %,2'$ $*$ #$%0'%#5-'$ &,7(,"#"%#$ p7' $," ", 2,7-"'2'$:

I,$ $-5*-#"%#$ #4#7(+,$ -+*$%0'" #$%# &,"&#(%,:

Ejemplo 3.27

!" #+ Y*#5, M:f #" #+ !4#7(+, M:Od= +,$ #>*-+-?0-,$ #" #$%0'%#5-'$ (*0'$ CH= HD / CX= XD$," '27-$-?+#$= (#0, #+ CW= WD ", +, #$= (*#$ +' #$%0'%#5-' W #$%6 2F?-+7#"%# 2,7-"'2'=

('0' &*'+>*-#0 4*5'2,0= (,0 +' X:

Ejemplo 3.28

!" +,$ 4*#5,$ 2#$&0-%,$ #" #+ H('0%'2, O:J 2#+ &'()%*+, '"%#0-,0 C2-+#7' 2#+ (0-$-,"#0,=?'%'++' 2# +,$ $#<,$= #%&:D %,2,$ +,$ #>*-+-?0-,$ 2# @'$A $," '27-$-?+#$= $'+B, #" #+ 4*#5, G'+&9"8E'+,7' &," ! %A / O / #" #+ 4*#5, 2# B,%'&-9" (,0 7'/,0)':

!" #+ 4*#5, G'+&9"8E'+,7'= $-#"2, ! %A / O=

5.'&+/% 9

E'+,7' G'+&9"

5.'&+/% 8E'+,7' A / O= A / O Q= A

G'+&9" A = Q A / O.! = A / O.!

2# $*$ %0#$ #>*-+-?0-,$ 2# @'$A #" #$%0'%#5-'$ (*0'$= $9+, #+ #>*-+-?0-, C C"/+85= C"/+85D#$ '27-$-?+#: I,$ #>*-+-?0-,$ C C"/+85= B"/,-"D / CB"/,-"= C"/+85D ", $," '27-$-?+#$=/' >*# +' #$%0'%#5-' B"/,-" #$%6 2F?-+7#"%# 2,7-"'2' (,0 C"/+85:

!" #+ 4*#5, 2# B,%'&-9" (,0 7'/,0)'= +,$ ."-&,$ #>*-+-?0-,$ '27-$-?+#$ $," '>*#++,$>*# -"&+*/#" '+5." B,%, .%-+= CH= X= HD / CH= W= WD= 7-#"%0'$ >*# +,$ 2#76$ #>*-+-?0-,$=CH= H= HD= CX= X= XD / CW= W= WD= ", $," '27-$-?+#$: !" #1#&%,= #" #$%# 4*#5,= #+ 4*5'2,0WJ %-#"# $*$ #$%0'%#5-'$ X / W 2,7-"'2'$ (,0 H= #+ 4*5'2,0 WO %-#"# $* #$%0'%#5-' H2,7-"'2' (,0 X= / #+ 4*5'2,0 WM %-#"# $* #$%0'%#5-' X 2,7-"'2' (,0 W:

Z"' -7(,0%'"%# (0,(-#2'2 2# #$%# &,"&#(%, #$ >*# #$ 7*/ '7(+-'7#"%# '(+-&'?+#: !"('0%-&*+'0= %,2,$ +,$ 4*#5,$ 1-"-%,$ %-#"#" '+5." !@ '27-$-?+#:

Equilibrio estricto

!+ #>*-+-?0-, 2# @'$A #$%0-&%, (*#2# -"%#0(0#%'0$# -"%*-%-B'7#"%# &,7, *" #>*-+-?0-, 2#@'$A >*# ", 2#4'0)' 2# $#0+, '*">*# $# A*?-#0' &,7#%-2, *" #00,0 7-".$&*+, '+ #$%-7'0+,$ ('5,$ 2#+ 4*#5,:

Definición 3.13

V'2, #+ 4*#5, 4% N& J= :::= & 5g $J= :::= $5P= 2#&-7,$ >*# #+ (#01-+ 2# #$%0'%#5-'$p7% Cp7J = :::= p75 D #$ *" !A.(,(;%(/ +! B&"C !"$%(=$/ $- &'2' *"' 2# $*$ #$%0'%#5-'$&,7(,"#"%#$ p7' #$ 0#$(*#$%' 9(%-7' ."-&' ' +' &,7?-"'&-9" p7.' 2# #$%0'%#5-'$ 2# +,$2#76$ 4*5'2,0#$:

C!$ 2#&-0= O'= COp' à BC& 'D= 3 'Cp7' = p7.'Db3 'Cp'= p7.'DD

;# %0'%' 2# *" #>*-+-?0-, 0,?*$%, '"%# #00,0#$ $*1-&-#"%#7#"%# (#>*#l,$ #" +,$ ('5,$:

Z"' &'0'&%#0)$%-&' &*0-,$' # -"%#0#$'"%# 2# +,$ #>*-+-?0-,$ 2# @'$A #$%0-&%,$= 7*/ 16&-+ 2#2#2*&-0 2# +' 2#1-"-&-9"= #$ >*# ">,/ ,&" !"$%&$!'(&" 0.%&" 0.!+!1 X/%)&% 0&%$! +!

!,,/": Z" '$(#&%, "#5'%-B, 2# #$%# &,"&#(%, 2# #>*-+-?0-, #$ >*# ", #<-$%# #" %,2,$ +,$ 4*#5,$ 1-"-%,$: !" ('0%-&*+'0= 1/ !*("$! @B !"$%(=$/ !1 !, -.!'/ +! ,&" )/1!+&"= '+ ",%#"#0 #$%# 4*#5, #>*-+-?0-,$ 2# @'$A #" #$%0'%#5-'$ (*0'$:

I,$ $-5*-#"%#$ #4#7(+,$ '/*2'" %'7?-F" ' -+*$%0'0 #+ &,"&#(%,:

Ejemplo 3.29!" #+ Y*#5, M:f= #" #+ !4#7(+, M:Od #+ ."-&, !@ #" #$%0'%#5-'$ (*0'$ #$%0-&%, #$ #+ CH= HD:

Ejemplo 3.30

!+ 4*#5, 2# +'$ 7,"#2'$ ", %-#"# !@ #$%0-&%,= (*#$ $* ."-&, !@= #+ (#01-+ CCJ / O= J/ OD=CJ/ O= J/ ODD= #$%6 &,"$%-%*-2, (,0 #$%0'%#5-'$ 7-<%'$ (0,(-'$: !1#&%-B'7#"%#= +' #$%0'%#5-'7-<%' CJ/ O= J/ OD 2# YJ ", #$ +' ."-&' 0#$(*#$%' 9(%-7' ' +' #$%0'%#5-' CJ/ O= J/ OD 2# YO= /'

>*# %'"%, +' #$%0'%#5-' !"#" &,7, +' #$%0'%#5-' !#$% $," %'7?-F" 0#$(*#$%'$ 9(%-7'$:

Ejemplo 3.31

V# +,$ ,%0,$ 4*#5,$ 2#$&0-%,$ #" #+ H('0%'2, O:J 2#+ &'()%*+, '"%#0-,0= %,2,$ %-#"#" !@#$%0-&%,: !" +,$ 4*#5,$ 2-+#7' 2#+ (0-$-,"#0,= ?'%'++' 2# +,$ $#<,$= G'+&9"8E'+,7' C$-#"82, ! 2-$%-"%, 2# A / OD / +' &'3' 2#+ &-#0B, $," #$%0-&%,$ %,2,$ +,$ !@ #" #$%0'%#5-'$(*0'$: !" #+ 4*#5, G'+&9"8E'+,7' C$-#"2, ! %A / OD ", #<-$%# "-"5." !@ #$%0-&%,: !" #+ 4*#5, 2# +'$ (#%-&-,"#$ 2# @'$A= $," #$%0-&%,$ %,2,$ +,$ !@ 2#+ %-(, C F = J. F D 2,"2#

Qa

F a

J= / $9+, #++,$:

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i5#"F0-8&, #" 1,07' #$%0'%F5-&'j= (#0, $) 2-0#7,$ >*# +' 0#$(*#$%'= #" %F07-",$ -"%*-%-B,$= ' 2-&A'(0#5*"%' #$ >*# =&"( $/+/" +,$ 4*#5,$ 1-"-%,$ #" 1,07' #$%0'%F5-&' $," %'+#$ >*# %,2,$ $*$

#>*-+-?0-,$ 2# @'$A $," #$%0-&%,$: H &,"%-"*'&-9" B'7,$ ' '"'+-3'0 2,$ &,"&#(%,$ 2#$,+*&-9" >*#= #" +' #$&'+' 2# #<-5#"&-'$= $# $-%.'" &#0&' 2#+ (*"%, &0)%-&, #" >*# +' #<-85#"&-' #$ +' 7'/,0 (,$-?+# &," +' &,"2-&-9" 2# >*# %'+#$ $,+*&-,"#$ #<-$%'" #" %,2,$ +,$ 4*#5,$ 1-"-%,$:

!+ (0-7#0, 2# #++,$= #+ #>*-+-?0-, (#01#&%, C%'7?-F" ++'7'2, #>*-+-?0-, (#01#&%, 2#7'", %#7?+,0,$'D $# 2#?# ' ;#+%#" / (*#2# -"%#0(0#%'0$# -"%*-%-B'7#"%# &,7, *" #>*-+-8?0-, 0,?*$%, '"%# #00,0#$ $*1-&-#"%#7#"%# (,&, (0,?'?+#$ '+ &,"4#%*0'0 *" 4*5'2,0 >*F#$%0'%#5-'$ 4*5'06" +,$ 2#76$ C&,7, $- +' 7'", 2# +,$ ,%0,$ 4*5'2,0#$= '+ $#l'+'0 +' #$%0'8%#5-' ' $#5*-0= $*10-#0' *" (#>*#l, %#7?+,0 >*# (*2-#0' A'&#0+' #+#5-0 (,0 #00,0 &*'+>*-#0,%0'D: !+ $#5*"2,= #+ #>*-+-?0-, (0,(-,= $# 2#?# ' _/#0$," / (*#2# -"%#0(0#%'0$# -5*'+ >*#

#+ '"%#0-,0= (#0, #<-5-#"2, '2#76$ >*# +,$ #00,0#$ >*# &,"2*&#" ' 0#$*+%'2,$ (#,0#$ $#'"7*&A, 7#",$ (0,?'?+#$:

Equilibrio perfecto (o perfecto de mano temblorosa)

I' 2#1-"-&-9" >*# $-5*# 2# #>*-+-?0-, (#01#&%, ", #$ +' ,0-5-"'+ 2# ;#+%#"= $-", *"' 2#1-"-8&-9" #>*-B'+#"%# #+'?,0'2' (,0 _/#0$,"= >*# ",$ 0#$*+%' 76$ &,"B#"-#"%# (,0 $#0B-0 2#?'$# ('0' +' (,$%#0-,0 2#1-"-&-9" 2# #>*-+-?0-, (0,(-,:

Definición 3.14

&: V'2, *" 4*#5, 1-"-%, #" 1,07' #$%0'%F5-&' 4% N& J= :::= & 5g $J= :::= $5P= / 2'2, *"".7#0, (,$-%-B, e= #+ (#01-+ 2# #$%0'%#5-'$ 7-<%'$ p% CpJ= :::= p'= :::= p5D #$ .1 !A.(,(#;%(/ e#0!%X!=$/ $- 2'2, *" 4*5'2,0 &*'+>*-#0' '= / *"' #$%0'%#5-' (*0' % &*'+>*-#0' 2#+ 4*5'2,0 '= +' (0,?'?-+-2'2 #$%-(*+'2' #" p' 2# >*# 4*#5*# % #$

87 !$%0-&%'7#"%# (,$-%-B':97 !$%0-&%'7#"%# 7#",0 >*# e= $- % ", #$ *"' 0#$(*#$%' 9(%-7' 2#+ 4*5'2,0 ' '+

B#&%,0 2# #$%0'%#5-'$ p.

! 2# +,$ ,%0,$ 4*5'2,0#$:;: V'2, *" 4*#5, 1-"-%, #" 1,07' ",07'+ 4% N& J= :::= & 5g $J= :::= $5P= #+ (#01-+ p #$*" !A.(,(;%(/ +! B&"C 0!%X!=$/ $- #<-$%# *"' $*&#$-9" Ne) P) %J= O=::: 2# ".7#0,$ #$%0-&8%'7#"%# (,$-%-B,$ / #<-$%# *"' $*&#$-9" Cp) Dp%J= O=::: 2# (#01-+#$= %'+#$ >*#

87 E'0' %,2, ) = #+ (#01-+ p) #$ *" #>*-+-?0-, e) 8(#01#&%, 2# 4:97 +-7

) räe) %Q / +-7

) räp) %p:

I, >*# A#7,$ ++'7'2, #>*-+-?0-, e8(#01#&%, ", #$ "#&#$'0-'7#"%# *" #>*-+-?0-, 2#@'$A= (#0, $) #$%6 &#0&' 2# *",: !+ $-5*-#"%# #4#7(+, &,7#"3'06 ' '&+'0'0 #$%,$ &,"&#(%,$K

Ejemplo 3.32

!" #+ $-5*-#"%# 4*#5,= #" #+ >*# CH= HD / CX= XD $," $*$ !@ #" #$%0'%#5-'$ (*0'$=

5.!'/ 67c

5.'&+/% 9

H X

5.'&+/% 8

H f= f O= O

X O= O O= O

&: !+ (#01-+ p% CCQ=r= Q=JD= CQ=r= Q=JDD #$ *" #>*-+-?0-, e8(#01#&%, ('0' e%Q=O= /'>*# &*'+>*-#0' 2# +,$ 4*5'2,0#$ 4*#5' &," (0,?'?-+-2'2 $*1-&-#"%#7#"%# (#>*#l' C7#8",0 >*# eD $* #$%0'%#5-' (*0' X >*# ", #$ 9(%-7' 10#"%# ' +' #$%0'%#5-' CQ=r= Q=JD 2#+ ,%0, 4*5'2,0: ;-" #7?'05,= p ", #$ #>*-+-?0-, e8(#01#&%, ('0' "-"5." e -5*'+ , 7#",0 >*#Q=J= /' >*# #+ 4*5'2,0 J 4*#5' X &," *"' (0,?'?-+-2'2 2#7'$-'2, '+%' C", #$%0-&%'7#"%#7#",0 >*# eD= ' (#$'0 2# >*# X ", #$ 9(%-7' 10#"%# ' CQ=r= Q=JD 2# $* '2B#0$'0-,: S?$F08

B#$# >*# p ", #$ *" #>*-+-?0-, 2# @'$A:

;: !+ (#01-+ CCQ=J= Q=rD= CQ=J= Q=rDD #$ *" #>*-+-?0-, e8(#01#&%, ('0' e%Q=rc C'*"8>*# ", +, #$ ('0' "-"5." e -5*'+ , 7#",0 >*# Q=rD:

=: !+ (#01-+ CCJ. d/ O= d/ OD= CJ. d/ O= d/ ODD= $-#"2, Qa da J= #$ *" #>*-+-?0-,d8(#01#&%,:

+: I' $*&#$-9" 2# (#01-+#$ e) 8(#01#&%,$ Np) P) %J=O=:::= 2,"2#

p) %CAJ.

e) O

=e) OB = AJ.

e) O

=e) OBD / e) % J/ O)

%-#"# (,0 +)7-%# #+ (#01-+ p% CCJ= QD= CJ= QDD% CH= HD= / #" &,"$#&*#"&-' HWE W: !" .1!A.(,(;%(/ 0!%X!=$/:

!: I' $*&#$-9" 2# (#01-+#$ Np) P) %J=O=:::= 2,"2#

p) %CA

e) O

= J.e) OB = A

e) O

= J.e) OBD / e) % J/ O)

%-#"# (,0 +)7-%# #+ (#01-+ p% CCQ= JD= CQ= JDD% CX= XD= (#0, "- +,$ (#01-+#$ 2# #$%' $*&#8$-9" $," e) 8(#01#&%,$= "- +, $," +,$ 2# "-"5*"' $*&#$-9" &*/, +)7-%# $#' #+ (#01-+ CX= XD:!" &,"$#&*#"&-'= #+ #>*-+-?0-, 2# @'$A HaE a: 1/ !" .1 !A.(,(;%(/ 0!%X!=$/7

Ejemplo 3.33

!" #+ $-5*-#"%# 4*#5,= #" #+ >*# CH= HD= CX= XD / CW= WD $," $*$ !@ #" #$%0'%#5-'$ (*0'$=

5.!'/ 67d

5.'&+/% 9

H X W

5.'&+/% 8

H f= f O= O J= J

X O= O O= O J= O

W J= J O= J J= J

&: Z%-+-3'"2, #+ 7-$7, 0'3,"'7-#"%, >*# #" #+ #4#7(+, '"%#0-,0= +' $*&#$-9" 2#(#01-+#$ e) 8(#01#&%,$ Np) P) %J=O=:::= 2,"2# p)

% aCJ. e) . eñ) = e) = eñ) D= CJ. e) . eñ) = e) = eñ) Db /e) % eñ) % J/ O) = %-#"# (,0 +)7-%# #+ (#01-+ p% CCJ= Q= QD= CJ= Q= QDD% CH= HD= / #" &,"$#8&*#"&-' HWE W: !" .1 !A.(,(;%(/ 0!%X!=$/:

;: @-"5*"' $*&#$-9" 2# (#01-+#$ Np) P) %J=O=::: #" #$%0'%#5-'$ 7-<%'$ &,7(+#%'$= 2,"2#p) % aCe) = J. e) . eñ) = eñ) D= Cd) = J.d) . dñ) = dñ) Db / +-7 e) % +-7 eñ) % +-7 d) % +-7 dñ) % Q=

/ &," +)7-%# #+ (#01-+ %% CCQ= J= QD= CQ= J= QDD% CX= XD= #$%6 &,"$%-%*-2' (,0 #>*-+-?0-,$e) K(#01#&%,$= /' >*# H 2#+ 4*5'2,0 O #$ +' ."-&' 0#$(*#$%' 9(%-7' ' Ce) = J. e) . eñ) = eñ) D2#+ 4*5'2,0 J= / (,0 %'"%, J. d) . dñ) %#"20)' >*# $#0 7#",0 >*# e) = +, >*# &,"%0'2-&# #+A#&A, 2# >*# +-7 e) % +-7 eñ) % +-7 d) % +-7 dñ) % Q: I, 7-$7, (*#2# '05*7#"%'0$#

'&#0&' 2#+ (#01-+ CW= WD: !" &,"&+*$-9"= 1( HaE a: 1( HPE P: "/1 !A.(,(;%(/" 0!%X!=$/"7

Equilibrio propio

Definición 3.15

&: V'2, *" 4*#5, 1-"-%, #" 1,07' #$%0'%F5-&' 4% N& J= :::= & 5g $J= :::= $5P= / 2'2, *"".7#0, (,$-%-B, e= #+ (#01-+ 2# #$%0'%#5-'$ 7-<%'$ p% CpJ= :::= p' = :::= p5D #$ .1 !A.(,(#;%(/ +! B&"C e#0%/0(/ $- 2'2, *" 4*5'2,0 &*'+>*-#0' '= / *" ('0 &*'+>*-#0' 2# #$%0'%#85-'$ (*0'$ % / %ñ 2#+ 4*5'2,0 'K

87 I'$ (0,?'?-+-2'2#$ #$%-(*+'2'$ #" p' 2# >*# 4*#5*# % , %ñ= .p= 'C %D / .p= 'C %ñD= $,"'7?'$ #$%0-&%'7#"%# (,$-%-B'$:

97 ;- 3 'C %= p.'Da3 'C %ñ= p.'D= #"%,"&#$ .p= 'C %Dm e ` .p= C %ñD C#$ 2#&-0= $- % +# 0#(,08%' 7#",0 *%-+-2'2 #$(#0'2' >*# %ñ= 4*#5' % &," (0,?'?-+-2'2 7#",0 , -5*'+ >*#e B#&#$ +' (0,?'?-+-2'2 &," >*# 4*#5' %ñD:

;: V'2, *" 4*#5, 1-"-%, #" 1,07' #$%0'%F5-&' 4% N& J= :::= & 5g $J= :::= $5P= #+ (#01-+ %

#$ *" !A.(,(;%(/ +! B&"C 0%/0(/ $- #<-$%# *"' $*&#$-9" Ne) P) %J= O=::: 2# ".7#0,$ #$%0-&8%'7#"%# (,$-%-B,$ / #<-$%# *"' $*&#$-9" Np) P) %J= O=::: 2# (#01-+#$= %'+#$ >*#

87 E'0' %,2, ) = #+ (#01-+ p) #$ *" #>*-+-?0-, e) 8(0,(-, 2# 4:97 +-7

) räe) %Q / +-7

) räp) %p:

Ejemplo 3.34

!" #+ 4*#5, 2#+ !4#7(+, M:MOK

&: !+ (#01-+ p% aCJ. d/ := d/ :D= CJ. d/ := d/ :Db= $-#"2, :n O / Qa da J= #$ *"#>*-+-?0-, d8(0,(-,: !" #1#&%,= &*7(+# +' &,"2-&-9" J (,0>*# %,2'$ +'$ #$%0'%#5-'$ 2#+(#01-+ $," 7-<%'$ &,7(+#%'$= / '2#76$ $'%-$1'&# +' &,"2-&-9" O (,0>*#= &*7(+-#"2, +,$('5,$ +' 0#+'&-9"

3 'CX= p.'D% Oa3 'CH= p.'D%fCJ. d/ :D!OCd/ :D% f. cd/ :

,&*00# >*# +'$ (0,?'?-+-2'2#$ &*7(+#" +' 0#+'&-9"

.p= 'CXD%d/ :m d ` .p= 'CHD%dCJ. d/ :D% C:. dDCd/ :D

;: I' $*&#$-9" 2# (#01-+#$ e) 8(0,(-,$ Np)

P) %J=O=:::= 2,"2#

p) %CAJ.

e) O

=e) OB = AJ.

e) O

=e) OBD

/ e) %J/ O) = %-#"# (,0 +)7-%# #+ (#01-+ p% CCJ= QD= CJ= QDD% CH= HD= / #" &,"$#&*#"&-' #+(#01-+ HWE W: !" .1 !A.(,(;%(/ 0%/0(/7

!$ 7*/ 16&-+ B#0= / $# 2#7,$%0'06 76$ '2#+'"%#= >*# $/+/ !A.(,(;%(/ 0%/0(/ !" 0!%#X!=$/: U"%*-%-B'7#"%#= $- #+ #>*-+-?0-, p #$ (0,(-,= ", $9+, #$ 0,?*$%, ('0' '+ 7#",$ *"'

$*&#$-9" 2# %#7?+,0#$ &'2' B#3 76$ (#>*#l,$= &,7, &*7(+#" +,$ #>*-+-?0-,$ (#01#&%,$=$-", >*# %'7?-F" #$ 0,?*$%, ('0' '+ 7#",$ *"' $*&#$-9" 2# %#7?+,0#$ i0'3,"'?+#$j C#$

2#&-0= %#7?+,0#$ &*/,$ #00,0#$ '$,&-'2,$ &*7(+#" +' &,"2-&-9" O 2# +,$ #>*-+-?0-,$e8(0,(-,$= >*# '$#5*0' >*# +,$ #00,0#$ 76$ 50'B#$ %-#"#" '$-5"'2' *"' (0,?'?-+-2'2 2# *"7#",0 ,02#" 2# 7'5"-%*2 >*# +,$ 7#",$ 50'B#$D:

S%0' (0,(-#2'2 >*# $# 2#2*&# -"7#2-'%'7#"%# 2# +' 2#1-"-&-9" #$ >*# $/+/ !A.(,(#;%(/ +! B&"C (1$!%(/% H!" +!=(%E =/1"$($.(+/ 0/% !"$%&$!'(&" )(*$&" =/)0,!$&": !" .1!A.(,(;%(/ 0%/0(/: !" #1#&%,= $- p% CpJ= :::= p' =:::= p5D #$ *" !@ / p' #$ *"' #$%0'%#5-'7-<%' &,7(+#%' ('0' %,2, ' C%,2' #$%0'%#5-' (*0' 2# ' $# 4*#5' &," (0,?'?-+-2'2 #$%0-&%'87#"%# (,$-%-B'D= #<-$%# *" B'+,0 e (,$-%-B, %'+ >*# 2-&A, (#01-+ #$ *" #>*-+-?0-, e8(0,(-,:W,"&0#%'7#"%#= ?'$%'0)' %,7'0 e -5*'+ ' J/ O= (*#$%, >*# &*'+>*-#0 #$%0'%#5-' (*0' % 2#+ 4*5'2,0 ' #$ 0#$(*#$%' 9(%-7' ' p.'= / (,0 %'"%, ", $# '&%-B' +' #<-5#"&-' 2# >*#

.p ='C %Dm e ` .p= 'C %ñD: [ #"%,"&#$ ?'$%'06 #+#5-0 +' $*&#$-9" Ne) P) %J= O=:::= 2,"2# e) %e

O) = / +'

$*&#$-9" Np) P) %J= O=:::= 2,"2# p) %p= ('0' >*# $# &*7(+'

+-7) rä

e) % Q / +-7) rä

p) % p

/ (,2'7,$ &,"&+*-0 >*# p #$ *" #>*-+-?0-, (0,(-,:I' 50'" -7(,0%'"&-' %#90-&' 2#+ &,"&#(%, 2# #>*-+-?0-, (0,(-, 0'2-&' #" 2,$ &'0'&8

%#0)$%-&'$ '2-&-,"'+#$ >*# %-#"#: I' (0-7#0' #$ >*# $- p #$ #>*-+-?0-, (0,(-, 2# *" 4*#5,#" 1,07' ",07'+ 4= p -"2*&# *" #>*-+-?0-, $#&*#"&-'+ C&,"&#(%, >*# $#06 2#1-"-2, #" #+W'()%*+, dD #" &*'+>*-#0 4*#5, #" 1,07' #<%#"$-B' &*/' 0#(0#$#"%'&-9" #" 1,07' ",07'+&,-"&-2' &," 4: I' $#5*"2' #$ >*# #<-$%# ('0' %,2, 4*#5, 1-"-%,= &,7, $# '1-07' #" #+(09<-7, %#,0#7':

Otra caracterización de los equilibrios perfecto y propio

I,$ %#,0#7'$ >*# $-5*#" #$%'?+#&#" &'0'&%#0)$%-&'$ 2# '7?,$ #>*-+-?0-,$ >*# 0#$*+%'"7*/ &,"B#"-#"%# ' #1#&%,$ 2# &6+&*+,= (,0 $* 1'&-+-2'2 2# '(+-&'&-9": E'0' +' 2#7,$8%0'&-9" 2# #$%,$ %#,0#7'$= 2#"%0, 2# *" %0'%'7-#"%, 0-5*0,$, 2# #$%,$ &,"&#(%,$= BF'$#k-%3?#05#0 COQQOD / ^'" V'77# CJrefD:

Teorema 3.11

V'2, *" 4*#5, 1-"-%, #" 1,07' #$%0'%F5-&' 4% N& J= :::= & 5g $J= :::= $5P= #+ (#01-+ 2##$%0'%#5-'$ p% CpJ= :::= p'= :::= p5D #$ *" #>*-+-?0-, (#01#&%, "( I ">,/ "( p #$ #+ +)7-%# 2#*"' $*&#$-9" Np) P) %J= O=::: 2# (#01-+#$= %'+#$ >*#

87 E'0' %,2, ) = #+ (#01-+ p) #$%6 &,"$%-%*-2, (,0 #$%0'%#5-'$ 7-<%'$ &,7(+#%'$:97 E'0' &*'+>*-#0 4*5'2,0 '= $* #$%0'%#5-' p' #" p #$ 0#$(*#$%' 9(%-7' ' +' &,7?-8

"'&-9" p) .' 2# #$%0'%#5-'$ 2# +,$ 2#76$ ('0' &*'+>*-#0 (#01-+ p) 2# +' $*&#$-9":

Teorema 3.12

V'2, *" 4*#5, 1-"-%, #" 1,07' #$%0'%F5-&' 4% N& J= :::= & 5g $J= :::= $5P= $#' #+ (#01-+ 2##$%0'%#5-'$ p% CpJ= :::= p'= :::= p5D *" !A.(,(;%(/ 0%/0(/= / $#'" Ne) P) %J= O=::: *"' $*&#8$-9" 2# ".7#0,$ #$%0-&%'7#"%# (,$-%-B,$ &," +)7-%# "*+, / Np) P) %J= O=::: *"' $*&#$-9"

2# #>*-+-?0-,$ e) 8(0,(-,$ &," +)7-%# p:

Teorema 3.13

V'2, *" 4*#5, 1-"-%, #" 1,07' #$%0'%F5-&' 4% N& J= :::= & 5g $J= :::= $5PK

&: ;- #+ (#01-+ #$%0'%F5-&, p% CpJ= :::= p'= :::= p5D #$ *" #>*-+-?0-, (#01#&%, #"%,"8

&#$ #$ *" #>*-+-?0-, '27-$-?+#:;: ;- #+ 4*#5, 4 #$ ?-(#0$,"'+ C5%OD / #+ (#01-+ #$%0'%F5-&, p%CpJ= :::= p'= :::= p5D#$ *" #>*-+-?0-, '27-$-?+# #"%,"&#$ #$ *" #>*-+-?0-, (#01#&%,:

=: ;- #+ (#01-+ #$%0'%F5-&, p% CpJ= :::= p'= :::= p5D #$ *" #>*-+-?0-, #$%0-&%,= #"%,"8&#$ %'7?-F" #$ *" #>*-+-?0-, (0,(-,:

+: ;- #+ (#01-+ #$%0'%F5-&, p% CpJ= :::= p'= :::= p5D #$ *" #>*-+-?0-, (0,(-,= #"%,"&#$%'7?-F" #$ *" #>*-+-?0-, (#01#&%,:

!: !<-$%# '+ 7#",$ *" (#01-+ 2# #$%0'%#5-'$ C(*0'$ , 7-<%'$D >*# #$ *" #>*-+-?0-,(0,(-,:

D!)/"$%&=(>1 +! ,/" &0&%$&+/" & I +

&: E,0 0#2*&&-9" '+ '?$*02,= $- *" #>*-+-?0-, p% CpJ= :::= p'= :::= p5D ", #$ '27-$-8?+#= #<-$%# *" 4*5'2,0 ' >*# 4*#5' *"' #$%0'%#5-' p' 2F?-+7#"%# 2,7-"'2' (,0 ,%0'#$%0'%#5-' pñ': HA,0' ?-#"= #" #$# &'$, p' ", #$ 0#$(*#$%' 9(%-7' ' "-"5." B#&%,0 2##$%0'%#5-'$ 7-<%'$ &,7(+#%'$ 2# +,$ 2#76$= (*#$ pñ' +# (0,(,0&-,"'06 *" ('5, #$(#0'2,#$%0-&%'7#"%# 7'/,0 >*# p' C#" %F07-",$ -"%*-%-B,$= p' 2#4' 2# $#0 9(%-7' #" &*'"%, $#(0,2*&# #+ 76$ 7)"-7, %#7?+,0 >*# A'&# (,$-?+# >*# +,$ 2#76$ 4*5'2,0#$ 4*#5*#"%,2'$ $*$ #$%0'%#5-'$ (*0'$= -"&+*$, '>*#++'$ &,"%0' +'$ &*'+#$ p' 2' ('5,$ #$%0-&%'87#"%# -"1#0-,0#$ >*# +,$ 2# pñ'D: E,0 %'"%,= ", #<-$%# "-"5*"' $*&#$-9" Cp)

.'D) %J=O=::: 2#(#01-+#$ 2# #$%0'%#5-'$ 7-<%'$ &,7(+#%'$ >*# &*7(+' +, #$%'?+#&-2, #" #+ R#,0#7'

M:JJ: H$) (*#$= #+ #>*-+-?0-, p ", #$ (#01#&%,:+: ;# 2#2*&# 2# 7'"#0' -"7#2-'%' 2# +' 2#1-"-&-9" 2'2' >*# %,2, #>*-+-?0-, e8

(0,(-, #$ e8(#01#&%,= /' >*# $- % ", #$ 0#$(*#$%' 9(%-7' '+ B#&%,0 2# #$%0'%#5-'$ p.' 2#+,$ ,%0,$ 4*5'2,0#$= +# 0#(,0%' ' ' 7#",0 *%-+-2'2 #$(#0'2' 10#"%# ' p.' >*# ,%0' #$%0'8%#5-' (*0' %ñ= +*#5, 4*5'06 % &," (0,?'?-+-2'2 7#",0 , -5*'+ >*# e B#&#$ +' (0,?'?-+-82'2 &," >*# 4*#5' %ñ C>*# #$ 7#",0 >*# +' *"-2'2D= / (,0 %'"%, 4*5'06 % &,"(0,?'?-+-2'2 #$%0-&%'7#"%# 7#",0 >*# e: [ 2# +, '"%#0-,0 $# 2#2*&# >*# %,2, #>*-+-8?0-, (0,(-, #$ (#01#&%,:

!" &*'"%, '+ '('0%'2, ?D= 7#0#&# +' (#"' A'&#0 ,?$#0B'0 >*# $- 5b O ", $# &*7(+# +'

(0,(,$-&-9"= &,7, $# 2#7,$%0'06 76$ '2#+'"%# #" #+ !4#7(+, M:Me= #" #+ >*# A'/ %0#$ 4*5'2,0#$ / #<-$%# *" #>*-+-?0-, '27-$-?+# >*# ", #$ (#01#&%,: E,0 .+%-7,= '*">*# ",A#7,$ -"%#"%'2, *"' 2#7,$%0'&-9" 2#%'++'2' / 0-5*0,$' 2#+ '('0%'2, &D= #$ 16&-+ '05*7#"8%'0 -"%*-%-B'7#"%# >*# %,2, #>*-+-?0-, #$%0-&%, #$ %'7?-F" (0,(-,: !" #1#&%,= '+ $#0 *"#>*-+-?0-, #$%0-&%, 0,?*$%, ' &*'+>*-#0 (#0%*0?'&-9" 2# +,$ ('5,$ $*1-&-#"%#7#"%# (#>*#8l'= %'7?-F" +, $#06 ' &*'+>*-#0 (#0%*0?'&-9" , %#7?+,0 #" #+ (0,&#$, 2# $#l'+'0 +' #$%0'%#85-' (*0' #+#5-2'= $-#7(0# >*# #$%# %#7?+,0 $#' +, $*1-&-#"%#7#"%# (#>*#l, &,7, ('0'>*# +'$ (0,?'?-+-2'2#$ 2# $#l'+'0 ,%0'$ #$%0'%#5-'$ (*0'$ (,0 #00,0 $#'" ' $* B#3 $*1-&-#"8%#7#"%# (#>*#l'$= +, >*# A'0)' >*# +,$ ('5,$ #$(#0'2,$ 2-1-0-#0'" 2# +,$ ('5,$ $-" #00,80#$ #" &'"%-2'2#$ -"$-5"-1-&'"%#$: E#0, $- #+ #>*-+-?0-, #$ 0,?*$%, '"%# &*'+>*-#0 %#7?+,0

$*1-&-#"%#7#"%# (#>*#l,= +, $#06 %'7?-F" ('0' +,$ &'$,$ #" >*# +'$ (0,?'?-+-2'2#$ 2##00,0 $# '4*$%'" ' +' ('*%' #<-5-2' (,0 +' 2#1-"-&-9" 2# #>*-+-?0-, (0,(-,:

(,0 #$%'0 W / V 2,7-"'2'$ (,0 X: H$) (*#$= CX= XD &*7(+# +' &'0'&%#0-3'&-9" 2#+ R#,0#87' M:JJ= / (,0 %'"%, #$ *" #>*-+-?0-, (#01#&%,: R'7?-F" (,20)'7,$ A'?#0 -"B,&'2, #+R#,0#7' M:JM= '('0%'2, ?D= /' >*# CX= XD #$ '27-$-?+# / $9+, A'/ 2,$ 4*5'2,0#$:

E'0' B#0 >*# #+ #>*-+-?0-, CX= XD ", #$ (0,(-,= (0,&#2'7,$ (,0 0#2*&&-9" '+ '?$*082,: ;#' *"' $*&#$-9" &*'+>*-#0' 2# (#01-+#$ Np) P) %J=O=::: e) 8(0,(-,$ &*/, +)7-%# #" #+(#01-+ CX= XD &,"1,07# e) %-#"2# ' &#0,: ;#' p) % Cp)

J: p) OD= 2,"2#

p) J% Ca) = J. a) .b) . c) = b) = c) D / p)

O% Cañ) = J. añ) .bñ) . cñ) = bñ) = cñ) D

3 OCp) J= HD% fa) ! OCJ. a) . b) . c) D!b) %O! ca) .b) . Oc)

3 OCp) J= XD%Oa) !OCJ. a) .b) . c) D!b) !Mc) %O. b) ! c)

3 OCp) J= WD%Ja) !OCJ. a) .b) . c) D!b) % O. a) .b) . Oc)

3 OCp) J= VD% Q

E*#$%, >*# 3 OCp) J= VDa3 OCp)

J= WDa3 OCp) J= HD= #$ (0#&-$, >*# c) m e) b) m eO

) a) : ;-"#7?'05,= #" B-0%*2 2# +' &'0'&%#0-3'&-9" 2#+ R#,0#7' M:JO ('0' +,$ #>*-+-?0-,$ (0,(-,$=X 2#+ 4*5'2,0 O A' 2# $#0 0#$(*#$%' 9(%-7' ' p)

J% Ca) = J. a) . b) . c) = b) = c) D ('0'%,2, ) #" #+ >*# e) $#' $*1-&-#"%#7#"%# (#>*#l,: E,0 %'"%,=

3 OCp) J= XD% Oa) ! OCJ. a) . b) . c) D!b) !Mc) b3 'Cp

) J= HD%

% fa) ! OCJ. a) . b) . c) D!b)

2# 2,"2# $# 2#2*&# >*# c) b ca) / M= / #$%, #$%6 #" &,"%0'2-&&-9" &," c) m eO) a) :

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Ejemplo 3.37

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(-, / (#01#&%,:!" +' ?'%'++' 2# +,$ $#<,$= %,2,$ $*$ !@ $," (0,(-,$= (*#$ +,$ 2,$ !@ #" #$%0'%#5-'$

(*0'$ $," #$%0-&%,$= 7-#"%0'$ >*# #+ #>*-+-?0-, #" #$%0'%#5-'$ 7-<%'$ &,7(+#%'$ #$ "#&#8$'0-'7#"%# (0,(-,:

!" #+ 4*#5, 2# +'$ 7,"#2'$ $* ."-&, !@= #+ (#01-+ CCJ/ O= J/ OD= CJ/ O= J/ ODD= '+ #$%'0

&,"$%-%*-2, (,0 #$%0'%#5-'$ 7-<%'$ &,7(+#%'$= #$ *" #>*-+-?0-, (0,(-,:

Ejemplo 3.38

V'2, #+ 4*#5, &," %0#$ 4*5'2,0#$= %,7'2, 2# ^'" V'77# CJrefDK

5.!'/ 6788

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X Q= J= Q J= Q= Q

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p)

% aCe) = J. e) D= CJ. eñ) = eñ) D= CJ. eññ) = eññ) Db= 2,"2# +-7 e) % +-7 eñ) % +-7 eññ) %Q: V-&A'$*&#$-9" %-#"# (,0 +)7-%# #+ (#01-+ F % CCQ= JD= CJ= QD= CJ= QDD% CX= U= tD= (#0, B#0#7,$>*# ", #$%6 &,"$%-%*-2' (,0 #>*-+-?0-,$ e) 8(#01#&%,$: !1#&%-B'7#"%#= $- +,$ 4*5'2,0#$ O /M 4*#5'" p)

.J% aCJ. eñ) = eñ) D= CJ. eññ) = eññ) Db= #+ 4*5'2,0 J (*#2# 0#$(,"2#0K

H= +, >*# +# (0,(,0&-,"'0)' *" ('5, -5*'+ '

3 JCH= p) .JD% CJ. eñ) DCJ. eññ) DCJD! CJ. eñ) Deññ) CJD! eñ) CJ. eññ) DCJD! eñ) eññ) CQD

X= +, >*# +# (0,(,0&-,"'0)' *" ('5, -5*'+ '

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k#$%'"2, 7-#7?0, ' 7-#7?0,K

3 JCH= p) .JD.3 JCX= p)

.JD%

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4Juegos dinámicos

con informacióncompleta

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4.1. INTRODUCCIÓN

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Significado de juego dinámico

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Elementos de un juego representado en forma extensiva.Ejemplos iniciales

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Ejemplo 4.1 :) ,-/6* +/ +05-$504)

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ENTRON

INCUMBRON

No entrar Entrar

Competir duro Competir suave

3, 7

0, 0 5, 5

Figura 4.1 Juego de disuasión 1.

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ENTRON

ENTRON ENTRON

INCUMBRON

No entrar Entrar

3, 7

22, 22 2 21, 14, 21, 4

A

A A

B

B B

Figura 4.2 Juego de disuasión 2.

Juegos dinámicos con información completa 221

Ejemplo 4.2 ;< ,-/6* +/< .2/590=5

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J1

J2

J1

S

S

S

T

T

T

1, 1

0, 3

2, 2

1, 4

Figura 4.3 Juego del trespiés.

Ejemplo 4.3 ;< ,-/6* +/< '0/390=5

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J1 S S S S S S S S J1 J1 J1J2 J2 J2 J2

T T T T T T T T

1, 1 0, 3 2, 2 1, 4 3, 3 2, 5 50, 50 49, 52

51, 51

Figura 4.4 Juego del ciempiés.

Ejemplo 4.4 >0</3$ +/< 92050*)/2* 2/9/.0+* ) ?/'/5

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J1

J1 J1J1 J1

J2

J2 J2 J2 J2J2 J2 J2 J2

J2Con Ca

Con Ca CaCon

Con Con Con ConCa Ca Ca Ca

Con Ca Ca Ca Ca Ca Ca Ca CaCon Con Con Con Con Con Con

2 6 1 5 6 10 5 9 1 5 0 4 5 9 4 82 1 6 5 1 0 5 4 6 5 10 9 5 4 9 8

Figura 4.5 Dilema del prisionero, repetido 2 veces.

Ejemplo 4.5 @-/6* +/< 2/9$2.*" A/2504) +05'2/.$

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Terminología básica adicional

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d E,$ '*),-).*5 +/ 0)1*23$'04) (#1./%#" 1#(1#$#"%'1 #+ &,",&/./#"%, 9*# &'4' 2*3'4,1 %/#"# N#" #+ .,.#"%, 4# 4#&/4/1 #" *" ",4,O 4#+ 4#$'11,++, (1#-/, 4#+ 2*#3,;

d J"' /5.2$./60$ (*1' 4# *" 2*3'4,1 #$ *" (+'" 4# '&&/7" &,.(+#%,; !$(#&/6/&' *"''&&/7" 6'&%/@+# 4# 4/&?, 2*3'4,1 ('1' &'4' *", 4# $*$ &,"2*"%,$ 4# /"6,1.'&/7"; J"9/210< #$%1'%B3/&, #$ *" -#&%,1 4# #$%1'%#3/'$8 *"' (,1 &'4' *", 4# +,$ 2*3'4,1#$;

d G,4, &'./", 9*#8 $/3*/#"4, +'$ 1'.'$ 4#+ 51@,+8 &,"4*&# 4#$4# #+ ",4, /"/&/'+ '*" ",4, %#1./"'+ #$ *"' 5/)+$7 2/5-<.$+*7 +/5$22*<<* * .2$E/'.*20$ (,$/@+# 4#+ 2*#3,;H'4' -#0 9*# $# 2*#3' #6#&%/-'.#"%# *" 2*#3, $# 1#&,11# *",8 : $7+, *",8 4# +,$ 4#$'11,A++,$ (,$/@+#$ N&*5+ $#' #$%# 4#$'11,++, 4#(#"4# 4# &7., ?':'" 2*3'4, +,$ 2*3'4,1#$ : 4#9*B 1#$*+%'4,$ $# ?':'" (1,4*&/4, #" +'$ 2*3'4'$ 4# '0'1O; >$) (*#$8 %,4,$ +,$ 4#$'11,++,$(,$/@+#$ 4#+ 2*#3, ?'" 4# #$%'1 ' +' -/$%' #" #+ 51@,+ 4#+ 2*#3,8 : ?': %'"%,$ &,., ",4,$%#1./"'+#$; H'4' (#16/+ #$%1'%B3/&, 4#%#1./"' *" 4#$'11,++, 4#+ 2*#3,8 $'+-, #" +, 9*# $#1#6/#1# ' +'$ 2*3'4'$ 4#+ '0'1; W@$B1-#$# 9*# +' ('+'@1' 1#$*+%'4, $# *$' '9*) &,., $/"7A"/., 4# 4#$'11,++, , %1':#&%,1/'8 : ", #" $* '&#(&/7" 4# ('3,$ 6/"'+#$;

!+ $/3*/#"%# #2#.(+, /+*$%1'%/-, ':*4'15 ' '&+'1'1 #$%,$ %B1./",$

Ejemplo 4.7

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Ejemplo 4.8

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Juegos con información perfecta y juegos con información imperfecta

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Definición 4.1

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Ejemplo 4.9

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Figura 4.9 Juego 2.5 en forma extensiva.

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!" +,$ #2#.(+,$ $/3*/#"%#$8 $# 4' +' 1#(1#$#"%'&/7" #" 6,1.' #$%1'%B3/&' 4# '+3*",$4# +,$ 2*#3,$ (1#$#"%'4,$ '"%#1/,1.#"%#8 : $# &'+&*+'" $*$ #9*/+/@1/,$ 4# C'$? #" #$%1'%#A3/'$ (*1'$;

Ejemplo 4.10!" #+ 2*#3, 4# 4/$*'$/7" L8 +'$ #$%1'%#3/'$ 4# !CGPWC $," N(!)#&# 8 >O8 N(!)#&# 8 VO8N *+ .!)#&# 8 >O : N *+ .!)#&# 8 VO8 : +'$ 4# UCHJKVPWC $," > : V; Y,1 %'"%,8 $*1#(1#$#"%'&/7" #" 6,1.' ",1.'+ #$Q

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jN *+ .!)#&# 8 VO8 >k;

Ejemplo 4.11

!" #+ 2*#3, 4#+ %1#$(/B$8 #+ &,"2*"%, 4# #$%1'%#3/'$ 4# ]I #$ 2 I% eG.G8 G. <8 <.G8<.<f : #+ 4# ]L #$ 2 L%eG8 <f; Y,1 %'"%,8 $* 1#(1#$#"%'&/7" #" 6,1.' ",1.'+ #$Q

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Ejemplo 4.12

<#' #+ 2*#3, 4# &'1%'$ 4#+ !2#.(+, I;IM NH'()%*+, IO8 9*# 1#(1,4*&/.,$ ' &,"%/"*'&/7"Q

J0 (Azar)

Blanca

Carlos

Carlos

Blanca

1/2

1/2

O-C

E-B

A

A

R

RA

RA

P

P

R

5, 52

5, 52

5, 52

10, 102

210, 10

25, 5


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4.2. EQUILIBRIO DE NASH PERFECTO EN SUBJUEGOS

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Los equilibrios de Nash de un juego representado en forma extensiva

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Subjuegos

H*'"4, #" *" (*"%, 4#+ 4#$'11,++, 4# *" 2*#3, #" 6,1.' #=%#"$/-' $# 4' +' &/1&*"$%'"&/'

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Definición 4.2

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R" G'.@/B" $# &,"$/4#1'" $*@2*#3,$ '9*#++,$ #" +,$ 9*# :' $7+, -' ' /"%#1-#"/1 *",4# +,$ 2*3'4,1#$;

F" E,$ 2*#3,$ #$%5%/&,$ 1#(1#$#"%'4,$ #" 6,1.' #=%#"$/-' ", %/#"#" "/"3F" $*@2*#3,

(1,(/,8 :' 9*# #+ F"/&, &,"2*"%, 4# /"6,1.'&/7" *"/%'1/, 9*# %/#"#" #$ #+ ",4,/"/&/'+;

Ejemplo 4.14

$% !" #+ 2*#3, 4# 4/$*'$/7" I8 #+ F"/&, $*@2*#3, (1,(/, &,11#$(,"4# ' +' ('1%# 4#+51@,+ 9*# &,./#"0' #" #+ ",4, 4# 4#&/$/7" 4# UCHJKVPWC 9*# ?': %1'$ +' 2*3'4'(!)#&# 4# !CGPWC;

ENTRON

INCUMBRON

No entrar Entrar

Competir duro Competir suave

3, 7

0, 0 5, 5

Figura 4.11 Subjuegos propios del juego de disuasión 1.

&% !" #+ 2*#3, 4# 4/$*'$/7" L8 #+ F"/&, $*@2*#3, (1,(/, &,11#$(,"4# /3*'+.#"%# '+' ('1%# 4#+ 51@,+ 9*# &,./#"0' #" #+ ",4, 4# 4#&/$/7" 4# UCHJKVPWC; E' _#6/"/A&/7" \;L #=&+*:# 9*# ?':' $*@2*#3,$ 9*# &,./#"&#" #" #+ &,"2*"%, 4# /"6,1.'&/7"N&," 4,$ ",4,$O 4# !CGPWC , #" '+3*", 4# $*$ 4,$ ",4,$;

ENTRON

ENTRON ENTRON

INCUMBRON

No entrar Entrar

3, 7

22, 22 2 21, 14, 21, 4

A

A A

B

B B

Figura 4.12 Subjuegos propios del juego de disuasión 2.

'% !" #+ 2*#3, 4#+ %1#$(/B$8 +,$ 4,$ $*@2*#3,$ (1,(/,$ &,11#$(,"4#" ' +' ('1%# 4#+51@,+ 9*# &,./#"0' #" +,$ 4,$ ",4,$ 4# 4#&/$/7" 4/$%/"%,$ 4#+ /"/&/'+; </" #.@'13,8 #+

4/+#.' 4#+ (1/$/,"#1,8 &,., %,4,$ +,$ 2*#3,$ #$%5%/&,$8 &'1#&# 4# $*@2*#3,$ (1,(/,$;

J1

J2

J1

S

S

S

T

T

T

1, 1

0, 3

2, 2

1, 4

Figura 4.13 Subjuegos propios del juego del trespiés.

Y,1 ,%1' ('1%#8 #+ 4/+#.' 4#+ (1/$/,"#1, 1#(#%/4, 4,$ -#&#$ %/#"# \ $*@2*#3,$ (1,(/,$

9*# #.(/#0'" #" &'4' *", 4# +,$ ",4,$ 4# 4#&/$/7" 4# ]I 4/$%/"%,$ 4#+ ",4, /"/&/'+8 %'+&,., $# -# #" +' [/3*1' \;I\;

J1

J1 J1J1 J1

J2

J2 J2 J2 J2J2 J2 J2 J2

J2Con Ca

Con Ca CaCon



2 6 1 5 6 10 5 9 1 5 0 4 5 9 4 82 1 6 5 1 0 5 4 6 5 10 9 5 4 9 8

Figura 4.14 Subjuegos propios del dilema del prisionero, repetido 2 veces.

Equilibrio de Nash perfecto en subjuegos. Resultado perfectoen subjuegos

!$%'.,$ :' #" &,"4/&/,"#$ 4# 4#6/"/1 #+ "*#-, &,"&#(%, 4# #9*/+/@1/,8 9*# 6*# (1,(*#$%,(,1 P#/"?'14 <#+%#" #" IibZ; !$%# '*%,1 1#&/@/7 #+ Y1#./, C,@#+ 4# !&,",.)' #" Iii\(,1 $*$ &,"%1/@*&/,"#$ ' +' %#,1)' 4# 2*#3,$8 #$(#&/'+.#"%# +'$ 1#6#1/4'$ ' B$%# : ,%1,$&,"&#(%,$ 4# #9*/+/@1/, N&,., #+ #9*/+/@1/, (#16#&%, 4# .'", %#.@+,1,$' 4#6/"/4, #" #+H'()%*+, MO;

Definición 4.3

<#' 1 *" 2*#3, #" 6,1.' #=%#"$/-'

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&% <#' 3 *" 1#$*+%'4, , 4#$'11,++, (,$/@+# 4# 1; _#&/.,$ 9*# 3 #$ *" O/5-<.$+*V/21/'.* /) W-&,-/6*5 NPY<O 4# 1 $/ 3 (*#4# ,@%#"#1$# &,., 1#'+/0'&/7" 4# *" (#16/+ ' 9*# #$ *" !CY<;

P#&*B14#$# 9*# '+3*",$ $*@2*#3,$ 1#(1#$#"%'" $/%*'&/,"#$ #" +'$ 9*# $7+, 9*#4' (,1 2*3'1 *" 2*3'4,1; H,"$/4#1'1#.,$ 9*# +,$ !C 4# %'+#$ $*@2*#3,$ $," +'$ #$%1'%#3/'$ 7(%/A.'$ 4# 4/&?, 2*3'4,1;

E,$ $/3*/#"%#$ #2#.(+,$ ",$ (#1./%/15" /+*$%1'1 '.@,$ &,"&#(%,$; Y,1 1'0,"#$ 4# $#"&/A++#0 ",$ ,&*('1#.,$ (1/"&/('+.#"%# 4# +,$ #9*/+/@1/,$ : 1#$*+%'4,$ #" #$%1'%#3/'$ (*1'$;

Ejemplo 4.15

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$*@2*#3,8 &," *" F"/&, 2*3'4,18 #+ F"/&, !C &,"$/$%# #" 9*# UCHJKVPWC 2*#3*#,+"4.)$# %6&v.8 : (,1 %'"%, $#15" !CY< 4#+ 2*#3, 3+,@'+ '9*#++,$8 : $7+, '9*#++,$8

(#16/+#$ 9*# $#'" !C 4#+ 2*#3, 3+,@'+ : &*:' 1#$%1/&&/7" '+ $*@2*#3, (1,(/, $#' (1#&/$'A.#"%# ,+"4.)$# %6&v.; g': 4,$ !C #" #$%1'%#3/'$ (*1'$ 4#+ 2*#3, 3+,@'+8 9*# $," '0% N(!)#&# 8 ,+"4.)$# %6&v.O : 'ñ0% N *+ .!)#&# 8 ,+"4.)$# 86#+O; _# #++,$8 $7+, '0(1,4*&# +' '&&/7" ,+"4.)$# %6&v. '+ 1#$%1/"3/1+, '+ $*@2*#3,8 : #" &,"$#&*#"&/'8 #+F"/&, !CY< #" #$%1'%#3/'$ (*1'$ #$ '0;

&% !" #+ 2*#3, 4# 4/$*'$/7" L8 #+ F"/&, $*@2*#3, (1,(/, #$ %'.@/B" '9*#+ 9*#&,./#"0' #" #+ ",4, 4# 4#&/$/7" 4# UCHJKVPWC8 1#(1#$#"%'4, #" +' [/3*1' \;IL;_/&?, $*@2*#3,8 &*:' 1#(1#$#"%'&/7" #" 6,1.' #$%1'%B3/&' #$

JKL:MNOPK

> V

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V I8 \ .I8 .I

%/#"# 4,$ !C8 9*# $," '0% N>8 VO : 'ñ0% NV8 >O; Y,1 ,%1' ('1%#8 +,$ !C #" #$%1'%#3/'$(*1'$ 4#+ 2*#3, 3+,@'+ $," jN(!)#&# 8 >O8 Vk8 jN *+ .!)#&# 8 >O8 >k : jN *+ .!)#&# 8 VO8 >k8%'+ &,., +,$ ?#.,$ &'+&*+'4, #" #+ !2#.(+, \;I^; o#'.,$ 9*B (#16/+#$ !C 4#+ 2*#3,3+,@'+ /"4*&#" *" !C 4#+ $*@2*#3, '+ $#1 1#$%1/"3/4,$ ' 4/&?, $*@2*#3,;

!+ (#16/+ jN(!)#&# 8 >O8 Vk /"4*&# #" #+ $*@2*#3, #+ (#16/+ N>8 VO8 9*# $) #$ !C;!+ (#16/+ jN *+ .!)#&# 8 >O8 >k /"4*&# #" #+ $*@2*#3, #+ (#16/+ N>8 >O8 9*# ", #$ !C;!+ (#16/+ jN *+ .!)#&# 8 VO8 >k /"4*&# #" #+ $*@2*#3, #+ (#16/+ NV8 >O8 9*# $) #$ !C;

!" &,"&+*$/7"8 +,$ (#16/+#$ jN(!)#&# 8 >O8 Vk : jN *+ .!)#&# 8 VO8 >k $," +,$ !CY< @*$&'A4,$;

Ejemplo 4.16

!" #+ 2*#3, 4#+ %1#$(/B$8 ?': 4,$ $*@2*#3,$ (1,(/,$8 %'+ &,., $# ?' 1#(1#$#"%'4, #" +'[/3*1' \;IM; !+ $#3*"4,8 9*# ++'.'1#.,$ $*@2*#3, L8 #$ ' $* -#0 $*@2*#3, 4#+ (1/.#1,N$*@2*#3, IO; !+ $*@2*#3, I &,./#"0' #" #+ (1/.#1 ",4, 4# 4#&/$/7" 4# ]L : #+ $*@2*#3,L &,./#"0' #" #+ $#3*"4, ",4, 4# 4#&/$/7" 4# ]I; !+ $*@2*#3, I8 &*:' 1#(1#$#"%'&/7"

#" 6,1.' ",1.'+ #$

@#

< G

@!< I8 \ ^8 M

G L8 L ^8 M

%/#"# *" F"/&, !C8 9*# #$ '0% NG8 GO8 ./#"%1'$ 9*# #+ F"/&, !C 4#+ $*@2*#3, L #$

,@-/'.#"%# G; Y,1 ,%1' ('1%#8 +,$ !C #" #$%1'%#3/'$ (*1'$ 4#+ 2*#3, 3+,@'+8 9*# $# ?'"&'+&*+'4, #" #+ !2#.(+, \;II $," NG.G8 GO : NG.<8 GO;

o#'.,$ 9*B (#16/+#$ !C 4#+ 2*#3, 3+,@'+ /"4*&#" *" !C #" &'4' *", 4# +,$ $*@2*#A3,$ '+ $#1 1#$%1/"3/4,$ ' 4/&?,$ $*@2*#3,$; !+ (#16/+ NG .G8 GO /"4*&# #" #+ $*@2*#3, I#+ (#16/+ NG8 GO8 9*# $) #$ !C : #" #+ $*@2*#3, L #+ (#16/+ G 9*# %'.@/B" #$ !C; Y,1 ,%1'('1%#8 #+ (#16/+ NG. <8 GO /"4*&# #" #+ $*@2*#3, I #+ (#16/+ N<8 GO8 9*# ", #$ !C : #" #+$*@2*#3, L #+ (#16/+ < 9*# %'.(,&, +, #$;

!" &,"&+*$/7"8 #+ (#16/+ NGAG8 GO #$ #+ !CY< @*$&'4,;

Observación sobre el significado de las estrategias (continuación)

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.,$ &,"&1#%'1 9*# +'$ 2*3'4'$ ' 1#'+/0'1 #" &,"2*"%,$ 4# /"6,1.'&/7" #=&+*/4,$ 4#+4#$'11,++, 4#+ 2*#3, ", $," "#&#$'1/'$ $/ #+ '"5+/$/$ $# +/./%' '+ &5+&*+, 4# +,$ !C8 (#1, $) +, $," $/ (1,6*"4/0'.,$ '+3, .5$8 (,1 #2#.(+, &'+&*+'"4, +,$ !CY<;

!" #6#&%,8 ,@$B1-#$# #+ 2*#3, 4#+ %1#$(/B$8 4#+ 9*# 1#(#%/.,$ $* 6,1.' #$%1'%B3/&'Q

@-/6* +/< .2/590=5" X*23$ /5.2$.=60'$

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G.G I8 I I8 I

G.< I8 I I8 I

<.G ^8 M L8 L

<.< ^8 M I8 \

_#$4# #+ (*"%, 4# -/$%' 4# +' 6,1.' #$%1'%B3/&'8 +'$ #$%1'%#3/'$ G .G : G.< 4# ]I$," #9*/-'+#"%#$ N' &*'+9*/#1 2*3'4,1 +# #$ /"4/6#1#"%# 9*# ]I 2*#3*# *"' * ,%1'O8 (,1 #$,'.@'$ /"%#1-/#"#" 4#+ ./$., .,4, #" +,$ !C 4#+ 2*#3,8 9*# $," NG.G8 GO : NG.<8GO8 : (,1 #$, (,41)' #+/./"'1$# +' 1#4*"4'"&/'8 1#6/1/B"4,",$ ' #++'$ &,., G8 &*:, $/3"/A6/&'4, $#1)' `]I 2*#3' G : $# '&'@' #+ 2*#3,a; !" #$# &'$,8 +' 6,1.' #$%1'%B3/&' 4#+ 2*#3,&," %1#$ #$%1'%#3/'$ ('1' ]I8 9*# ++'.'1)'.,$ 6,1.' #$%1'%B3/&' 1#4*&/4'8 $#1)'Q

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!$ #-/4#"%# 9*# #$%# 2*#3, 1#4*&/4, %/#"# *" F"/&, !C8 9*# #$ NG8 GO; !$%# '"5+/$/$89*# ", 4/$%/"3*# #"%1# G.< : G.G8 #$ &,11#&%, $/ $7+, #$%'.,$ /"%#1#$'4,$ #" +,$ !C4#+ 2*#3,; </" #.@'13,8 #" #+ !2#.(+, \;Ib $# ?' /4#"%/6/&'4, #+ (#16/+ NG.G8 GO &,.,#+ F"/&, !CY< 4#+ 2*#3,8 +, 9*# 4#.*#$%1' 9*# +' 4/$%/"&/7" #"%1# G. < : G.G #$"#&#$'1/' $/ #$%'.,$ /"%#1#$'4,$ #" +,$ !CY< 4#+ 2*#3,;

Ejemplo 4.17

!" #+ 2*#3, 4#+ 1#('1%, &," !%\8 ?': &/"&, $*@2*#3,$ (1,(/,$8 #" %,4,$ +,$ &*'+#$ 2*#3' $,+'.#"%# ]L8 *", %1'$ &'4' *"' 4# +'$ (,$/@+#$ 2*3'4'$ 4# ]I; <# .*#$%1'" #" +'[/3*1' \;IZ;

Sí Sí Sí Sí SíResto Resto Resto Resto Resto

J1

J2J2J2J2J2

4 0 3 1 2 2 1 3 0 40 4 1 3 2 2 3 1 4 0

0 1 2 3 4

Figura 4.15 Subjuegos propios del juego del reparto con 4 monedas.

E,$ !C 4# 4/&?,$ $*@2*#3,$ $,"8 4# /09*/#14' ' 4#1#&?'Q

_#+ (1/.#1 $*@2*#3, N%1'$ +' 2*3'4' ^ 4# ]IOQ 5.%)+;_#+ $#3*"4, $*@2*#3, N%1'$ +' 2*3'4' I 4# ]IOQ 5.%)+;_#+ %#1&#1 $*@2*#3, N%1'$ +' 2*3'4' L 4# ]IOQ 5.%)+ : 29 ;_#+ &*'1%, $*@2*#3, N%1'$ +' 2*3'4' M 4# ]IOQ 29 ;_#+ 9*/"%, $*@2*#3, N%1'$ +' 2*3'4' \ 4# ]IOQ 29 ;

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

>$) (*#$8 NL8 5.%)+A 5.%)+A 5.%)+A29 A29 O : NL8 5.%)+A 5.%)+A29 A29 A29 O $," +,$ F"/&,$!CY< #" #$%1'%#3/'$ (*1'$ 4#+ 2*#3,; W@$B1-#$# 9*# $/ $# (,"# #" (15&%/&' &*'+9*/#1'4# #$%,$ 4,$ #9*/+/@1/,$8 #+ 4#$'11,++, 1#$*+%'"%# #$ 9*# ]I 2*#3' L N,61#&# L .,"#4'$O: ]L 4/&# 29 , 5.%)+8 : #+ 1#$*+%'4, #" %B1./",$ 4# ('3,$ #$ 9*# '.@,$ 1#&/@#" L

.,"#4'$;

Ejemplo 4.18

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Sí Sí Sí Sí Sí No No No No No

J1

J2J2J2J2J2

4 0 3 0 2 0 1 0 0 0

0 0 1 0 2 0 3 0 4 0

0 1 2 3 4

Figura 4.16 Subjuegos propios del juego del ultimátum con 4 monedas.

E,$ !C 4# 4/&?,$ $*@2*#3,$ $,"8 4# /09*/#14' ' 4#1#&?'Q

_#+ (1/.#1 $*@2*#3, N%1'$ +' 2*3'4' ^ 4# ]IOQ 29 : *+;

_#+ $#3*"4, $*@2*#3, N%1'$ +' 2*3'4' I 4# ]IOQ 29 ;

_#+ %#1&#1 $*@2*#3, N%1'$ +' 2*3'4' L 4# ]IOQ 29 ;

_#+ &*'1%, $*@2*#3, N%1'$ +' 2*3'4' M 4# ]IOQ 29 ;

_#+ 9*/"%, $*@2*#3, N%1'$ +' 2*3'4' \ 4# ]IOQ 29 ;

G,4,$ +,$ !C 4#+ 2*#3, 3+,@'+ &*:' 1#$%1/&&/7" ' &'4' *", 4# +,$ $*@2*#3,$ (1,(/,$$#' *" !C 4# 4/&?, $*@2*#3, ?'" 4# $#1 (#16/+#$ '0% N%:I 8 %:L O &*:' $#3*"4' &,.(,"#"A%# %:L $#' , @/#" 29 A29 A29 A29 A29 , @/#" *+A29 A29 A29 A29 ; >4#.5$8 +' (1/.#1' &,.(,"#"%# %:I#$ "#&#$'1/'.#"%# *"' 1#$(*#$%' 7(%/.' 4# ]I ' %:L 8 : #" &,"$#&*#"&/' #$ , @/#" +'#$%1'%#3/' ^ 4# ]I , @/#" +' #$%1'%#3/' I 4# ]I;

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Rm ", $#1)' %'.@/B" #" #+ #2#.(+, '"%#1/,1 *" !CY< #+ (#16/+ NL8 *+A *+A 29 A29 A29 O 9*#(,41)' $#1 &,"$#&*#"&/' 4#+ $/3*/#"%# '"*"&/, 4# ]L8 (1#-/, '+ &,./#"0, 4#+ 2*#3,Q `<7+,'&#(%'1B 4/&/#"4, 29 +' ,6#1%' 4# ]I8 $/ 4/&?' ,6#1%' #$ L , .5$aS E' 1#$(*#$%' #$ 9*# ",8:' 9*# #$# '"*"&/,A'.#"'0' ", 1#$*+%' &1#)@+#8 : ", #$ &1#)@+# (,19*# ]L $# #$%5 &,.(1,A

.#%/#"4, ' 4#&/1 *+ '*" &*'"4, ]I +# ,61#&/#1' I8 +, &*'+ +# (#12*4/&'1)' N&," *+ ,@%#"A41)' *" ('3, 4# ^ ./#"%1'$ 9*# &," 29 ,@%#"41)' IO;

Comentario acerca del significado de ENPS

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Teorema 4.1 Q/*2/3$ +/ /Y05./)'0$ +/ ;KVW

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4.3. JUEGOS DINÁMICOS CON INFORMACIÓN COMPLETAY PERFECTA. INDUCCIÓN HACIA ATRÁS

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Definición 4.4

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Teorema 4.2

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4, ('1' #+ 2*#3,O8 +, 9*# 3'1'"%/0' 9*# ' #$ *" !CY< 4#+ 2*#3, 1;Y,1 F+%/.,8 $/ "/"3F" 2*3'4,1 %/#"# .5$ 4# *"' '&&/7" 7(%/.' #" &'4' #%'(' 4#+(1,&#$, :8 (,1 %'"%,8 #+ .B%,4, 4# /"4*&&/7" ?'&/' '%15$ /4#"%/6/&' *" F"/&, (#16/+ 4##$%1'%#3/'$ ('1' #+ 2*#3,8 %'+ !CY< #$ F"/&,;

Ejemplos de aplicación de la inducción hacia atrás

G'+ &,., +' ?#.,$ 4#6/"/4,8 +' /"4*&&/7" ?'&/' '%15$ ", #$ '(+/&'@+# "/ '+ 2*#3, 4# 4/$*'A

$/7" L8 4#+ !2#.(+, \;I8 "/ '+ 4/+#.' 4#+ (1/$/,"#1, 1#(#%/4,8 4#+ !2#.(+, \;\8 4#@/4, '9*# '.@,$ 2*#3,$ $," 4# /"6,1.'&/7" /.(#16#&%'; <) +, #$ '+ 1#$%, 4# +,$ 2*#3,$ 4#6/"/4,$#" #$%# &'()%*+,8 (,1 $#1 %,4,$ 4# /"6,1.'&/7" (#16#&%'; !" +'$ 6/3*1'$ 4# +,$ #2#.(+,$9*# $/3*#"8 $# $#n'+'15" +'$ '&&/,"#$ 7(%/.'$ #"31,$'"4, +'$ 1'.'$ 4# #$'$ '&&/,"#$8 :$# /"4/&'15" +,$ ('3,$ 1#$*+%'"%#$ 4# &'4' (,4' &,+,&'"4, *"' &'2' &," +,$ ('3,$ 2*"%, '+&,11#$(,"4/#"%# ",4,;

Ejemplo 4.19

$% !" #+ 2*#3, 4# 4/$*'$/7" I8 4#+ !2#.(+, \;I8 #+ F"/&, ",4, 4# 4#&/$/7" 9*# $7+,

(1#&#4# ' ",4,$ %#1./"'+#$ #$ #+ 4# UCHJKVPWC; <* '&&/7" 7(%/.' #$ ,+"4.)$# %6&0v.8 : +' (,4' &,11#$(,"4/#"%# &,"4*&# '+ 2*#3, 1#4*&/4, 9*# '('1#&# #" +' ('1%# N'O 4#+' [/3*1' \;IX $,@1# +' +)"#' 4# (*"%,$; >?,1'8 #+ F"/&, ",4, 4# 4#&/$/7" 9*# 9*#4' #$#+ /"/&/'+8 : $* '&&/7" 7(%/.' #$ (!)#&# 8 &," +, &*'+ #+ ",4, /"/&/'+ ('$' ' $#1 ",4,%#1./"'+8 &," ('3,$ NZ8 ZO8 : '&'@' #+ (1,&#$,; !" +' ('1%# N@O $# ,@$#1-' #+ (1,&#$,&,.(+#%,8 9*# (#1./%# '(1#&/'1 9*# #+ F"/&, 1#$*+%'4, (#16#&%, #" $*@2*#3,$ #$ (!0

)#&# r,+"4.)$# %6&v.8 9*# #+ -#&%,1 4# ('3,$ 1#$*+%'"%# #$ NZ8 ZO8 : 9*# #+ F"/&, !CY<#$ N(!)#&# 8 ,+"4.)$# %6&v.O;&% <#' #+ 2*#3, 4#+ %1#$(/B$ 4#+ !2#.(+, \;L8 &*:, 51@,+ ?#.,$ &,+,&'4, '9*) #" 6,1A.' -#1%/&'+ (,1 &,"-#"/#"&/' 4#+ 4/@*2,; Y1,&#4/#"4, 4# .'"#1' '"5+,3' ' +' 4#$&1/%'

#" #+ '('1%'4, '"%#1/,18 %#"41)'.,$ #+ 1#$*+%'4, 4# +' (1/.#1' (,4' #" N'O8 #+ 4# +'$#3*"4' #" N@O : #+ 4# +' %#1&#1' #" N&O;

ENTRON ENTRON

INCUMBRON INCUMBRON

No entrar No entrar

Competir duro

Competir duro

Entrar Entrar

Competir suave

Competir suave

3, 7 3, 7

0, 0 0, 0

5, 5 5, 5

5, 5

5, 5 5, 5

(a)(a) (b)

Figura 4.17 Inducción hacia atrás en el juego de disuasión 1.

TTT

TT

T

TTT

SSS

SS

S

SSS

1, 11, 11, 1

0, 30, 30, 3

2, 22, 22, 2

0, 30, 3

2, 22, 22, 2

1, 41, 41, 4

1, 1

(a) (c)

J1J1J1

J2J2

J2

J1J1J1

(b)

Figura 4.18 Inducción hacia atrás en el juego del trespiés.

Y,1 %'"%,8 #+ F"/&, 1#$*+%'4, (#16#&%, #" $*@2*#3,$ #$ G8 #+ -#&%,1 4# ('3,$ 1#$*+%'"A%# #$ NI8 IO8 : #+ F"/&, !CY< #$ NG .G8 GO

Ejemplo 4.20

!" #+ 2*#3, 4#+ &/#.(/B$ 4#+ !2#.(+, \;M8 '"5+,3, '+ 4#+ %1#$(/B$8 #+ (1,&#$, 4# /"4*&A&/7" ?'&/' '%15$ &,"4*&#8 #=(1#$'4, 4# .,4, 1#$*./4, N&,+,&'"4, #" *"' &'2' $,@1#

&'4' ",4, 9*# $# &,"-/#1%# #" %#1./"'+8 +,$ ('3,$ 6/"'+#$ #" &'$, 4# 9*# '?) %#1./"'1'#+ 2*#3,O8 ' +' $/3*/#"%# $/%*'&/7"Q

J1 S S S S S S S S J1 J1 J1J2 J2 J2 J2

T T T T T T T T

1, 1

1, 1

0, 3

0, 3

2, 2

2, 2

1, 4

1, 4

3, 3

3, 3

2, 5

2, 5

50, 50

50, 50

49, 52

49, 52

51, 51

Figura 4.19 Inducción hacia atrás en el juego del ciempiés.

4,"4# #+ F"/&, 1#$*+%'4, (#16#&%, #" $*@2*#3,$ #$ G8 #+ -#&%,1 4# ('3,$ 1#$*+%'"%# #$ NI8 IO8: #+ F"/&, !CY< #$ #+ (#16/+ N =6>&# ? .! @6&/A6$.# !+8+7 =6>&# ? .! @6&/A6$.# !+8+O;

Ejemplo 4.21

!" #+ 2*#3, 4#+ 1#('1%,8 &," !%\ .,"#4'$8 #+ (1,&#$, 4# /"4*&&/7" ?'&/' '%15$ &,"4*A&#8 #=(1#$'4, 4# .,4, 1#$*./4,8 ' +'$ 4,$ $/%*'&/,"#$ (,$/@+#$ $/3*/#"%#$8 $#3F" $#' +''&&/7" 7(%/.' #+#3/4' (,1 #+ 2*3'4,1 L %1'$ +' '&&/7" L 4#+ 2*3'4,1 IQ


J1

J2J2J2J2J2

4 0 3 1 2 2 1 3 0 40 4 1 3 2 2 3 1 4 0

0 1 2 3 4

0, 4 1, 3 2, 2

2, 2

1, 3 0, 4


J1

J2J2J2J2J2

4 0 3 1 2 2 1 3 0 4

0 4 1 3 2 2 3 1 4 0

0 1 2 3 4

0, 4 1, 3 2, 2

2, 2

1, 3 0, 4

(a)

(b)

Figura 4.20 Inducción hacia atrás en el juego del reparto, con n %4 monedas.

!" +' $/%*'&/7" N'O #+ 1#$*+%'4, (#16#&%, #" $*@2*#3,$ #$ L r 5.%)+8 #+ -#&%,1 4#('3,$ 1#$*+%'"%# #$ NL8 LO8 : #+ !CY< #$ #+ (#16/+ NL8 5.%)+05.%)+05.%)+029029 O8 ./#"%1'$9*# #" +' $/%*'&/7" N@O #+ 1#$*+%'4, (#16#&%, #" $*@2*#3,$ #$ Lr 29 8 #+ -#&%,1 4# ('3,$1#$*+%'"%# #$ NL8 LO8 : #+ !CY< #$ #+ (#16/+ NL8 5.%)+05.%)+029029029 O; !" &,"&+*$/7"8 +,$4,$ !CY< $," #+ (#16/+ NL8 5.%)+05.%)+05.%)+029029 O : #+ (#16/+ NL8 5.%)+05.%)+029029029 O;

Ejemplo 4.22

!" #+ 2*#3, 4#+ *+%/.5%*.8 &," !% \ .,"#4'$8 #+ (1,&#$, 4# /"4*&&/7" ?'&/' '%15$&,"4*&#8 #=(1#$'4, 4# .,4, 1#$*./4,8 ' +'$ 4,$ $/%*'&/,"#$ (,$/@+#$ $/3*/#"%#$8 $#3F"$#' +' '&&/7" 7(%/.' #+#3/4' (,1 #+ 2*3'4,1 L %1'$ +' '&&/7" ^ 4#+ 2*3'4,1 IQ

0, 0 3, 1 2, 2 1, 3

3, 1

0, 4


J1

J2J2J2J2J2

4 0 3 0 2 0 1 0 0 00 0 1 0 2 0 3 0 4 0

0 1 2 3 4

4, 0 3, 1 2, 2

4, 0

1, 3 0, 4


J1

J2J2J2J2J2

4 0 3 0 2 0 1 0 0 00 0 1 0 2 0 3 0 4 0

0 1 2 3 4

(a)

(b)

Figura 4.21 Inducción hacia atrás en el juego del ultimátum, con n %4 monedas.

!" +' $/%*'&/7" N'O #+ 1#$*+%'4, (#16#&%, #" $*@2*#3,$ #$ Ir 29 8 #+ -#&%,1 4# ('3,$1#$*+%'"%# #$ NM8 IO8 : #+ !CY< #$ #+ (#16/+ NI8 *+0 29029029029 O8 ./#"%1'$ 9*# #" +' $/%*'A&/7" N@O #+ 1#$*+%'4, (#16#&%, #" $*@2*#3,$ #$ ^r 29 8 #+ -#&%,1 4# ('3,$ 1#$*+%'"%# #$N\8 ^O8 : #+ !CY< #$ #+ (#16/+ N^8 29029029029029 O; !" &,"&+*$/7"8 +,$ 4,$ !CY< $," #+(#16/+ NI8 *+029029029029 O : #+ (#16/+ N^8 29029029029029 O;

Ejemplo 4.23

!" #+ 2*#3, 4#+ 1#('1%, #" -#1$/7" &,"%/"*'8 4#+ !2#.(+, \;Z8 #+ (1,&#$, 4# /"4*&&/7"?'&/' '%15$ #$ #+ $/3*/#"%#Q

H,.#"0'.,$ #" +,$ ",4,$ 4# 4#&/$/7" 4# ]L; <#' *" ",4, &*'+9*/#1'8 %1'$ ?'@#1#$&1/%, ]I #+ "F.#1, # ; E' '&&/7" 7(%/.' 4# ]L $#15 29 8 $/ # b Z^8 $#15 5.%)+ $/ # a Z^ :$#15" '.@'$ $/ # % Z^; Y,4'"4, '?,1' #+ 51@,+8 %#"41#.,$ *" 51@,+ 1#4*&/4, #" #+ 9*##+ ",4, 9*# $/3*# ' &*'+9*/#1 2*3'4' 4# ]I #$ *" ",4, %#1./"'+8 : +,$ ('3,$ &,11#$(,"A4/#"%#$ ' #$# ",4, $," NI^^. # 8 # O %$ # b Z^8 N# 8 I^^. # O $/ # a Z^8 : NZ^8 Z^O $/# %Z^; !" &,"$#&*#"&/'8 +' 2*3'4' 7(%/.' F"/&' 4# ]I #$ # % Z^; >$) (*#$8 +,$ F"/&,$1#$*+%'4,$ (#16#&%,$ #" $*@2*#3,$ $," Z^r 29 : Z^r 5.%)+; !+ -#&%,1 4# ('3,$ 1#$*+A%'"%# #$ NZ^8 Z^O #" '.@,$ &'$,$8 : +,$ 4,$ F"/&,$ !CY< $," NZ^8 `29 $/ # n Z^8 : 5.%)+

#" &'$, &,"%1'1/,aO : NZ^8 `29 $/ # b Z^8 : 5.%)+ #" &'$, &,"%1'1/,aO; !" +' [/3*1' \;LL/"%#"%' 1#(1#$#"%'1$# 3156/&'.#"%# +' $/%*'&/7";

Sí

Sí

SíSí

Sí

Resto

Resto

Resto

Resto

Resto

J1

J2

J2

J2

J2

J2

r, 100 r 2

100 r, r 2

50, 50

50, 50

r 05

r 505

r 1005

r > 50r < 50

100r

2 r 100 2 r r

r r 100 2

r r 100 250 50

50 50

1000

1000

0100

0100

Figura 4.22 Inducción hacia atrás en el juego del reparto, versión continua.

4.4. JUEGOS DINÁMICOS CON INFORMACIÓN COMPLETA PEROIMPERFECTA. INDUCCIÓN HACIA ATRÁS GENERALIZADA

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

Inducción hacia atrás generalizada

Definición 4.5

_'4, *" 2*#3, 1 6/"/%, #" 6,1.' #=%#"$/-' &," /"6,1.'&/7" &,.(+#%'8 (#1, ", "#&#A$'1/'.#"%# (#16#&%'8 ++'.'.,$ $<6*20.3* +/ 0)+-''04) U$'0$ $.2B5 6/)/2$<0D$+* '+9*# (1,&#4# '$)Q

!" <# /4#"%/6/&'" %,4,$ +,$ $*@2*#3,$ 9*# $# (1,4*&#" #" F+%/., +*3'1 N#$ 4#&/18'9*#++,$ 9*# &,./#"0'" #" +,$ ",4,$ 4# 4#&/$/7" +, .5$ &#1&'",$ (,$/@+# ' +,$ ",4,$

%#1./"'+#$; !$%,$ $*@2*#3,$ (*#4#" %#"#1 *", , -'1/,$ 2*3'4,1#$; <# &'+&*+'" +,$ !C4# 4/&?,$ $*@2*#3,$;

#" > &,"%/"*'&/7"8 $/ $7+, #=/$%# *" F"/&, !C #" #$%1'%#3/'$ (*1'$ #" &'4' $*@A 2*#3,8 $# #+/./"' &'4' *", 4# #$,$ $*@2*#3,$8 $'+-, $* ",4, 4# &,./#"0, 9*# #$1##.(+'0'4, (,1 #+ ",4, %#1./"'+ 4#+ 2*#3, 3+,@'+ '+ 9*# $# ?'@1)' ++#3'4, 4# ?'@#1$# 2*3'4, #+ (#16/+ !C &,11#$(,"4/#"%# ' #$# $*@2*#3,8 : $# +# '%1/@*:#" +,$ ('3,$ 4#4/&?, (#16/+; _# #$%' .'"#1' $# ?'" (,4'4, +'$ 1'.'$ 4#+ 51@,+ &,11#$(,"4/#"%#$ ' +,$$*@2*#3,$ 6/"'+#$ 4#+ 2*#3, 3+,@'+ /"/&/'+8 : ",$ #"&,"%1'.,$ &," *" 51@,+ .5$ &,1%,;

(" <# 1#(/%# &," #+ 51@,+ 1#4*&/4, +, 4/&?, #" +'$ #%'('$ '"%#1/,1#$8 : $# &,"%/"F'&," #$%# (1,&#$, ?'$%' 9*# $# ++#3' '+ ",4, /"/&/'+ 4#+ 2*#3, 4# ('1%/4';

>&'@'4, #+ (1,&#$,8 %#"41#.,$ *",$ ('3,$ '$,&/'4,$ '+ ",4, /"/&/'+ 4#+ 2*#3,8 :*"'$ 1'.'$ 4#+ 51@,+ $#n'+'4'$ &,., &,.(,"#"%#$ 4# +,$ !C 4# &'4' $*@2*#3,; Y*#$@/#"8 #+ F"/&, 4#$'11,++, 4#+ 2*#3, N&'./", 4#$4# #+ ",4, /"/&/'+ ?'$%' *" ",4, %#1./"'+O

&,"$/$%#"%# #" 1'.'$ $#n'+'4'$ #$ #+ F"/&, 1#$*+%'4, (#16#&%, #" $*@2*#3,$8 : +,$ ('3,$'$,&/'4,$ '+ ",4, /"/&/'+ $," +,$ 9*# &,11#$(,"4#1)'" ' #$# 4#$'11,++, 4#+ 2*#3,; Y,1 ,%1'('1%#8 #+ F"/&, (#16/+ 4# #$%1'%#3/'$ #" #+ 9*# +' #$%1'%#3/' 4# &'4' 2*3'4,1 &,"$/$%# #" 2*3'1 +' '&&/7" /"4/&'4' #" &'4' *", 4# $*$ &,"2*"%,$ 4# /"6,1.'&/7"8 #$ #+ F"/&, #9*/+/A@1/, 4# C'$? (#16#&%, #" $*@2*#3,$;

P&5/2?$'04) R"FG

!" !" #+ &'$, 4# 9*# #=/$%'" 3Z<.09</5 ;K /) $<6-)* * ?$20*5 +/ <*5 5-&,-/6*58 #+(1,&#$, $7+, &'.@/' #" 9*# ?': 9*# &,"$/4#1'1 %,4'$ +'$ (,$/@/+/4'4#$8 #$ 4#&/18 $#6,1.'" %'"%,$ 51@,+#$ 1#4*&/4,$ &,., &,.@/"'&/,"#$ ?':' 4# !C #" +' #%'(' '&%*'+;

>+ 6/"'+8 ,@%#"41#.,$ 1#$*+%'4,$ (#16#&%,$ #" $*@2*#3,$ N%,4,$O : #9*/+/@1/,$ 4#C'$? (#16#&%,$ #" $*@2*#3,$;

#" !" #+ &'$, 4# 9*# #=/$%'" !C #" #$%1'%#3/'$ ./=%'$8 #+ (1,&#$, ", -'1)' $'+-, #" 9*#4#@#1#.,$ '%1/@*/1 '+ ",4, 4# &,./#"0, 4# +,$ $*@2*#3,$ &," !C #" #$%1'%#3/'$ ./=A%'$ +,$ ('3,$ #$(#1'4,$ &,11#$(,"4/#"%#$ ' #$# !C;

Teorema 4.3

</ *" 2*#3, '4./%# +' /"4*&&/7" ?'&/' '%15$ 3#"#1'+/0'4'8 : %,4,$ : &'4' *", 4# $*$$*@2*#3,$ 6/"'+#$ N%'"%, #" #+ 2*#3, 3+,@'+ &,., #" +,$ 1#4*&/4,$O '4./%#" *" !C

F"/&,8 #+ 1#$*+%'4, .#4/'"%# /"4*&&/7" 4#+ 2*#3, #$ #+ F"/&, 1#$*+%'4, (#16#&%, #"$*@2*#3,$ : +'$ #$%1'%#3/'$ 3#"#1'4'$ ' ('1%/1 4# +'$ '&&/,"#$ %,.'4'$ (,1 &'4' 2*3'A4,1 #" &'4' *", 4# $*$ &,"2*"%,$ 4# /"6,1.'&/7" &,"$%/%*:#" #+ Z)0'* #9*/+/@1/, 4#C'$? (#16#&%, #" $*@2*#3,$;

H,., #" #+ &'$, 4# +' /"4*&&/7" ?'&/' '%15$ ('1' 2*#3,$ &," /"6,1.'&/7" (#16#&%'8 ",(,4#.,$ '$#3*1'1 9*# 4/&?, #9*/+/@1/, $# (1,4*0&' #" #$%1'%#3/'$ (*1'$8 ' ", $#1 9*#'n'4'.,$ '+3*"' #=/3#"&/' '4/&/,"'+; Y,1 ,%1' ('1%#8 &'@# (1#3*"%'1$#8 '+ /3*'+ 9*# $#?/0, #" #+ &'$, 4# /"6,1.'&/7" (#16#&%'8 $/ #+ (1,&#$, 4# /"4*&&/7" ?'&/' '%15$ 3#"#1'+/A0'4' $/3*# $/#"4, -5+/4, &*'"4, #" '+3F" ",4, 4# 4#&/$/7" ?':' *" "F.#1, /"6/"/%, 4#

'&&/,"#$ 6'&%/@+#$ N(#1, .'"%#"/#"4, 6/"/%' +' +,"3/%*4 4# &*'+9*/#1 4#$'11,++, (,$/@+#4#+ 2*#3,O; E' 1#$(*#$%' #$ '"5+,3' ' +' 9*# $# 4/, #"%,"&#$Q #+ (1,&#$, $/3*# $/#"4,

-5+/4, $/#.(1# 9*# #=/$%'" +'$ '&&/,"#$ 7(%/.'$ : +,$ #9*/+/@1/,$ 4# C'$? #" +,$ &*'+#$$# @'$'; !" #+ !2#.(+, \;Lh $# '"'+/0' *" 2*#3, &," #$('&/,$ &,"%/"*,$ 4# '&&/,"#$6'&%/@+#$;

Ejemplos de aplicación de la inducción hacia atrás generalizada

Ejemplo 4.24

<#' #+ $/3*/#"%# 2*#3, #=(1#$'4, #" 6,1.' #=%#"$/-' N[/3*1' \;LMOQ

J1

J1 J1

J2 J2

I DC

6, 4 1, 2

5, 4

2, 3 4, 23, 10, 6

I ID D

i id d

Figura 4.23 Juego con información imperfecta.

</ '"'+/0'.,$ #+ 2*#3,8 -#.,$ 9*# %/#"# 4,$ $*@2*#3,$ (1,(/,$; !" #+ (1/.#1, 4##++,$ 2*#3' $7+, ]L N%1'$ +' 4#&/$/7" U 4# ]IO8 ./#"%1'$ 9*# #+ ,%1, N%1'$ +' 4#&/$/7" _ 4#]IO #$ *" 2*#3, 4# 4#&/$/,"#$ $/.*+%5"#'$ : (,1 %'"%, &," *" &,"2*"%, 4# /"6,1.'&/7"", *"/%'1/, ('1' *", 4# +,$ 2*3'4,1#$8 #" #$%# &'$, ('1' ]I; !$ 2*$%'.#"%# #$%# $#3*"4,$*@2*#3, #+ 9*# ",$ /"4/&' 9*# *" .,4, '(1,(/'4, 4# 4#%#1./"'1 +,$ !CY< 4#+ 2*#3,#$ #+ '+3,1/%., 4# /"4*&&/7" ?'&/' '%15$ 3#"#1'+/0'4,;

E' '(+/&'&/7" 4# +' /"4*&&/7" ?'&/' '%15$ 3#"#1'+/0'4' $*(,"# #+ $/3*/#"%# 4#$'11,++,Q

!" >"'+/0'1 +,$ $*@2*#3,$ 6/"'+#$; !" #$%# &'$, +,$ F"/&,$ 9*# #=/$%#" $,"Q

d !+ $*@2*#3, 9*# &,./#"0' %1'$ +' 4#&/$/7" 4# U 4# ]I; !" #$%# $*@2*#3,8 ]L 4#&/4/A

15 +' '&&/7" $8 (*#$ +# (#1./%# ,@%#"#1 *" ('3, N/3*'+ ' \O $*(#1/,1 '+ 4# 8 N/3*'+ ' LO;d !+ $*@2*#3, 9*# &,./#"0' %1'$ +' 4#&/$/7" _ 4# ]I8 9*# #$ *" 2*#3, &," /"6,1A.'&/7" /.(#16#&%' &," +' $/3*/#"%# 6,1.' #$%1'%B3/&'Q

@#

$ 8

@!Uñ L8 M M8 I

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Figura 4.24 Inducción hacia atrás generalizada en el juego con información imperfecta.

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Ejemplo 4.25

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ENTRON ENTRON

No entrar No entrar Entrar Entrar

3, 7 3, 71, 4 4, 2

(a) (b)

Figura 4.25 Juegos reducidos correspondientes al EN (B, A) y al EN (A, B).

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INCUMBRON INCUMBRON

No entrar No entrar Entrar Entrar

3, 7 3, 7

22, 22 22, 222 21, 1 2 21, 14, 2 4, 21, 4 1, 4

A A

A AA A

B B

B BB B

1, 4 4, 2

4, 23, 7

(a) (b)

Figura 4.26 Inducción hacia atrás generalizada en el juego de disuasión 2.

Ejemplo 4.26

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2 6 1 5 6 10 5 9 1 5 0 4 5 9 4 82 1 6 5 1 0 5 4 6 5 10 9 5 4 9 8

22

55

22

61

16

Figura 4.27 Inducción hacia atrás generalizada en el dilema del Prisionero, repetido 2 veces.

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Ejemplo 4.27

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Fútbol

Fútbol Fútbol



1 1 1 1 0 0 0 0 0 0 0 0 2 2 2 23 7 2 6 1 5 0 4 1 5 0 4 2 6 1 51 0 5 4 1 0 5 4 1 0 5 4 1 0 5 4

Figura 4.28. Inducción hacia atrás generalizada.

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Ejemplo 4.28

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Un modelo simplificado

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4.6. APLICACIONES. EL MODELO DE LEONTIEF

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5Juegos estáticoscon información

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5.1. INTRODUCCIÓN

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Jugadas de azar en un nodo terminal

Ejemplo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

J1

J2 J2Con Ca

Con Ca CaCon

1 5 01 0 5

4 104 10

Azar

1/ 3 p 5 2 /3

Figura 5.1 Inserción de una jugada de azar en un nodo terminal del dilema del prisionero.

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1 5 0 61 0 5 6

Figura 5.2 Representación equivalente con pagos esperados en el nodo terminal.

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Jugadas de azar en un nodo de decisión no terminal

a) Resultado de las jugadas de azar desconocido por todos los jugadores

Ejemplo 5.2

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Juegos estáticos con información incompleta 277

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Azar

INCUMBRONINCUMBRON

ENTRONENTRON

No entrar

Competir duro

Competir duro

No entrar Entrar

Competir suave

Competir suave

Entrar

3, 35, 5 2 24, 20, 0

1, 3

3, 7

F(1/2) D (1/2)

Figura 5.3 Inserción al inicio de una jugada de azar, con resultados desconocidos.

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4, 42 22, 1

2, 5

Figura 5.4 Representación equivalente con pagos esperados.

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b) Resultado de las jugadas de azar conocido por todos los jugadores

Ejemplo 5.3@$"%.5,-,2$% '/$-' +"' (,0+,H' D'-.'&.3" ," ,# 7+,8$ '"*,-.$-T *$5$ ,% .8+'#4 %'#D$0+, '/$-' '26$% 7+8'5$-,% $6%,-D'"4 '"*,% 5, 5,&.5.- %+ 7+8'5'4 ,# -,%+#*'5$ 5, #' 7+8'5' 5, 'A'-< E' -,(-,%,"*'&.3" ," 1$-2' ,>*,"%.D' ,%T

Azar

INCUMBRONINCUMBRON

ENTRONENTRON

No entrar

Competir duro

Competir duro

No entrar Entrar

Competir suave

Competir suave

Entrar

3, 35, 5 2 24, 20, 0

1, 3

3, 7

F(1/2) D (1/2)

Figura 5.5 Inserción al inicio de una jugada de azar, con resultados conocidos por ambos.

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!QUVWQ *.,", G ,%*-'*,8.'% (+-'%4 0+, %$" -' *(.$"$4-' *(.$"$ 4 -' *(.$"$4/(.$"$ 4/(.$"$4-' *(.$"$ : /(.$"$4/(.$"$ 4 2.,"*-'% 0+, XQ@YRZVWQ *.,", $*-'% &+'*-$4 0+,%$" !'0%*.1$ 23$'4!'0%*.1$ 23$'4 !'0%*.1$ 23$'4!'0%*.1$ +3"v*4 !'0%*.1$ +3"v*4

!'0%*.1$ 23$' : !'0%*.1$ +3"v*4!'0%*.1$ +3"v*< !# 7+,8$ (+,5, -,%$#D,-%, %." 5.1.;&+#*'5 ($- ."5+&&.3" /'&.' '*-C%4 0+, (-$($-&.$"' #$% %.8+.,"*,% -,%+#*'5$%T

^".&$ ,0+.#.6-.$ 5, Q'%/ (,-1,&*$ ," %+67+,8$%T

?/(.$"$4/(.$"$ 4 !'0%*.1$ +3"v*4!'0%*.1$ +3"v*B

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/(.$"$ r!'0%*.1$ +3"v*4 &$" D,&*$- 5, ('8$% 1."'#,% ?N4 NB

^".&$ -,%+#*'5$ (,-1,&*$ ," %+67+,8$% %. ,# -,%+#*'5$ 5, #' 7+8'5' 5, 'A'- ,% [T

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/(.$"$ r!'0%*.1$ +3"v*4 &$" D,&*$- 5, ('8$% 1."'#,% ,%(,-'5$% ?G4 GB

R,-,&, #' (,"' $6%,-D'- 0+, ," ,%*, ,7,2(#$ #'% ,%*-'*,8.'% 5, &'5' 7+8'5$- /'"5,6.5$ 5,%&-.6.-%, &$2$ '&&.$",% 0+, 5,(,"5," 5, #$% -,%+#*'5$% 5, #' 7+8'5' 5, 'A'-4 ,%5,&.-4 &$2$ 1+"&.$",% ?$ 2,7$- 5.&/$4 '(#.&'&.$",%B 0+, '%.8"'" ' &'5' -,%+#*'5$ 5, #' 7+8'5' 5, 'A'- +"' '&&.3" 5.%($".6#, 5, 5.&/$ 7+8'5$-< O5,2C%4 5,6.5$ ' 0+, *$5$% #$% 7+8'5$-,% &$"$&," ,# -,%+#*'5$ 5, #' 7+8'5' 5, 'A'- '"*,% 5, *,",- 0+, *$2'- 5,&.%.$",%4#'% ,%*-'*,8.'% : #$% ,0+.#.6-.$% "$ 5,(,"5," 5, #' 5.%*-.6+&.3" 5, (-$6'6.#.5'5 5, #$%-,%+#*'5$% 5, #' 7+8'5' 5, 'A'-<

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c) Resultado de las jugadas de azar conocido por algunos jugadores, pero no todos

P'2$% ' ,>(#$-'-4 ," 5$% ,7,2(#$% %,"&.##$%4 +"$ ,%*C*.&$ : ,# $*-$ 5."C2.&$4 #'% &$"%,;

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Ejemplo 5.4

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

Azar

J2 J2J2 J2

J1 J1

ALTA ( 2/ 3) p 5 BAJA (1 1/3)2 5 p

Con Ca CaCon


1 5 0 4 1 5 0 101 0 5 4 1 0 5 10

Figura 5.6 Inserción de azar, con resultado que sólo J1 conoce, en un caso estático.

5$"5, %, /' %+(+,%*$ 0+, #' 7+8'5' 5, 'A'- %, 2'*,-.'#.A' 2,5.'"*, ,# #'"A'2.,"*$ 5,+" 5'5$ ,0+.#.6-'5$4 $6%,-D'"5$ %. /' %'#.5$ +"' (+"*+'&.3" '#*' ?0+, %.8".1.&'2':$- 0+, I4 #$ 0+, $&+--, &$" (-$6'6.#.5'5 I/ JB $ +"' (+"*+'&.3" 6'7' ?K 3 I4 #$ 0+,$&+--, &$" (-$6'6.#.5'5 K/ JB< W*-$ 2$5$ 5, 5,%&-.6.- #' %.*+'&.3" %,-)' 5,&.- 0+, %,D' ' 7+8'- +" 7+,8$ ,%*C*.&$ 5, 5$% 7+8'5$-,%4 5$"5, '26$% *.,"," #'% '&&.$",% 1'&*.;

6#,% !'()*+"$ : !"##"$ 4 0+, ,%, 7+,8$ D' ' %,- $ 6.," ,# 5.#,2' 5,# (-.%.$",-$4 &+:'1$-2' "$-2'# ,%

/01&'+) 2

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@'##'- L4 N 65 6

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8.'% (+-'%4 !'()*+"$ : !"##"$ 4 : %, D, $6#.8'5$ ' -'A$"'- ," *F-2."$% 5, ('8$% ,%(,-';5$%< E' 1$-2' "$-2'# 5,# 7+,8$ 0+, '&'6'2$% 5, ,%(,&.1.&'- ,% #' %.8+.,"*,T

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@'##'-;@$"1,%'- K/ J4 KK/ J KJ/ J4 c/ J

@'##'-;@'##'- L4 N S4 S

5$"5, #$% ('8$% 0+, '('-,&," ," #' *'6#' %$" ('8$% ,%(,-'5$%4 &$--,%($"5.,"*,% ' #'%.*+'&.3" (-,D.' ' &+'#0+.,- '&&.3" 5, #$% 7+8'5$-,% $ 5,# 'A'-< @$2$ D,2$%4 #$% 9".&$%

!Q ," ,%*-'*,8.'% (+-'% %$" #$% (,-1.#,% ?!'()*+"$4!'()*+"$ 4 !'()*+"$ B : ?!'()*+"$4!"##"$ 4 !"##"$ B<

Ejemplo 5.5

@$"%.5,-,2$% '/$-' +"' D'-.'&.3" ," ,# 7+,8$ 5,# !7,2(#$ N<J4 5, 2$5$ 0+, *$5$ %,'.8+'#4 %'#D$ 0+, '/$-' %3#$ ,# 7+8'5$- K ?!QUVWQB *.,", &$"$&.2.,"*$ 5,# -,%+#*'5$ 5,#' 7+8'5' 5, 'A'-4 2.,"*-'% 0+, ,# $*-$ 7+8'5$- ?XQ@YRZVWQB %3#$ &$"$&, #'% (-$6';6.#.5'5,% 0+, 8$6.,-"'" 5.&/' 7+8'5' ?-,%+#*'5$% ($%.6#,% M : [ &$" (-$6'6.#.5'5 K/ I

&'5' +"$4 ," ,%*, &'%$B< O5,2C%4 '26$% 7+8'5$-,% %'6," 0+, ,# 7+8'5$- K *$2'-C %+5,&.%.3" &$"$&.,"5$ ,# -,%+#*'5$ 5, #' 7+8'5' 5, 'A'-< E' -,(-,%,"*'&.3" ," 1$-2' ,>;*,"%.D' ,%T

Azar

INCUMBRONINCUMBRON

ENTRONENTRON

No entrar

Competir duro

Competir duro

No entrar Entrar

Competir suave

Competir suave

Entrar

3, 35, 5 2 24, 20, 0

1, 3

3, 7

F (1/2) D (1/2)

Figura 5.7 Inserción de azar, con resultado que sólo ENTRON conoce, en un caso dinámico.

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Otros ejemplos introductorios

Ejemplo 5.6 ?0<&*9& *"%(#"-".&'&

[$% #.&.*'"*,% '&+5,"4 ('-' &$2(-'- +" $67,*$ ?($- ,7,2(#$4 +"' (.,A' 5, '-*,B4 ' +"'%+6'%*' 0+, *.,", #'% %.8+.,"*,% -,8#'%T

d e'" 5, ,"*-,8'- ," %$6-, &,--'5$ %+ (+7' $ #.&.*'&.3"4 0+, (+,5, %,- L4 L4N $ K<d =, '6-," #$% %$6-,% : %, '57+5.&' ,# $67,*$ ' '0+,# #.&.*'"*, 0+, ,%&-.6.3 +"'

(+7' 2C% '#*'< =. #'% (+7'% %$" .8+'#,%4 %, '57+5.&' ' +"$ 5, ,##$% '# 'A'- &$" (-$6'6.#.;5'5 K/ I<

d !# #.&.*'"*, ' 0+.," %, '57+5.0+, ,# $67,*$ /' 5, ('8'- #' (+7' 0+, /.A$<

=+(3"8'%, 0+, #$% ('8$% $ 8'"'"&.'% 5,# 7+,8$ %$" #$% 6,",1.&.$% $6*,".5$%4 : 0+,,# (-.2,- #.&.*'"*, `K *.,", +"' D'#$-'&.3" (-.D'5' ?&'"*.5'5 2C>.2' 0+, ,%*'-)' 5.%;

(+,%*$ ' ('8'- ($- #' (.,A'B 5, L4N4 0+, ,% 5, 5$2.".$ (96#.&$4 2.,"*-'% 0+, ,# %,8+";5$4 `I4 *.,", +"' 5, 5$% ($%.6#,% D'#$-'&.$",% (-.D'5'%4 L $ K4 0+, %3#$ F# &$"$&,4 (,-$' #'% &+'#,% #$% 5,2C% '*-.6+:," (-$6'6.#.5'5,% .8+'#,%<

Ejemplo 5.7 @9)&* *0<&*9&*

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",2$% 0+, *'26.F" ,%*C ," ,# ."*,-D'#$ fL4Kg4 : *$5$% #$% 5,2C% #, '%.8"'" +"'5.%*-.6+&.3" 5, (-$6'6.#.5'5 +".1$-2,< E$% ('8$% $ 8'"'"&.'% 5,# 7+,8$ %$" #$% 6,",1.;&.$%4 : #'% -,8#'% 5,# 7+,8$ %$" #'% %.8+.,"*,%T

5*$+16( KT =, '6-," #$% %$6-,%4 %, '57+5.&' ,# $67,*$ ' +" #.&.*'"*, '# 'A'- ?*,".,"5$*$5$% #' 2.%2' (-$6'6.#.5'5B4 : %, #, /'&, ('8'- ' F%*, %+ (-$(.' (+7'<

5*$+16( IT =, '6-," #$% %$6-,% : %, '57+5.&' ,# $67,*$ ' '0+,# #.&.*'"*, 0+, ,%&-.6.3+"' (+7' 2C% '#*'< =. /': D'-.'% ?($- ,7,2(#$4 7B (+7'% .8+'#,% 0+, %$" #'% 2C% '#*'%4 %,'57+5.&' ,# $67,*$ ' +"$ 5, ,%*$% 7 7+8'5$-,% '# 'A'-4 &$" (-$6'6.#.5'5 K/ 7 5, 0+, %,'&+'#0+.,-' 5, ,##$%< !# #.&.*'"*, 0+, -,%+#*, 8'"'5$- ('8' +"' &'"*.5'5 8 ,"*-, L : K45,&.5.5' '# 'A'- ?5, '&+,-5$ &$" #' 5.%*-.6+&.3" 5, (-$6'6.#.5'5 +".1$-2,B<

Ejemplo 5.8 A, 80$1+ *$,."##+ '$ #& B$)'&'

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2,"*.-< E'% 8'"'"&.'%4 0+, 5,(,"5," *'"*$ 5, %+% 5,&#'-'&.$",% &$2$ 5,# /,&/$ 5, 0+,F%*'% %, '7+%*," ' #' D,-5'54 %$" #'% %.8+.,"*,%T

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_+, '#8+"$ 5, ,##$% /':' $6*,".5$ @OVO4 ," &+:$ &'%$ Ì -,&.6, L : `K -,&.6, G<

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Teoría de la decisión bayesiana

E' *,$-)' 5, #' 5,&.%.3" %, $&+(' 5, &32$ +" 5,&.%$- 5,6,-)' ,#,8.- +"' '&&.3" &$"&-,*',"*-, +" &$"7+"*$ 5, '&&.$",% ($%.6#,%4 &+'"5$ ,# -,%+#*'5$ 5, %+ ,#,&&.3" *'26.F" 5,;(,"5, 5,# ,%*'5$ 0+, '#&'"&, #' "'*+-'#,A'<

E$% ,#,2,"*$% 0+, &'-'&*,-.A'" +" (-$&,%$ 5, 5,&.%.3" %$"T!# 5,&.%$-4 0+, ,% ,# '8,"*, 0+, 5,6, *$2'- #' 5,&.%.3" : 0+, *-'*' 5, '#&'"A'- +"$%

$67,*.D$%<!# &$"7+"*$ 9 5, '&&.$",% $ '#*,-"'*.D'% 5.%($".6#,% ('-' ,# 5,&.%$-<!# &$"7+"*$ : 5, ,%*'5$% 5, #' "'*+-'#,A' ?1'&*$-,% $ D'-.'6#,% "$ &$"*-$#'5'% ($-

,# 5,&.%$- 0+, 5,1."," ,# ,"*$-"$ 5,# (-$6#,2'B<E'% &$"%,&+,"&.'% $ -,%+#*'5$% 0+, %, %.8+," 5, #' ,#,&&.3" 5, +"' '#*,-"'*.D' : #'

(-,%,"*'&.3" 5, +" ,%*'5$ 5, #' "'*+-'#,A'<

=,' #' 1+"&.3"

3 T 9#: r !?"4 ;B r 3?"4 ;B

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5.2. JUEGOS BAYESIANOS ESTÁTICOS. EQUILIBRIO BAYESIANODE NASH

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Tipos, conjeturas y pagos. Estrategias. Ejemplos

=. ," +" 7+,8$ ,%*C*.&$ &$" ."1$-2'&.3" &$2(#,*'4 #' -,(-,%,"*'&.3" ," 1$-2' "$-2'#,>.8)' &$"&-,*'- ,# "92,-$ ( 5, 7+8'5$-,%4 #'% '&&.$",% 91 5.%($".6#,% : #$% ('8$% 31?"K4<<<4 "14 <<<4 "(B ('-' &'5' 7+8'5$- 14 : '5,2C%4 #' ,%*-+&*+-' *,2($-'# 5,# 7+,8$ %+($")' 0+,#$% 7+8'5$-,% *$2'6'" %.2+#*C",'2,"*, %+% 5,&.%.$",% : ' &$"*."+'&.3" -,&.6)'" #$% (';8$%4 ," +" 7+,8$ 6':,%.'"$ /': 0+, 'H'5.- D'-.$% ,#,2,"*$% : '2(#.'- #' 5,%&-.(&.3"4 *'#&$2$ /'&, #' %.8+.,"*, 5,1.".&.3"<

Definición 5.1

!" +" 7+,8$ <&=$*"&,+ ,%*C*.&$4 &'5' 7+8'5$- 1 *.,",4 '5,2C% 5, '&&.$",% 91 : ('8$%

314 +" &$"7+"*$ 5, 9"(+* < 1 : +"' %+($%.&.3" $ .+,8$90)& %1 %$6-, ,# *.($ 5, #$% $*-$%

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@<*$)B&.";, GD2H

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ID =. < 14 &$"7+"*$ 5, *.($% 5, 14 ,% +".*'-.$4 #' 9".&' 5.%*-.6+&.3" 5, (-$6'6.#.5'5($%.6#, %$6-, ,%*, &$"7+"*$ ,% #' *-.D.'# 0+, ,%(,&.1.&' 0+, %+ 9".&$ ,#,2,"*$ *.,", (-$;6'6.#.5'5 K< !" ,%, &'%$4 *$5$% #$% 7+8'5$-,% &$"$&," ,# *.($ 5, 14 0+, ,% ,# 9".&$ ,#,;2,"*$ 5, < 1< \$- *'"*$4 #' 2'",-' 5, 2$5,#.A'- ,# /,&/$ 5, 0+, +" 7+8'5$- 1 "$ *,"8'"."8+"' ."1$-2'&.3" (-.D'5' &$"%.%*, ," '%.8"'-#, +" &$"7+"*$ 5, *.($% < 1 +".*'-.$<

@$"$&.5$ %+ *.($ . 14 #' &$"7,*+-' 5,# 7+8'5$- 1 %$6-, #$% *.($%

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Definición 5.2

!" +" 7+,8$ 6':,%.'"$ ,%*C*.&$4

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Ejemplo 5.10

!" ,# 7+,8$ 5,# !7,2(#$ N<G4 #$% &$"7+"*$% 0+, 5,%&-.6," #'% '&&.$",%4 *.($% : ,%*-'*,;8.'%4 '%) &$2$ #'% &$"7,*+-'% : ('8$%4 %$"T

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3K? 8 4 Dm OEUOB%3K? 8 4 Dm ZOÒB%3K?[\B? 8 4 DB

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Ejemplo 5.11

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`+8'5$-,%T C % qK4 Ir< E#'2,2$% `. '# 7+8'5$- 1

O&&.$",%T 9K% 9I% qL4 L4N4 Kr 5,"$*'5'% "1

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%K?K/ L4NB%(-$6?vI% K/ vK% L4NB% %?vI%KB%K/ I@$"7,*+-' 5, ÌT %I?L4N/ LB%(-$6?vK% L4N/ vI% LB% %?vK%L4NB%K

%I?L4N/ KB%(-$6?vK% L4N/ vI% KB% %?vK%L4NB%K

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vK."K %. "Kb "I

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!%*-'*,8.'% 5, `KT =+% '&&.$",%4 ,% 5,&.-4 = K% 9K% qL4 L4N4 Kr<

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Ejemplo 5.12

!" #' %+6'%*' 5, #' D,-%.3" K 5,# !7,2(#$ N<] *,",2$%T

`+8'5$-,%T C % qK4 I4 <<<4 (r< E#'2,2$% `. '# 7+8'5$- 1

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[.%*-.6+&.3" 5, (-$6'6.#.5'5 ' (-.$-.T %?v Ea vB% v

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Definición 5.3

!" ,# 7+,8$ 6':,%.'"$ ,%*C*.&$

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2'>"1 à 91

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," #' D'-.'6#, 5, 5,&.%.3" "1<<C [,&.2$% 0+, ,# (,-1.# &'% ?+FK 4 +FI 4 <<<4 +F( B ,% +" JK0"#"<)"+ L&=$*"&,+ '$

M&*N ?!ZB ," ,%*-'*,8.'% (+-'% %. ('-' &'5' 7+8'5$- 1 #' ,%*-'*,8.' +F1 ,% +"' -,%(+,%*'3(*.2' ,%(,-'5' ' #' &$26."'&.3" &'.$% ?+FK 4 <<<4 +F1.K4 +F1!K4 <<<4 +(B<

!% 5,&.-4 &'5' *.($ 5,# 7+8'5$- 1 2'>.2.A' %+ ('8$ ,%(,-'5$ 7+8'"5$ +F1 ?. 1B4 $ 5.&/$'9" 5, $*-$ 2$5$4 a5'5$ +" 7+8'5$- &+'#0+.,-' 14 : +" *.($ . 1 &+'#0+.,-' 5, F%*,4 #' ,%*-';*,8.' +F1 ,%(,&.1.&'5' ('-' ,%*, 7+8'5$- ," ,# (,-1.# &'4 /' 5, '%.8"'- ' ,%, *.($ +"' '&&.3"+F1 ?. 1B 0+, %,' -,%(+,%*' 3(*.2'4 ," *F-2."$% ,%(,-'5$%4 '# &$"7+"*$ 5, &$26."'&.$",% 5,'&&.$",% '%.8"'5'% ' #$% *.($% ($%.6#,% 5, #$% $*-$% 7+8'5$-,% ($- %+% -,%(,&*.D'% ,%*-'*,;8.'% ," &'b<

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Cálculo del EB en los ejemplos anteriores

P'2$% ' &'#&+#'- #$% !Z 5, #$% ,7,2(#$% '"*,-.$-,%4 : #$ /'-,2$% 5, 2'",-' (-$8-,%.D'5,%5, +" (+"*$ 5, D.%*' 5.5C&*.&$4 5, 2$5$ 0+, ," #$% (-.2,-$% &'%$% /'-,2$% +"' '(-$;

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Ejemplo 5.13

=,' ,# 7+,8$ 5,# !7,2(#$ N<G4 &+:' -,(-,%,"*'&.3" ," 1$-2' ,>*,"%.D' -,(,*.2$% ' &$";*."+'&.3"

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Figura 5.8 Juego bayesiano del Ejemplo 5.4.

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Ejemplo 5.14

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Ejemplo 5.15

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E

vK."K %. "Kb "I

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Figura 5.9 Juego bayesiano de la subasta simplificada del Ejemplo 5.6.

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Ejemplo 5.16

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Ejemplo 5.17

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Azar Azar

1 11 1

2 2 2 22 2 2 2

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C

C CC CC CC C

CC C

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X (1/2) X (1/2)

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Figura 5.10 Juego sencillo de la verdad.

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Ejemplo 5.18

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Ejemplo 5.19

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Azar

Azar Azar

J1 J1J1 J1

1,1 0,5 1,1 0,5 1,1 0,5 1,1 0,5

5,0 4,4 5,0 10,10 5,0 4,4 5,0 10,10

Urna 1 (1/2)

B (1/2) B (3/4)

Urna 2 (1/2)

N (1 /2) N ( 1/ 4)


J2 J2 J2 J2J2 J2 J2 J2


Figura 5.11 Juego bayesiano del Ejemplo 5.4, pero con otra jugada de azar.

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Teorema de existencia

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Equilibrios bayesianos en estrategias mixtas

e'%*' '/$-'4 ," #$% 7+,8$% 6':,%.'"$% /,2$% '"'#.A'5$ %3#'2,"*, ,%*-'*,8.'% (+-'%4 0+,%$" -,8#'% 5, 5,&.%.3" 0+, '%.8"'" ' &'5' *.($ +"' '&&.3"< !" ('-*.&+#'-4 ," +" 7+,8$

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Definición 5.4!" +" 7+,8$ 6':,%.'"$ ,%*C*.&$ 1.".*$4 A B% q C m 9K4 9I4 <<<4 9(m < K4 < I4 <<<4 < (m %K4 %I4 <<<4

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Ejemplo 5.20

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!%*, 7+,8$ 6':,%.'"$ (+,5, D.%+'#.A'-%, '%)T

Azar

Cara ( 1/ 2) p 5 Cruz (1 1/2) p2 5

Cara Cara

2 1, 1

2 1, 1

2 1, 1Cara Cara

1, 12Cruz Cruz

Cruz Cruz

1, 12

1, 12

1, 12

2 1, 1

J2 J2

J ( Cara)1 1t 5 J ( Cruz)1 1t 5

Figura 5.12 Juego bayesiano sin EB en estrategias puras.

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D.I.P.

B.E.

E.S.

D.I.C.

Dinámicos coninformación completa

Dinámicos con información perfecta

Bayesianos estáticos

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Figura 5.13 Jerarquía de las distintas categorías de juegos.

5.3 APLICACIONES: DUOPOLIO DE COURNOT CON INFORMACIÓNINCOMPLETA

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Comparación en un caso particular sencillo

?"%c4 M%L4 eK%K4 eL%L : %%I/ JB

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Teorema 5.5

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Figura 5.14 Posibilidades correspondientes al caso ai b v i .

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El principio de equivalencia de ingresos

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Ejemplos introductorios y de motivación

Ejemplo 6.1

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Figura 6.1 Juegos de la disuasión 3a y 3b.

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Juegos dinámicos con información incompleta 345

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Ejemplo 6.2

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Figura 6.2 Juego de la disuasión 3c.

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Ejemplo 6.3

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Figura 6.3 Juego del Trespiés 2.

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Ejemplo 6.4

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Figura 6.4 Juego bayesiano estático.

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Ejemplo 6.5

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Ejemplo 6.6

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6.2. EL EQUILIBRIO BAYESIANO PERFECTO

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Representaciones de un juego. Representación multiagente

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Definición 6.1

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Ejemplo 6.7

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Estrategias mixtas y estrategias de comportamiento

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0,5."," &$6$ '>+,##$% ," #$% >+,8 &+'"0$ +" 2+3'0$- B' ' 0,&.0.- ," +" &$"2+"*$ 0,."5$-6'&.7"8 -,&+,-0' *$0' #' ."5$-6'&.7" 0, #' >+, /' 0.%(+,%*$ (-,B.'6,"*, ," ,# 2+,A

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Conjuntos de información dentro y fuera de la trayectoria de equilibrio

K. ," +" 2+,3$ ," 5$-6' ,1*,"%.B' &$" ."5$-6'&.7" .6(,-5,&*' ! ;0,# >+, > ,% #' 5'6.#.'0, *$0$% %+% &$"2+"*$% 0, ."5$-6'&.7"< "$ ,1.%*," 2+3'0'% 0, '='-8 &+'#>+.,- (,-5.# ,"

,%*-'*,3.'% (+-'% "% ;"M8 ???8 ",8 ???8 "$< 0,*,-6."' &$6(#,*'6,"*, ,# 0,%'--$##$ 0,# 2+,3$?!" ,%, &'%$8 0'0$ +" &$"2+"*$ 0, ."5$-6'&.7" 48 $ :.," ,% '#&'"='0$ &$" %,3+-.0'0 ," ,#0,%'--$##$ 0,# 2+,3$ $ :.," ,% %,3+-$ >+, "$ %, '#&'"='? C$- ,2,6(#$8 ," ,# 2+,3$ 0,#Q-,%(.E% N *$0$% #$% &$"2+"*$% 0, ."5$-6'&.7" %$" '#&'"='0$% ($- #' *-'9,&*$-.' 0,*,-6.A"'0' ($- ,# ,>+.#.:-.$ "% ;K.K8 K<?

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*'# &$"2+"*$ 0, ."5$-6'&.7" ,%*4 0,"*-$ 0, #' *-'9,&*$-.' 0,*,-6."'0' ($- ,%, (,-5.#? C$-,2,6(#$8 ," ,# 2+,3$ :'9,%.'"$ 0,# !2,6(#$ Y?H8 *$0$% #$% &$"2+"*$% 0, ."5$-6'&.7"84M% [ 3 M\ 9 4N% [ 3 N\ 0, cM8 9 4P% [ 3 P8 3 H8 3 8 3 Y\ 0, cN8 ,%*4" 0,"*-$ 0, #' *-'9,&*$-.'0,*,-6."'0' ($- &+'#>+.,- (,-5.# ,%*-'*E3.&$?

@' 0,5.".&.7" %.3+.,"*, &$"&-,*' ,%*' .0,' ," -,#'&.7" ' #$% (,-5.#,% 0, ,>+.#.:-.$?

Definición 6.2

K,' ! +" 2+,3$ 0."46.&$ &$" ."5$-6'&.7" .6(,-5,&*'8 9 4 à > +" &$"2+"*$ 0, ."5$-A6'&.7"? K,' p% ;pM8 ???8 p,8 ???8 p$< +" (,-5.# 0, ,>+.#.:-.$ ," !?

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U,-,&, #' (,"' $:%,-B'-_

m+, ," &+'#>+.,- 2+,3$ ,%*4*.&$ 5.".*$8 &$" $ %." 2+3'0' 0, '='-8 9 ('-' &+'#>+.,-(,-5.# 0, ,>+.#.:-.$8 *$0$% #$% &$"2+"*$% 0, ."5$-6'&.7" ,%*4" 0,"*-$ 0, #' *-'9,&*$A-.' 0, 0.&/$ ,>+.#.:-.$?

m+, ," &+'#>+.,- 2+,3$ 5.".*$8 ,%*4*.&$ $ 0."46.&$8 %. p% ;pM8 ???8 p,8 ???8 p$< ,% +"(,-5.# 0, ,>+.#.:-.$ ;%,' 0, ,%*-'*,3.'% 6.1*'% (-$(.'6,"*, 0.&/'% $ ,%*-'*,3.'% 0,

&$6($-*'6.,"*$< ," ,# &+'# *$0'% #'% ,%*-'*,3.'% p, %$" 6.1*'% &$6(#,*'%8 *$0$% #$%&$"2+"*$% 0, ."5$-6'&.7" ,%*4" 0,"*-$ 0, #' *-'9,&*$-.' 0, 0.&/$ ,>+.#.:-.$?

El equilibrio bayesiano perfecto

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Cine Fútbol

Jugadora 2 Jugadora 2

Jugador 1

1, 2 2, 10, 00, 0

x3 x2

x1

Cine CineFútbol Fútbol

Figura 6.7 Batalla de los sexos.

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@'% (-71.6'% 0,5.".&.$",% 5$-6'#.='-4" ,%*'% .0,'%?

Definición 6.3

K,' ! +" 2+,3$ 0."46.&$ &$" ."5$-6'&.7" .6(,-5,&*'8 9 > #' 5'6.#.' 0, *$0$% #$%&$"2+"*$% 0, ."5$-6'&.7" 0, !?

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3 à4

k4; 3 <%M<?

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i+ F ,/ ;pñ,8 p.,<n 48 ka8

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Ejemplo 6.9

a) K,'" #$% 2+,3$% 0, #' 0.%+'%.7" P' 9 P:8 0,5.".0$% 9 -,(-,%,"*'0$% ," ,# !2,6(#$ Y?M?!" &+'"*$ '# 2+,3$ 0, #' 0.%+'%.7" P'8 *.,", P "$0$% 0, 0,&.%.7"8 3 M8 3 N 9 3 P8 9 0$%

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58!%;O*9)+, NH>

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Definición 6.4

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Ejemplo 6.10

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Figura 6.8 Juego de la disuasión 3a. Único EBP.

Ejemplo 6.11

a) K,' ,# 2+,3$ 0, #' 0.%+'%.7" P:? D,6$% B.%*$ ," ,# !2,6(#$ Y?f ;'< >+, #$% 0$% !I0,# 2+,3$ 0, #' 0.%+'%.7" P:8 #$% (,-5.#,% "% ;+",8 /"< 9 "ñ% ; %'8 /- <8 %$" %,&+,"&.'#A6,"*, -'&.$"'#,% ('-' #$% %.%*,6'% 0, &$"2,*+-'% k_ 1; 3 N<% W8 1; 3 P<% Ma 9 kñ_

1; 3 N<%M8 1; 3 P<% Wa8 -,%(,&*.B'6,"*,? L'=$"'"0$ '"4#$3'6,"*, ' &$6$ %, /' /,&/$," ,# !2,6(#$ Y?MW8 %, 0,6+,%*-' >+, #' ,B'#+'&.7" ;"8 k< ,% +" !VC?C$- $*-' ('-*,8 B,'6$% >+, *'6:.E" #' ,B'#+'&.7" ;"ñ8 kñ< ,% +" !VC? !" ,5,&*$8 %,

&+6(#, #' &$"0.&.7" ;:< 0, 6$0$ *-.B.'# 9' >+, ,# &$"2+"*$ 0, ."5$-6'&.7" 4N ,%*4 5+,-'0, #' *-'9,&*$-.' 0, ,>+.#.:-.$8 9 %, &+6(#, #' &$"0.&.7" ;&< 9' >+, #' &$"2,*+-' %$:-, 4N

,% '&$-0, &$" ,# (,-5.#8 *'6:.E" 0, 6$0$ *-.B.'# ;&+'#>+.,- &$"2,*+-' /+:.,-' %.0$ B4#.A0'<8 9' >+, 4N ,%*4 5+,-' 0, #' *-'9,&*$-.' 0, ,>+.#.:-.$ 9 ,# (,-5.# "$ .6($", "."3+"'-,%*-.&&.7" ' #' &$"2,*+-' %$:-, 4N? !" &$"&#+%.7"8 '6:'% ,B'#+'&.$",%8 >+, %, 6+,%*-'"," #' X.3+-' Y?f8 %$" !VC?

21, 2 5, 50, 04, 2

Ei Ne

Cd Cd

Esi

Cs Cs

3, 7 x2 x3

INCUMBRON INCUMBRON INCUMBRON INCUMBRON

ENTRON ENTRON

21, 2 5, 50, 04, 2

Ei Ne

Cd Cd

Esi

Cs Cs

3, 7 x2

x1 x1

x3

x2 x3

p ( ) 05 x2 p ( ) 15 x3

( , ) es un EBP s

x2 x3

p ( ) 15 x2 p ( ) 05 x3

( , ) es un EBP s

Figura 6.9 Juego de la disuasión 3b. Dos EBP.

8I K,' ,# 2+,3$ 0, #' 0.%+'%.7" P&? L,5,-.-,6$% #' 0.%&+%.7" ' #' &$"2,*+-' %$:-, ,#&$"2+"*$ 4P% [ 3 P8 3 H\8 ,% 0,&.-8 '# ('- ; 18 M. 1< 0$"0, 1 ,% #' (-$:':.#.0'0 0, >+,SIGTUVLRI ,%*E ," ,# "$0$ 3 P? D,6$% B.%*$ ," ,# !2,6(#$ Y?f ;:< >+,_

!# !I "% `;+ 8 +",<8 /"a ,% %,&+,"&.'#6,"*, -'&.$"'# &$" -,%(,&*$ ' #' &$"2,*+-'k_ ; 1%W8 M. 1%M<8 9

!# !I "ññ% `; %'8 +",<8 /- a ,% %,&+,"&.'#6,"*, -'&.$"'# &$" -,%(,&*$ ' #' &$"2,*+A-' kññ_ ; 1% M8 M. 1% W<?I$% >+,0' ($- *'"*$ 'B,-.3+'- %. '#3+"' 0, #'% ,B'#+'&.$",% ;"8 k< 9 ;"ññ8 kññ< ,% +" !VC$ +" !VCb?

!" &+'"*$ ' #' ,B'#+'&.7" ;"8 k<8 #' &$"2,*+-' k_ ; 1%W8 M. 1%M< &+6(#, #' &$"A0.&.7" ;:< 0, #' b,5.".&.7" Y?H8 (+,%*$ >+, ,# (,-5.# ,%*-'*E3.&$ " $:#.3' ' ('%'- ($- 3 H8#$ >+, .6(#.&' 1%W? !1(-,%'0$ ," *E-6."$% 0, '&*+'#.='&.7" :'9,%.'"'8 ,%*'"0$ ,#&$"2+"*$ 0, ."5$-6'&.7" 4P% [ 3 P8 3 H\ ," #' *-'9,&*$-.' 0, ,>+.#.:-.$8 #' (-$:':.#.0'0 1(+,0, &'#&+#'-%, '%)_

1% k43; 3 P<%(-$:; 3 P/ "<

(-$:;4P/ "<%W

M%W

F0,64%8 &+6(#, #' &$"0.&.7" ;&< 0, 6$0$ *-.B.'# 0,:.0$ ' >+, "$ ,1.%*," &$"2+"*$%0, ."5$-6'&.7" >+, ,%*E" 5+,-' 0, #' *-'9,&*$-.' 0, ,>+.#.:-.$? !" &$"%,&+,"&.'8 :* %O*Q:$*9)+, R $S kI %! $, 63M?

o ($- J#*.6$8 ," &+'"*$ ' #' ,B'#+'&.7" ;"ññ8 kññ<8 #' &$"2,*+-' kññ_ ; 1% M8 M. 1%W<&+6(#, #' &$"0.&.7" ;:< 0, #' b,5.".&.7" Y?H 0, 6'",-' B'&)'8 (+,% 4P ,%*4 5+,-' 0, #'*-'9,&*$-.' 0, ,>+.#.:-.$? K." ,6:'-3$8 "$ &+6(#, #' &$"0.&.7" ;&< ($->+, ,# (,-5.# ,%A*-'*E3.&$ "ññ $:#.3' ' SIGTUVLRI ' &$"2,*+-'-8 %. *+B.,-' >+, 2+3'-8 >+, ,%*4 &$"&,-*,=' ," 3 H8 #$ >+, .6(#.&' 1% W8 ," &$"*-'0.&&.7" &$" #' &$"2,*+-' kññ? !" &$"%,A&+,"&.'8 :* %O*:$*9)+, ; $ññS kññI %! $, 63MGS <%;' ,' %! $, 63M? U,-,&, #' (,"' /'&,-"$*'- +"' $:%,-B'&.7" 0, U'%AG$#,##8 p/."%*$" 9 q-,," ;Mff^< >+, #.6.*' ,# ."*,-E% 0,#&$"&,(*$ !VCb? F (,%'- 0, >+, #' ,B'#+'&.7" ;"ññ8 kññ< ,% +" !VCb8 ,# (,-5.# "ññ "$ ,% +",>+.#.:-.$ 0, I'%/ (,-5,&*$ ," %+:2+,3$%8 9' >+, ."0+&, ,# (,-5.# ;+",8 /- < ," ,# %+:2+,A3$ >+, %, .".&.' ," 3 N 9 ,%*, (,-5.# "$ ,% !I 0,# %+:2+,3$? !# 5'##$ -'0.&' ," "$ ,1.3.-"."3+"' &$"%.%*,"&.' ,"*-, ,# (,-5.# ,%*-'*E3.&$ 9 #'% &$"2,*+-'% ," &$"2+"*$% 0, ."5$-A6'&.7" 5+,-' 0, #' *-'9,&*$-.' 0, ,>+.#.:-.$? !" #' X.3+-' Y?MW %, 6+,%*-'" #$% 0$%,>+.#.:-.$% 0,# 2+,3$?

INCUMBRON INCUMBRONINCUMBRON INCUMBRON

ENTRON ENTRON

21, 2 21, 25, 5 5, 50, 0 0, 04, 2 4, 2

Ei Ei

Ne Ne E E

Cd Cd Cd Cd

Esi Esi

Cs CsCs Cs

3, 7 3, 7 x3 x3

x2 x2

x1 x1

x4 x4

( , ) es un EBP s ( , ) es EBPD, pero no EBP s

ENTRON ENTRON

p 0,5 p 1,51 12 5 p 1 02 5 p

Figura 6.10 Juego de la disuasión 3c. Un EBP y un EBPD.

Ejemplo 6.12

!" ,# 2+,3$ 0, &'-*'% '"'#.='0$ ," #$% !2,6(#$% Y?] 9 Y?d ;:<8 %':,6$% >+, ,# J".&$ !I,% p% ;pM8 pN<8 0$"0, pM ,% #' ,%*-'*,3.' 0, &$6($-*'6.,"*$ 0, V#'"&' g%. #' &'-*' ,%$-$ $ &$('8 ?1&"(*) &$" %,3+-.0'08 9 %. ,% ,%('0' $ :'%*$8 ?1&"(*) &$" (-$:':.#.0'0 M/ P9 @'(,)*)"' &$" (-$:':.#.0'0 N/ Ph8 ':-,B.'0' ;;M8 W<8 ;M/ P8 N/ P<<8 9 pN ,% #' ,%*-'*,3.' 0,&$6($-*'6.,"*$ 0, G'-#$% g @'G&A') #' '(+,%*' &$" (-$:':.#.0'0 N/ P 9 H*"*) &$" (-$A:':.#.0'0 M/ Ph8 ':-,B.'0' ;N/ P8 M/ P<?

O'6$% ' :+%&'- +" %.%*,6' 0, &$"2,*+-'% k *'# >+, ;p8 k< %,' +" !VC? !" #' (-4&*.A

&'8 (+,%*$ >+, %7#$ ,1.%*, +" &$"2+"*$ 0, ."5$-6'&.7" "$ +".*'-.$8 ,# 4P% [ 3 P8 3 H\ ,">+, 0,&.0, G'-#$%8 :+%&'6$% +"' &$"2,*+-' ; 18 M. 1< 0$"0, 1 ,% #' (-$:':.#.0'0 ,%*.A

6'0' ($- G'-#$% 0, ,"&$"*-'-%, ," ,# "$0$ 3 P8 >+, %.3+, ' #' 2+3'0' 0, '='- $-$A&$(' 9' #' 2+3'0' ?1&"(*) 0, V#'"&'? !# &$"2+"*$ 4P% [ 3 P8 3 H\ ,%*4 ," #' *-'9,&*$-.' 0, ,>+.A#.:-.$8 (+,%*$ >+, #' (-$:':.#.0'0 ' (-.$-. 0, '#&'"='-#$ %. %, 2+,3' 0, '&+,-0$ &$" p ,%(-$:;4P/ p<% ;M/ N<;M<! ;M/ N<;M/ P<%N/ Pb W? !" B.-*+0 ($- *'"*$ 0, #' &$"0.&.7" ;:<8'(#.>+,6$% #' -,3#' 0, V'9,% ('-' &'#&+#'- 1_

1%k43; 3 P<%

(-$:; 3 P/ p<

(-$:;4P/ p<%

M/ N

N/ P%

P

H

F%) (+,%8 #' (-$:':.#.0'0 :+%&'0' ,% 1%P/ H? !" &$"&#+%.7"8 :* %O*:$*9)+, RpS kI8(',(% %! :* 9',J%A$;* R 1%P/ H8 M. 1% M/ H<8 %! $, 63M? !" #' X.3+-' Y?MM %,6+,%*-' ,# !VC 0,# 2+,3$?

J0 (Azar)

Blanca

Carlos

Carlos

Blanca

1/2

1/2

O-C

E-B

A(1)

A(1/3)

R(0)

RA(2/3)

RA(2/3)

P(1/3)

P(1/3)

R(2/3)

5, 52

5, 52

5, 52

10, 102

210, 10

25, 5

x1

x3

x2

x4

x3

x4

p 3/45

1 1/42 5 p

Figura 6.11 EBP del juego de cartas.

C$0-)' (,"%'-%, >+,8 0'0$ +" (,-5.# ,%*-'*E3.&$ p8 #' -,3#' 0, V'9,% &$6$ 6E*$0$('-' '&*+'#.='- ,# &4#&+#$ 0, #'% (-$:':.#.0'0,% ," +" &$"2+"*$ 0, ."5$-6'&.7" 0'0$ 4 ;,%

0,&.-8 ('-' 0,*,-6."'- %+% &$"2,*+-'%< %7#$ ,% '(#.&':#, &+'"0$ 0.&/$ &$"2+"*$ 0, ."5$-A6'&.7" 4 ,%*4 0,"*-$ 0, #' *-'9,&*$-.' 0, ,>+.#.:-.$8 '# %,- '##) (-$: ;4/ p+, '('-,&, ," ,# '('-*'0$ ;:< ," #' b,5.".&.7" Y?H? K." ,6:'-3$8 ,# %.3+.,"*, ,2,6(#$8*$6'0$ 0, U'%AG$#,##8 p/."%*$" 9 q-,," ;Mff^< 0,6+,%*-' >+, ," $&'%.$",% %) ,% '(#.A

&':#, #' -,3#' 0, V'9,%8 &$"B,".,"*,6,"*, -,."*,-(-,*'0' ," *E-6."$% -,#'*.B$%8 ('-' 0,A*,-6."'- #'% &$"2,*+-'% '(-$(.'0'% 5+,-' 0, #' *-'9,&*$-.' 0, ,>+.#.:-.$?

Ejemplo 6.13

G$"%.0,-,6$% ,# %.3+.,"*, 2+,3$ ," 5$-6' ,1*,"%.B'8 '# >+, ##'6'-,6$% c+,3$ Y?M_

Figura 6.12 Juego 6.1.

@' -,(-,%,"*'&.7" ," 5$-6' "$-6'#8 &$" ('3$% ,%(,-'0$%8 0,# 2+,3$ '"*,-.$- ,%

#>

G b

#BF N8 MW N8 MW

V W8 ^8 Y

9 #$% !I ," ,%*-'*,3.'% (+-'% %$" "% ;F8 G< 9 "ñ% ;V8 b<? F6:$% %$" (,-5,&*$% ,"%+:2+,3$%8 9' >+, "$ ,1.%*, "."3J" %+:2+,3$ (-$(.$? O'6$% ' '"'#.='- '6:$% ,>+.#.A:-.$% 9 0,6$%*-'- >+, %7#$ ,# %,3+"0$ ,% +" ,%&,"'-.$ 0, !VC?

!# ,>+.#.:-.$ " ;>+,8 0.&/$ %,' 0, ('%$8 *.,", +"' ,%*-'*,3.' 0E:.#6,"*, 0$6."'0'8 #'G 0, cN< .6($", $:B.'6,"*, +"' &$"2,*+-' '# &$"2+"*$ 0, ."5$-6'&.7" 4M% [ 3 M8 3 N\ 0,

cM8 9 "$ ,% $*-' >+, #' &$"2,*+-' ;<% M/ N8 M. <% M/ N< 0,-.B'0' 0, #' (-$(.' 2+3'0' 0,'='-? !" &+'"*$ '# &$"2+"*$ 0, ."5$-6'&.7" 4N% [ 3 P8 3 H\ 0, cN8 >+, ,%*4 5+,-' 0,#' *-'9,&*$-.' 0, ,>+.#.:-.$8 ($0-)'6$% '%.3"'-#, +"' &$"2,*+-' &$6$ ;) %f/ MW8M. ) % M/ MW<?

C'-' ,%*, %.%*,6' 0, &$"2,*+-'% ,% 54&.# B,- >+, " ,% %,&+,"&.'#6,"*, -'&.$"'#? !",5,&*$8 #' '&&.7" F 0, cM ,% 7(*.6'8 0'0' #' '"*,-.$- &$"2,*+-' ," 4M8 %+(+,%*$ >+, cN B'' -,%($"0,- &$" G8 (+,% #, (-$($-&.$"' ' cM +" ('3$ 0, N 5-,"*, ' +" ('3$ 0, W >+,$:*,"0-)' &$" V? o '0,64%8 #' '&&.7" G 0, cN ,% -,%(+,%*' 7(*.6'8 0'0' #' '"*,-.$-&$"2,*+-' ," 4N8 ' #' '&&.7" F 0, cM8 (+,% #, (-$($-&.$"' +" ('3$ ,%(,-'0$ 0, ^8 6'9$->+, ,# >+, $:*,"0-)' &$" b?

K." ,6:'-3$8 ,% &#'-$ >+, 5'##' #' &$"%.%*,"&.'8 ," ('-*.&+#'- #' &$"0.&.7" ;&< 0, #'b,5.".&.7" Y?H8 -,#'*.B' ' &$"2+"*$% 0, ."5$-6'&.7" 5+,-' 0, #' *-'9,&*$-.' 0, ,>+.#.:-.$?

!" ,5,&*$8 %. cN %, B.,-' $:#.3'0$ ' 2+3'- ;&$"*-' #$ (-,B.%*$ ," ,>+.#.:-.$8 >+, ,% >+,cM 2+,3' F 9 %, '&':' ,# 2+,3$< 0,:,-)' -'=$"'- >+,8 (+,%*$ >+, cM "$ /' B.%*$ ,#-,%+#*'0$ 0, #' 2+3'0' 0, '='- '"*,% 0, 2+3'- V8 #' (-$:':.#.0'0 ) 0, ,%*'- ," 3 P ,% M/ N8B'#$- >+, B.,", 0,*,-6."'0$ ($- ,# B'#$- 1% M/ N 0, #' 2+3'0' 0, '='-? F%) (+,%;) % M/ N8 M. ) % M/ N< ,% #' J".&' &$"2,*+-' &$"%.%*,"*, &$" #' 2+3'0' F 0, cM? C$-$*-' ('-*,8 0'0' ,%*' &$"2,*+-'8 #' '&&.7" 0, -,%(+,%*' 7(*.6' 0, cN ' #' '&&.7" F 0,cM ,% b8 (+,% #, (-$($-&.$"' +" ('3$ ,%(,-'0$ 0, Y 5-,"*, ' ^ >+, $:*,"0-)' &$" G?

C$- *'"*$8 '&':'6$% 0, 0,6$%*-'- >+, " "$ ,% +" ,%&,"'-.$ 0, !VC? O,'6$% >+, #',B'#+'&.7" ;"ñ% ;V8 b<8 k% `;<% M/ N8 M. <%M/ N<8 ;) % M/ N8 M. ) %M/ N<a< %) ,% +"!VC? !# -'=$"'6.,"*$ -,&.E" /,&/$ %$:-, #'% &$"2,*+-'% %.3+, %.,"0$ B4#.0$ ;<%M/ N($->+, #$ ,1.3, #' 2+3'0' 0, '='-8 9 ) %M/ N ($->+, &+'"0$ cM 2+,3' V "$ /' $:%,-B'0$,# -,%+#*'0$ 0, #' 2+3'0' 0, '='-<8 9 /,6$% B.%*$ >+, b ,% #' '&&.7" 7(*.6' 0, cN 0'0'#' &$"2,*+-' ;) %M/ N8 M. ) % M/ N< ," 4N? K$#$ 5'#*' B,- >+, #' '&&.7" V 0, cM ,% 7(*.A6'8 0'0' #' &$"2,*+-' ;<%M/ N8 M. <% M/ N< ," 4M8 9 %+(+,%*$ >+, cN B' ' -,%($"0,-&$" b? C+,% :.,"8 #$ ,% ($->+, #, (-$($-&.$"' +" ('3$ ,%(,-'0$ 0, ^ 5-,"*, ' +" ('3$0, N >+, $:*,"0-)' &$" F?

!" &$"&#+%.7"8 :* %O*:$*9)+,

R"ñ%;V8 b<8 k%`;<%M/ N8 M.<%M/ N<8 ;) %M/ N8 M.) %M/ N<a<

!W %! $, 63MS @)%,A;*! P$% :* %O*:$*9)+, R"% ;F8 G<8 k% `;<%M/ N8 M. <%M/ N<8;) % f/ MW8 M. ) %M/ MW<a< ,' :' %! R*$,P$% <'(%@'! @*,A%,%; P$% %! $, 63MGIH!" #' X.3+-' Y?MP %, 6+,%*-' ,# J".&$ !VC?

Azar

J2 J2

J1 J1

C

A B

CD

B A

D

5, 2 5, 100, 5 0, 5

2, 10 2, 10

I ( 1/2) p 5 D (1 1/2)2 5 p

x1

x3

x2

x4

x1

x3

x2

x4

q 1/25

r 1/25

1 1/22 5q

1 1/22 5r

Figura 6.13 EBP del Juego 6.1.

Ejemplo 6.14

K,' ,# &':'##$ 0, K,#*,"8 0,5.".0$ ," ,# !2,6(#$ Y?^? D,6$% B.%*$ >+, #$% ,>+.#.:-.$% ,",%*-'*,3.'% (+-'% %$" "% ;V8 Gñ8 S< 9 "ñ% ;G8 Gñ8 b<? F6:$% %$" (,-5,&*$% ," %+:2+,A3$%8 (+,% "$ ,1.%*," %+:2+,3$% (-$(.$%? C'-' ,%*+0.'- %+% !VC8 0,"$*'-,6$% 4M% [ 3 M\84N% [ 3 N\ 9 4P% [ 3 P8 3 H\ ' #$% &$"2+"*$% 0, ."5$-6'&.7" 0, cM8 cN 9 cP8 -,%(,&*.B'A

6,"*,8 9 0,"$*'-,6$% k43% ;<8 M. << ' #'% &$"2,*+-'% 0, cP ," 4P8 0, 6$0$ >+, < ,% #'(-$:':.#.0'0 ;&$"0.&.$"'0' ($- ,%*'- ," 4P< 0, ,%*'- ," ,# "$0$ 3 P?

!# (-.6,-$ 0, %+% !I8 "% ;V8 Gñ8 S< "$ (+,0, %,- ,%&,"'-.$ 0, +" !VC ". 0, +"!VCb (+,% #' ,%*-'*,3.' Gñ 0, cN "$ %,-)' -,%(+,%*' 7(*.6' ' #' &$6:."'&.7" ;V8 S< 0,cM 9 cP ," ,# &'%$ 0, *,",- >+, 2+3'- ," 4N%[ 3 N\? !" ,5,&*$8 #' -,%*-.&&.7" 0, ;V8 S< ' #'('-*, >+, -,%*' ($- 2+3'- 0,# 2+,3$ ' ('-*.- 0, 4N% [ 3 N\ ,% S8 9 0, ,##$ %, 0,0+&, >+, cN/+:.,%, $:*,".0$ +" ('3$ 0, H 2+3'"0$ Vñ ," #+3'- 0,# ('3$ 0, M >+, $:*.,", &$" Gñ?F%) (+,%8 "$ /' -,%($"0.0$ 7(*.6'6,"*, ," 4N% [ 3 N\ &$6$ ,1.3, #' -'&.$"'#.0'0 %,A&+,"&.'#8 '+">+, 4N ,%*E 5+,-' 0, #' *-'9,&*$-.' 0, ,>+.#.:-.$ 9 "$ .".&., +" %+:2+,3$?

!" &+'"*$ '# %,3+"0$ !I8 "ñ% ;G8 Gñ8 b<8 B'6$% ' (-$($",- +"' &$"2,*+-' %$:-, ,#&$"2+"*$ 0, ."5$-6'&.7" 4P >+, /'3' %,&+,"&.'#6,"*, -'&.$"'# '# ,>+.#.:-.$? I$% B'#,&+'#>+.,- &$"2,*+-' k4

3% ;<8 M. << 0$"0, Wm <m M/ P? !" ,5,&*$8 ('-' &+'#>+.,-' 0,

,%'% &$"2,*+-'% b ,% #' '&&.7" 7(*.6' 0, cP ," 4P ;$:*.,", +" ('3$ .3+'# ' M. < &$" b9 +" ('3$ .3+'# ' N< &$" S<8 9 '0,64% Gñ ,% -,%(+,%*' 7(*.6' 0, cN ' ;G8 b< 9 G ,%-,%(+,%*' 7(*.6' 0, cM ' ;Gñ8 b<? F%) (+,%8 " &+6(#, #' -'&.$"'#.0'0 %,&+,"&.'# &$"-,%(,&*$ ' &+'#>+.,-' 0, ,%'% &$"2,*+-'%? C$- $*-' ('-*,8 &+'#>+.,-' 0, ,##'% &+6(#, #'&$"0.&.7" ;:< 0, #' 0,5.".&.7" 0, !VC *-.B.'#6,"*,8 9' >+, 4P "$ ,%*4 ," #' *-'9,&*$-.'0, ,>+.#.:-.$8 9 &+6(#, #' &$"0.&.7" ;&< 0, 0.&/' 0,5.".&.7" *'6:.E" *-.B.'#6,"*,8 9'>+,8 ,%*'"0$ 5+,-' 0, #' *-'9,&*$-.' 0, ,>+.#.:-.$8 "'0' #' -,%*-."3,?

!" &$"&#+%.7"8 :* %O*:$*9)+, ;"ñ% ;G8 Gñ8 b<8 kñ% `k43% ;<8 M. <<8 0$"0,

Wm <m M/ Pa< %! $, 63M? !" #' X.3+-' Y?MH %, 6+,%*-' ,# !VC $:*,".0$?

I ID D

3, 2, 2 4, 4, 00, 0, 0 0, 0, 1

1, 1, 1J1

J3

J2

J3

C

B

C

B

x1

x3 x3

x2

x4 x4

q 1/3< 1 2/32 <q

( ', ') es un EBP s

Figura 6.14 EBP del caballo de Selten.

Ejemplo 6.15

K,' ,# 2+,3$ B,"*' 0, +" &$&/, +%'0$8 0,5.".0$ ," ,# !2,6(#$ Y?Y? b.%*."3'6$% 0$%($%.:.#.0'0,% %,3J" #' -,#'&.7" >+, ,1.%*' ,"*-, ,# B'#$- 2*8= >+, #' B,"0,0$-' '*-.:+9,' +" &$&/, 0, '#*' &'#.0'08 9 ,# B'#$- r<%<2*8/ ! ;M. <<2;8/ 8 >+, ,% ,# B'#$- ,%(,-'A0$ ' (-.$-. ($- ,# &$6(-'0$- ('-' +" &$&/, &+9' &'#.0'0 0,%&$"$&,? R :.," 2*8= mr<8 $:.," 2*8= br<?

r O'6$% ' 0,6$%*-'- >+, ," ,# (-.6,- &'%$ ;2*8= mr<< ,1.%*, '#3J" !VC &+9$-,%+#*'0$ &$"&#+9, &$" #' B,"*' 0, &+'#>+.,- *.($ 0, &$&/,? !" ,5,&*$8 %,' vW +" (-,&.$

0, B,"*' 5.2$ *'# >+, 2;8/ m vW 9 '0,64% 2*8= m vWmr< ;,% 0,&.-8 6'9$- $ .3+'# >+, ,#B'#$- >+, O '*-.:+9, '# *.($ '#*$8 6'9$- $ .3+'# >+, ,# B'#$- >+, G '*-.:+9, '# *.($ :'2$

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Figura 6.15 Precios de venta en equilibrio para los coches de calidad alta.

6.3. EL EQUILIBRIO SECUENCIAL Y EL EQUILIBRIO PERFECTODE MANO TEMBLOROSA

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Definición 6.5

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Definición 6.6

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Existencia del equilibrio y relaciones entre los distintos conceptosde equilibrio

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Teorema 6.1

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Ejemplos ilustrativos

Ejemplo 6.16

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Ejemplo 6.17

a) K,' ,# 2+,3$ 0, #' 0.%+'%.7" P:? D,6$% B.%*$ ," ,# !2,6(#$ Y?MM ;'< >+, #'% 0$%,B'#+'&.$",% ;"8 k< 9 ;"ñ8 kñ< %$" !VC? @$% (,-5.#,% %$" "% ;+",8 /"< 9 "ñ% ; %'8 /- <8 9#$% %.%*,6'% 0, &$"2,*+-'% %$" k_ 1; 3 N<%W8 1; 3 P<%Ma 9 kñ_ 1; 3 N<% M8 1; 3 P<%Wa?

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2+,3$ 0, #' 0.%+'%.7" P'? C+,% :.,"8 ,% 54&.# B,- >+, ,# 6.%6$ -'=$"'6.,"*$ >+, %, /'/,&/$ ," ,# !2,6(#$ Y?MY ('-' ,%*' ,B'#+'&.7" %.-B, '>+) ('-' 0,6$%*-'- >+, " ,% +",>+.#.:-.$ (,-5,&*$ 0, 6'"$ *,6:#$-$%' 0,# 2+,3$8 9 ($- *'"*$ " ,% ,%&,"'-.$ 0, +" ,>+.#.A:-.$ %,&+,"&.'#?

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Ejemplo 6.18

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Ejemplo 6.19

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Una exploración más detallada de las relaciones entre los distintosconceptos de equilibrio

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Nash

Nash ENPS EBP Secuencial Perfecto

Admisible Perfecto Propio Estricto

Forma estratégica

Forma extensiva

Existen para todo juego finito No siempre existen

Figura 6.16 Escalas lógicas de los refinamientos del EN.

6.4. JUEGOS DE SEÑALIZACIÓN

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Definición 6.7

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Azar

Prob ( )t p1 5

E E

Prob ( ) 1t p2 5 2

R R R R R R

m0 m0m1 m1

m2

m1

t 1 t 2

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m1

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a2

a2

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Figura 6.17 Esquema estructural de un juego de señalización en forma extensiva.

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El equilibrio bayesiano perfecto en el juego básico de señalización

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Teorema 6.2

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G$"%.%*,"&.' 0,"*-$ 0, #' *-'9,&*$-.' 0,# ,>+.#.:-.$? !" &+'#>+.,- &$"2+"*$ 0,."5$-6'&.7" 4ñ0 %.*+'0$ ," #' *-'9,&*$-.' 0, ,>+.#.:-.$8 #'% &$"2,*+-'% 0, k %$" &$"%.%A*,"*,% &$" #'% ,%*-'*,3.'% 0, p? F%) ,% ($- 0,5.".&.7" 0, k0?

G$"%.%*,"&.' 5+,-' 0, #' *-'9,&*$-.' 0,# ,>+.#.:-.$? K, &+6(#, 0, 6'",-' *-.B.'#?

F('-*'0$ &< OE'%, X+0,":,-3 9 Q.-$#, ;MffM8 K,&&.7" d?P<8 0$"0, *'6:.E" %, 0'+" ,2,6(#$8 ('-' +" 2+,3$ 64% &$6(#,2$8 0, +" !VC >+, "$ ,% ,>+.#.:-.$ %,&+,"&.'#?

G$" ,# 5." 0, 5'&.#.*'- #' 0.%&+%.7" %$:-, #$% !VC 0, +" 2+,3$ 0, %,Z'#.='&.7"8 9 #'.0,"*.5.&'&.7" 0, '#3+"$% 0, #$% !VC 64% ."*,-,%'"*,%8 %, 0,5."," #$% &$"&,(*$% 0, ,>+.A#.:-.$ '3-+('0$- 9 ,>+.#.:-.$ %,('-'0$-?

Definición 6.8

K,' ! 2+,3$ 0, %,Z'#.='&.7"?

*I b,&.6$% >+, +"' ,%*-'*,3.' (+-' "+ 0,# ,6.%$- ,% +"' %!A;*A%&)* (% *&;$<*Q9)+, %. *$0$% #$% *.($% 0, ,6.%$- 0,&.0," ,# 6.%6$ 6,"%'2,8 ,% 0,&.-8 %. ,1.%*, +"6,"%'2, 0 à K *'# >+, "+ ;( <%0 ('-' *$0$ *.($ ( à B ?

8I b,&.6$% >+, +"' ,%*-'*,3.' (+-' "+ 0,# ,6.%$- ,% +"' %!A;*A%&)* (% !%<*;*Q9)+, %. *.($% 0.%*."*$% 0, ,6.%$- 0,&.0," 6,"%'2,% 0.%*."*$%8 ,% 0,&.-8 %. ( Ç ( ñ .6(#.&'"+ ;( <Ç "+ ;( ñ<?

9I b,&.6$% >+, +" (,-5.# 0, ,>+.#.:-.$ p% ;p+ 8 p @< ,% +" %P$):)8;)' *&;$<*(';%. p+

,% +"' ,%*-'*,3.' 0, '3-+('&.7"8 9 >+, ,% +" %P$):)8;)' !%<*;*('; ' ()!9;)@),*Q('; %. p+ ,% +"' ,%*-'*,3.' 0, %,('-'&.7"?

b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

O,'6$% +" ,2,6(#$ .#+%*-'*.B$?

Ejemplo 6.20

!%*, 2+,3$8 '# >+, ##'6'-,6$% 2+,3$ 0, #' >+,-,##'8 ,% +"' '0'(*'&.7" 0, +" ,2,6(#$ 0,U$--$v ;MffH< 9 %, &$"5$-6' '# %.3+.,"*, -,#'*$_

B)*" "') -'0*$-*-& 1&) $'A7,A'$G,*L 2$ 1)&6'",&$*7 MNOP &6)'G' 2$* G*$(,-*- <L<2' 12'-' "') 0,7 & G,$G& 0,7 '2)&"L *7 -'0*$-*$(' MNQP 1*)* <2' C"(' )'(,)' 7*<2')'77*L R '7 -'0*$-*$(' *G'1(* )'(,)*)7* & $& 7& *G'1(*L '$ G2R& G*"& "' G'7';)*'7 I2,G,&S T* 1)&;*;,7,-*- * 1),&), -' <2' 4*R* 4*;,-& $'A7,A'$G,*L <2' '" -' -&0,.$,& 1E;7,G&L '" OUQL 1')& '7 -'0*$-*-& "V "*;' ", 4* ",-& $'A7,A'$('S 5, "' G'7';)*'7 I2,G,&L "' "*;)W 7* X')-*-L *2$<2' )'"27(*)W G&"(&"& M2$ G&"(' -' -&" 0,7 '2)&"

1*)* G*-* 2$&L R 2$* ,$-'0$,Y*G,J$ -' Z % [S\\\ '2)&"L <2' '7 -'0*$-*-& 4*;)W-' 1*A*) *7 -'0*$-*$(' ", )'"27(* G271*;7'PS

!" #' X.3+-' Y?Md %, -,(-,%,"*' ," 5$-6' ,1*,"%.B' ,%*, 2+,3$ 0, %,Z'#.='&.7"8 ':-,AB.'"0$ 6,0.'"*, I 9 b #'% ,1(-,%.$",% %'A7,A'$(' 9 ],7,A'$('8 9 6,0.'"*, F 9 IF #'%

,1(-,%.$",% ?G'1(& 9 %& *G'1(&?

A A

A A

NA NA

NA NA

J1J2 J2

J1J2 q 15 q 55

q 15 q 55

J2

Azar

2 1 2 5

2 1 2 5

2 I 2 2 2 I 2 2

2 2 2 2

1 5

1 5

I 22 I 22

2 2 2 2

N ( 1/2) p 5

D ( 1/2) p 5

Figura 6.18 El juego de la querella.

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2+"*$% 0, ."5$-6'&.7" 0, cN >+, %.3+,"8 -,%(,&*.B'6,"*,8 ' #$% 6,"%'2,% <%M 9 <% ^?V+%>+,6$% #$% !VC 0,# 2+,3$?

G$6,"&,6$% ($- #$% ,>+.#.:-.$% '3-+('0$-,% ," ,%*-'*,3.'% (+-'%_

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;M/ N<% ;S.H</ N% M/ N >+, #, -,($-*'-)' IF? C$- $*-' ('-*,8 <%M ('-' &+'#>+.,- *.($,% #' -,%(+,%*' 7(*.6' 0, cM '"*, &+'#>+.,- ,%*-'*,3.' 0, cN >+, -,%($"0' '# 6,"%'2,<%M &$" F? @' -,(+,%*' 7(*.6' 0, cN '# 6,"%'2, <%^ 5+,-' 0,# ,>+.#.:-.$ *'6:.E"%,-)' F8 >+, 0$6."' ,%*-.&*'6,"*, ' IF ," ,%, &$"2+"*$ 0, ."5$-6'&.7"? F%) (+,% ,#(,-5.# p% ;;p+ ;I<% M8 p+ ;b<% M<8 ;p @;M<%F8 p @;^<%F<< ,% +" ,>+.#.:-.$ '3-+('A0$-? @'% &$"2,*+-'% >+, #$ -,%('#0'" %$" ;M/ N8 M/ N< ," ,# &$"2+"*$ 0, ."5$-6'&.7" >+,%.3+, '# 6,"%'2, <%M8 9 ; 18 M. 1<8 %.,"0$ 1 &+'#>+.,- B'#$- 0, `W8 Ma8 ," ,# &$"2+"*$0, ."5$-6'&.7" >+, %.3+, '# 6,"%'2, <%^?

K,' <%^ #' J".&' $5,-*'A6,"%'2, >+, -,'#.=' cM8 %,' &+'# %,' %+ *.($? !" ,%,&'%$8 #' -,%(+,%*' 7(*.6' ($- ('-*, 0, cN '# 6,"%'2, 0, ,>+.#.:-.$ <%^ ,% F8 (+,%

0$6."' ,%*-.&*'6,"*, ' IF? K." ,6:'-3$8 ," ,%, &'%$ ' cM #, ."*,-,%'-)' 0,%B.'-%, 0,#,>+.#.:-.$ ($*,"&.'# 2+3'"0$ <%M (+,% #, -,($-*' ('3$% ,%*-.&*'6,"*, 6'9$-,% /'3' #$>+, /'3' cN ," -,%(+,%*'? F%) (+,%8 "$ ,1.%*, ,>+.#.:-.$ '3-+('0$- 0$"0,p+ ;I<%p+ ;b<% ^?

O,'6$% '/$-' #$% ,>+.#.:-.$% 0.%&-.6."'0$-,% ," ,%*-'*,3.'% (+-'%_

K,' <% ^ #' $5,-*'A6,"%'2, >+, -,'#.=' cM &+'"0$ %+ *.($ ,% I8 9 <%M #' $5,-*'A6,"%'2, >+, -,'#.=' &+'"0$ %+ *.($ ,% b? !% 0,&.-8 p+ ;I<%^ 9 p+ ;b<% M? !" ,%, &'%$8#' -,%(+,%*' 7(*.6' ($- ('-*, 0, cN ,% F *-'% &+'#>+.,- $5,-*'8 ,% 0,&.-8p @;M<%p @;^<%F? K." ,6:'-3$8 p+ _ ;p+ ;I<% ^8 p+ ;b<%M< "$ ,% -,%(+,%*' 7(*.6'

($- ('-*, 0, cM ' p @_ ;p @;M<%F8 p @;^<%F<8 9' >+, p+ ;I<% M #, (-$($-&.$"'-)' +"('3$ 0, .M ," #+3'- 0, .^?

K,' <% M #' $5,-*'A6,"%'2, >+, -,'#.=' cM &+'"0$ %+ *.($ ,% I8 9 <%^ #' $5,-*'A6,"%'2, >+, -,'#.=' &+'"0$ %+ *.($ ,% b? !% 0,&.-8 p+ ;I<%M 9 p+ ;b<% ^? !" ,%, &'%$8#' -,%(+,%*' 7(*.6' ($- ('-*, 0, cN ,% IF *-'% #' $5,-*' <%M 9 F *-'% #' $5,-*' <% ^?!% 0,&.-8 p @;M<%IF 9 p @;^<%F? K." ,6:'-3$8 p+ _ ;p+ ;I<%M8 p+ ;b<%^< *'6($&$,% -,%(+,%*' 7(*.6' ($- ('-*, 0, cM ' p @_ ;p @;M<%IF8 p @;^<%F<8 9' >+, p+ ;b<% M#, (-$($-&.$"'-)' +" ('3$ 0, .N ," #+3'- 0, .^?

!" &$"&#+%.7" "$ ,1.%*, "."3J" ,>+.#.:-.$ %,('-'0$-8 9 ,# J".&$ ,>+.#.:-.$ '3-+('A

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6.5. APLICACIONES: EL MODELO DE SPENCE DE SEÑALIZACIÓNEN EL MERCADO LABORAL

!# 6$0,#$ 0, K(,"&, (#'"*,' -.3+-$%'6,"*, #' .0,' 0, >+, ,# ".B,# 0, ,%*+0.$% 0, +"*-':'2'0$- (+,0, '&*+'-8 '"*, #'% ,6(-,%'% >+, ($0-)'" &$"*-'*'-#$8 &$6$ %,Z'# ."0.&'0$A

-' 0, %+ (-$0+&*.B.0'0 #':$-'#? K, *-'*' 0, +"$ 0, #$% *-':'2$% (.$",-$% ," ,# &'6($ '/$-'##'6'0$ 0, #' %,Z'#.='&.7"? X+, (+:#.&'0$ ," Mf]P 9 ($- *'"*$ "$ ,%*4 ,1(-,%'0$ ," #$%

*E-6."$% 6$0,-"$% 0, #' *,$-)' 0, 2+,3$%? b, /,&/$8 ,% '"*,-.$- ' #' ,#':$-'&.7" 0, #$%-,5."'6.,"*$% 0,# ,>+.#.:-.$ 0, I'%/ ,%*+0.'0$% ," ,%*, &'()*+#$? K." ,6:'-3$8 (+,%*$>+, ($", 0, 6'".5.,%*$ '#3+"'% .0,'% 9 &+,%*.$",% :4%.&'% 0, #$% 2+,3$% 0."46.&$% &$"."5$-6'&.7" ."&$6(#,*' ;-,B.%.7" :'9,%.'"'8 &$6($-*'6.,"*$ 5+,-' 0,# ,>+.#.:-.$8 ,*&?</' ,%*.6+#'0$ ,# ,%*+0.$ 9 &-,'&.7" 0, '#3+"$% 0, ,%$% -,5."'6.,"*$%?

O'6$% ' 0,%'--$##'- ," 5$-6' %.6(#.5.&'0' +"' B,-%.7" 0, ,%*, 6$0,#$8 :'%'0' ," ,#*-'*'6.,"*$ 0, Q.-$#, ;Mfdd<8 >+, #$ &$"B.,-*, ," +" 2+,3$ :4%.&$ 0, %,Z'#.='&.7" &$"0$% *.($% 0, ,6.%$-8 +" &$"2+"*$ &$"*."+$ 0, 6,"%'2,% 9 0$% '&&.$",% 0,# -,&,(*$-? @$%-,%+#*'0$% 0, ,%*' '0'(*'&.7" 0,# 6$0,#$ %$" ,>+.B'#,"*,% ' #$% 0, $*-'% '0'(*'&.$",%64% #.*,-'#,% 0,# 6$0,#$ $-.3."'#8 ," #'% >+, B'-.$% -,&,(*$-,% &$6(.*," ," +" 2+,3$,%*4*.&$ *-'% -,&.:.- ,# 6,"%'2, 0,# ,6.%$-?

Descripción del juego

+7 I2A*-&) O '" 2$ ()*;*I*-&) <2' '"G&A' 2$ $,X'7 -' '"(2-,&" 'n \ R -'0*$-* 2$"*7*),& ^n \ *7 I2A*-&) QL <2' '" 2$* '01)'"*S T* '01)'"* *G'1(* & )'G4*Y* 7* &6')(*S+7 ()*;*I*-&) (,'$' 2$* 1)&-2G(,X,-*- 1L <2' 12'-' "') *7(*L 1* L & ;*I*L 1;? _7 "*;' -'<2C (,1& '"8 1')& 7* '01)'"* "J7& G&$&G' 7* 1)&;*;,7,-*- * 1),&),8 <8 -' <2' "' ()*(' -'7(,1& -' *7(* 1)&-2G(,v,-*- ? T* '7'GG,J$ -'7 $,v'7 -' '"(2-,&" 1&) '7 ()*;*I*-&) (,'$' G&".('" -' *-<2,",G,J$ G;'8 1< <2' -'1'$-'$ -' '"' $,v'78 1')& (*0;,C$ -' "2 (,1& '6'G(,v&?

O'6$% ' ,1(-,%'- ,# 2+,3$ '"*,-.$- &$6$ 2+,3$ :4%.&$ 0, %,Z'#.='&.7"8 9 ' &$"&-,*'-#'% -,3#'% 0,# 2+,3$ ;/'&.,"0$ '#3+"'% %.6(#.5.&'&.$",%< 9 #$% ('3$% $ 3'"'"&.'%?

!# 2+3'0$- M ,% ,# ,6.%$-? K+ ,%('&.$ 0, *.($% ,% B % [ 1*8 1;\ 0$"0, 1*b 1;? !#.3,+" 6,"%'2, ;'8 ^< à K % @!# @! >+, $:%,-B' ,# 2+3'0$- N8 >+, ,% ,# -,&,(*$-8 ,# &+'#,#.3, +"' '&&.7" * à [ ?8 @+ \8 0$"0, F %.3".5.&' ?G'1(* 9 L! %.3".5.&' @'G4*Y*8 9 %,'&':' ,# 2+,3$? F #$% 6,"%'2,% ;'8 ^< #,% ##'6'-,6$% *'6:.E" &$"*-'*$%? !# &$%*, 0,'0>+.%.&.7" 0, ,%*+0.$% ('-' #$% 0$% *.($% ,% G;'8 1*<% 'G* 9 G;'8 1;<% 'G;8 %.,"0$Wa G*a G;? @' 5+"&.7" 0, (-$0+&*.B.0'0 A;'n 1< &$."&.0, &$" 18 0$"0, 1 ,% ,# *.($ ;(-$A0+&*.B.0'0< 0,# ,6.%$-n ($- *'"*$ #' (-$0+&*.B.0'0 ,%(,-'0' ' (-.$-.8 >+, 0,"$*'6$% &$A6$ r<8 ,% <1*! ;M. << 1;? F%) (+,%8 0'0$ ,# 6,"%'2, ;'8 ^<8 ,# B,&*$- 0, ('3$% %. ,#-,&,(*$- ,#.3, L! ,% ;W8 W< 9 %. ,# -,&,(*$- ,#.3, F ,%_

;^. 'G*8 1*.^< %. ,# *.($ 0,# ,6.%$- ,% 1*?

;^. 'G;8 1;.^< %. ,# *.($ 0,# ,6.%$- ,% 1;?

R:%E-B,%, >+, ," ,# (#'"*,'6.,"*$ '"*,-.$- /'9 B'-.'% %+($%.&.$",% .6($-*'"*,%_

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@' (-$0+&*.B.0'0 A;'n 1< 0,# *-':'2'0$- ('-' #' ,6(-,%' %7#$ 0,(,"0, 0, %+ *.($ 9"$ 0,# ".B,# 0, ,%*+0.$% '0>+.-.0$? !% +"' /.(7*,%.% %.6(#.5.&'0$-'?

!" #' X.3+-' Y?Mf ,%*4 #' -,(-,%,"*'&.7" ;'(-$1.6'0'< 0,# 2+,3$ ," 5$-6' ,1*,"%.B'?

Azar

Prob ( ) p qa 5

E E

Prob ( ) 1 p qb 5 2

E EE EE E

e0 e0e1 e1

e e

R

R R

R R R

A

A

A

AA ARE

RE

RE

RERE RE

R R A

A

RE

RE

w w

w ec2 a w ec2 b0 0

p w2a p w2b0 0

Figura 6.19 Juego del modelo de Spence en forma extensiva.

Análisis previo para el caso más simple de información perfecta

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m 1*8 6.,"*-'%>+, '# *-'*'- &$" +" ,6.%$- 0, :'2' (-$0+&*.B.0'0 1; ,#,3.-)' %+ '&&.7" F 0, '&,(*'&.7" %.9 %7#$ %. $:%,-B'-' +" 6,"%'2, ;'8 ^< 0,# ,6.%$- ! *'# >+, ^m 1;? !" -,%(+,%*' ' ,%,

(#'" 0, '&&.7" 0, L8 +" ,6.%$- ! 0, *.($ 1* ,#,3.-)' +" 6,"%'2, ;'8 ^< >+, 6'1.6.='-'^. 'G*8 #$ >+, &$"%,3+.-)' ,#.3.,"0$ ^% 1* 9 '% W8 9 +" ,6.%$- ! 0, *.($ 1; ,#,3.-)'+" 6,"%'2, ;'8 ^< >+, 6'1.6.='-' ^. 'G;8 #$ >+, &$"%,3+.-)' ,#.3.,"0$ ^% 1;9 '%W?

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

0, *-':'2'0$- %." (,-2+0.&'- '# $*-$ ;'&,(*'"0$ #' ,6(-,%' ('3'-#, +" %'#'-.$ 6'9$-<8(,-2+0.&'-)' ' #' ,6(-,%'?

!" #' X.3+-' Y?NW %, /' -,(-,%,"*'0$ ,# -,%+#*'0$ 0, #' ."0+&&.7" /'&.' '*-4%?

Azar

Prob ( ) p qa 5

E E

Prob ( ) 1 p qb 5 2

E EE EE E

e0 e0e1 e1

e e

R

R R

R R R

A

A

A

AA A

A

RE

RE

RE

RERE RE

RE

R R

A

A

RE

RE

w p5 a w p5 b

w ec2 a w ec2 b0 0

p w2a p w2b0 0

Figura 6.20 Inducción hacia atrás en el caso de información perfecta.

R*-$ 6$0$ J*.# 0, B.%+'#.='- ,%*, -,%+#*'0$ ,% -,(-,%,"*'-#$ ," ,# (#'"$ ;'8 ^< 0, #'%B'-.':#,% ".B,# 0, ,%*+0.$% 9 %'#'-.$8 &$6$ %, /'&, ," #' X.3+-' Y?NM8 0$"0, %, 6+,%*-'"#'% &+-B'% 0, ."0.5,-,"&.' 0, '6:$% *.($% 0, 2+3'0$-?

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Figura 6.22 Equilibrios agrupadores en el juego del modelo de Spence.

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Figura 6.23 Algunas desviaciones significativas desde equilibrios agrupadores.

Equilibrios separadores en estrategias puras

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Equilibriosseparadores

w

g( ; )e pb

pa

wq

pb

ee p c ( )/5 2a b b p e p c ( )/5 2 a b a p

I ( )a K

I ( )b K

I ( ) {( , )/ }a b a b p e w w ec p5 2 5

I ( ) {( , )/ }b b b b p e w w ec p5 2 5

B

Q C

g( ; )e p

Y Zm*a

m*b

Figura 6.24 Equilibrios separadores en el juego del modelo de Spence.

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Análisis del caso particular en el que no es posible la señalización

U,-,&, #' (,"' ,%*+0.'- ,%*, &'%$8 *'6:.E" ' 6$0$ 0, -,5,-,"&.'? K+($"3'6$% >+, #'

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Información perfecta

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Azar

Prob ( ) p qa 5

E E

Prob ( ) 1 p qb 5 2

R R

R R

w0 w1

pa pb

(a) (b)

A

AA

ARE

RERE

RE

R R

R R

w0 w1

pa pb

A

AA

ARE

RERE

RE

pb pa pa

pb0 00 0

p pa b2 p pb a20 00 00 0

w

pa

p*a

wq

pb

p*b

I ( )b pb

I ( )b pa

I ( )a pa

I ( )a pb

g ( ; )e p

Figura 6.25 Caso sin variable señalizadora nivel de estudios. Información perfecta.

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Información imperfecta. Equilibrios agrupadores y separadores en estrategias puras

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Equilibrios agrupadoressin señalización (BQ)

w

pa

wq

pb

g( ; )e p

Equilibrios agrupadorescon señalización (BQC)

Equilibrios agrupadorescon señalización (BYZ)

B

Q C

Y Z

Figura 6.26 Equilibrios sin señalización versus equilibrios con señalización.

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Relación entre el modelo de Akerlof y el modelo de Spence

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7.1. INTRODUCCIÓN

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%#1./"'+#$ +,$ ",4,$ 6*# (1#&#4#" /".#4/'%'.#"%# ' 4/&9' #%'('> L &,"%/"*'&/@" 1#(#%/A1)'.,$ +' (1#3*"%' '"%#1/,15 (#1, 1#7#1/4' ' +,$ $*<2*#3,$ 6*# '9,1' 1#(1#$#"%'" +' ;+%/.'#%'('> !" +' .#4/4' #" 6*# (,4'.,$ 1#$(,"4#1 ' +' (1#3*"%' '"%#1/,1 (,41#.,$ 4'1 *"('$, 9'&/' '%18$ #" #+ 81<,+ = 1#4*&/1 :$%# *" (,&, .8$> L$) $#3*/1)'.,$ 9'$%' ++#3'1 '+",4, /"/&/'+>

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

U"%*/%/-'.#"%#5 +' 1'0@" 4# #$%' 4/-#1$/4'4 4# &,.(,1%'./#"%,$ $# #"&*#"%1' #" +'&1##"&/' 4# +,$ 2*3'4,1#$ #" 6*# #+ &,.(,1%'./#"%, (1#$#"%# (*#4# 4# '+3*"' .'"#1''7#&%'1 '+ &,.(,1%'./#"%, 7*%*1, 4# +,$ /"4/-/4*,$5 4# %'+ .'"#1' 6*# $# (*#4# &,"4/&/,A

"'1 &," #+ 7/" 4# $'&'1 ('1%/4, 4# :+ #" #+ 7*%*1,> ?/&9, 4# ,%1, .,4,5 $/ #+ &,.(,1%'A./#"%, 7*%*1, $# (*#4# -#1 /"7+*/4, (,1 #+ &,.(,1%'./#"%, (1#$#"%#5 #"%,"&#$ *"

/"4/-/4*, $# -#18 #" &/#1%' .#4/4' #$%/.*+'4, ' #-/%'1 '6*#++'$ '&&/,"#$ 6*# (*#4'"3#"#1'1 *" &'$%/3, 7*%*1, = ' 1#'+/0'1 '6*#++'$ 6*# (*#4'" -#1$# 1#&,.(#"$'4'$ .8$'4#+'"%#>

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

Factor de descuento, valor presente descontado y pago medio

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!+ -'+,1 , *%/+/4'4 6*# +,$ '3#"%#$ #&,"@./&,$ '%1/<*=#" ' 1#&/</1 *"' &'"%/4'4 4#4/"#1, I, &*'+6*/#1 ,%1, 9#&9, '+ 6*# '%1/<*='" *%/+/4'4J %/#"# *" &'18&%#1 %#.(,1'+5 4#A

(#"4# 4#+ .,.#"%, #" 6*# 9'=' 4# (1,4*&/1$# +' 1#&#(&/@"> !" ('1%/&*+'15 +' *%/+/4'4 6*#*" '3#"%# '%1/<*=# #" #+ .,.#"%, (1#$#"%# " ' &,<1'1 .8$ %'14# *"' 4#%#1./"'4' &'"%/A4'4 4# 4/"#1, 4#(#"4# 4#+ .,.#"%, " ñ #" 6*# $# -'=' ' #7#&%*'1 #+ &,<1,>

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

(1,.#$' ", $# &*.(+'> T' $#3*"4' 1'0@" (1,&#4# 4#+ 9#&9, 4# 6*# 4/$(,"#1 '9,1' 4#+4/"#1, ",$ 4' .8$ ,(,1%*"/4'4#$ 4# '&%*'&/@" 6*# 4/$(,"#1 4# :+ #" #+ 7*%*1,5 '*"6*#

Juegos repetidos 407

$#' $#3*1, 6*# #$' (1,.#$' 4# 4/$(,"/</+/4'4 7*%*1' $# -' ' &*.(+/1> !" -/1%*4 4# #$%'$#3*"4' 1'0@"5 /"&+*$, '+ (1#$%'1 4/"#1, '+ 3,</#1", I(,1 #2#.(+,5 &,.(1'"4, T#%1'$ 4#+[#$,1,J5 6*# &,"++#-' #" +,$ (')$#$ 4#$'11,++'4,$ *" 1/#$3, .)"/., 4# (#14#1 $* 4/"#1,5$# #N/3# *" %/(, 4# /"%#1:$ 6*# &,.(#"$# 4# +' (:14/4' 4# ,(,1%*"/4'4#$ 6*# +' 4/$(,"/A</+/4'4 4#+ 4/"#1, <1/"4'>

\,1.'+/0'1#.,$ #+ (1/.#1 7'&%,1 .#4/'"%# *"' (1,<'</+/4'4 b = #+ $#3*"4, .#4/'"%#*"' %'$' 4# (1#7#1#"&/' %#.(,1'+ , %/(, 4# /"%#1:$ a I#" %'"%, (,1 *",J5 4,"4# Xm bm ]5Xm a>

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F,"$/4#1'"4, &,"2*"%'.#"%# '.<'$ 1'0,"#$5 1#$*+%' 6*# #+ '3#"%# #" &*#$%/@" #$ /"A4/7#1#"%# #"%1# +' (1,.#$' I&," (1,<'</+/4'4 b 4# &*.(+/./#"%,J 4# &,<1'1 F #" #+ /"$A%'"%# " ! ]5 = %#"#1 +' $#3*1/4'4 4# &,<1'1 bF #" #$# ./$., /"$%'"%#> B,1 ,%1' ('1%#5 #$

/"4/7#1#"%# #"%1# &,<1'1 bF #" #+ /"$%'"%# " ! ] I$*(*#$%, 6*# #+ &*.(+/./#"%, 4#+ &,<1,#$ $#3*1,J = &,<1'1 bF/ I]!aJ #" #+ /"$%'"%# " >

!" 1#$*.#"5 = ++'.'"4, d '+ &,&/#"%# b/ I]!aJ5 &,"&+*/.,$ 6*# #+ '3#"%# #$ /"4/7#A1#"%# #"%1# &,<1'1 F #" " ! ] = &,<1'1 Fñ% dF #" " >

Definición 7.1

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?/&9, 4# ,%1, .,4,5 #+ 7'&%,1 4# 4#$&*#"%, d #$ #+ -'+,15 #" *"/4'4#$ 4# ('3, '&%*'A+#$5 4# *"' *"/4'4 4# ('3, 6*# $# 9'18 #7#&%/-' *" (#1/,4, .8$ %'14#>

T' 4#7/"/&/@" '"%#1/,1 ",$ (1,(,1&/,"' *" .,4#+, $/.(+# I*" ;"/&, 7'&%,1 .*+%/(+/&'A%/-,J 4# (1#7#1#"&/' %#.(,1'+ 6*# /"&,1(,1' +'$ 4,$ 1'0,"#$ <8$/&'$5 '"%#1/,1.#"%# #N(+/A&'4'$5 (,1 +'$ &*'+#$ *" <#"#7/&/, , *%/+/4'4 #" #+ (1#$#"%# #$ (1#7#1/4, ' *" <#"#7/&/,",./"'+.#"%# /3*'+5 (#1, ' 1#&/</1 #" #+ 7*%*1,> T,$ -'+,1#$ #N%1#.,$5 d%X = d% ]5&,11#$(,"4#"5 1#$(#&%/-'.#"%#5 '+ &'$, #" 6*# #+ '3#"%# $@+, -'+,1' #+ (1#$#"%# I/.('A&/#"&/' #N%1#.'J = '+ &'$, #" 6*# -'+,1' #" /3*'+ .#4/4' #+ 7*%*1, 6*# #+ (1#$#"%#I('&/#"&/' #N%1#.'J> ?'4, *" 7'&%,1 4# 4#$&*#"%, d5 $# ++'.' %'$' 4# 4#$&*#"%, '$,&/'4'5

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!$ 78&/+ '9,1'5 <'$8"4,$# #" +' ?#7/"/&/@" Y>]5 &,.('1'1 4,$ ('3,$ &*=' 7#&9' 4/7/#1##" *" ";.#1, &*'+6*/#1' 4# /"%#1-'+,$ %#.(,1'+#$ <8$/&,$>

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F/ d$ ' ('3'1 #" #+ /"$%'"%# " !$

ñññ

F/ d^ ' ('3'1 #" #+ /"$%'"%# " !^F/ d ' ('3'1 #" #+ /"$%'"%# " !]F ' ('3'1 #" #+ /"$%'"%# "

F,., .*&9'$ 4#&/$/,"#$ #&,"@./&'$5 %'"%, /"4/-/4*'+#$ &,., 4# 31*(,5 /"-,+*&1'"

+' #N/$%#"&/' 4# 7+*2,$ 4# ('3,$5 &,"$%/%*/4,$ (,1 *"' $#&*#"&/' 4# ('3,$ 1#'+/0'4,$ #"4/7#1#"%#$ 7#&9'$5 #$ (1#&/$, $'<#1 &,.('1'1 *",$ 7+*2,$ &," ,%1,$ #" %:1./",$ 4# +'$(1#7#1#"&/'$ 4#+ 4#&/$,1> !$' &,.('1'&/@" ",$ +' (,$/</+/%' #+ &,"&#(%, 4# 7'&%,1 4# 4#$A&*#"%,> L4#.8$5 #$ &,"-#"/#"%# 4/$(,"#1 4# *"' #$(#&/# 4# *"/4'4 4# .#4/4'5 #$ 4#&/154# *"' 1#7#1#"&/' 6*# (#1./%' $/$%#.'%/0'1 ,14#"'4'.#"%# +'$ &,.('1'&/,"#$ '+*4/4'$>!$' 1#7#1#"&/' #$ #+ -'+,1 (1#$#"%#5 6*# $# 4#7/"/18 ' &,"%/"*'&/@">

Definición 7.2

D#' _%" " %]5^5>>> *"' $#&*#"&/' , 7+*2, 4# ('3,$5 7/"/%' , /"7/"/%'>

/5 ?'4, *" 7'&%,1 4# 4#$&*#"%, d5 #+ 6/0%' ('#&#-)# +#&.%-)/+% 4# +' $#&*#"&/'4# ('3,$ _%" " %]5^5>>> $# 4#",%' aBb_%" " %]5^5>>>5 dc5 = $# 4#7/"# &,.,

aBb_%" " %]5^5>>>5 dc%%]!p%^! d^%V!ñ!d" .]%" !ñ%;"

%" d" .]

45 ?'4, *" 7'&%,1 4# 4#$&*#"%, d5 #+ (/$% 3#+*% 4# +' $*&#$/@" 4# ('3,$_%" " %]5^5>>> #$ #+ ('3, 7/2, %& %'+ 6*# +' $#&*#"&/' &,"$%'"%# _%&" `" %]5^5>>>5 4,"4# %&" %%& O " 5%/#"# #+ ./$., -'+,1 (1#$#"%# 4#$&,"%'4, 6*# +' $*&#$/@" 4# ('3,$ /"/&/'+ _ %" " %]5^5>>>>

d#&,14#.,$ +' 7@1.*+' 4# +' $*.' 4# +,$ %:1./",$ 4# *"' (1,31#$/@" 3#,.:%1/&' 4#1'0@" d = (1/.#1 %:1./", /3*'+ ' +' *"/4'45 (*#$ ",$ $#18 ;%/+ #" +'$ 4/$&*$/,"#$ 6*#$/3*#"> ?'4' +' $*&#$/@" 3#,.:%1/&' _'" " %]5^5>>> 4,"4# '" %d" .]

O " M

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=: D/ d% ]5 #+ ('3, .#4/, %e 4#+ 7+*2, , $#&*#"&/' 7/"/%' 4# ('3,$ _%" " %]5^5>>>5( #$$/.(+#.#"%# +' .#4/' '1/%.:%/&' 4# +,$ -'+,1#$ 4# 4/&9' $#&*#"&/'5 =' 6*#5

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%e% I]. dJaBb_%" " %]5^5>>>5ä5 dc

Ejemplo 7.1

T' ['<+' Y>] /+*$%1' +,$ -'+,1#$ (1#$#"%#$ = ('3,$ .#4/,$ 4# '+3*"'$ $#&*#"&/'$ 4#('3,$5 ('1' 4/$%/"%,$ 7'&%,1#$ 4# 4#$&*#"%,> T'$ &*'%1, (1/.#1'$ $#&*#"&/'$ $," 7/"/%'$ =

+'$ &*'%1, ;+%/.'$ /"7/"/%'$>

Tabla 7.1

?/.)%'+# +#&."#-)%

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A/0%' ('#&#-)# B/$% 3#+*%

da ] _%" " %]5^5 >>>5(

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Tabla 7.1 (Continuación)

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Vectores de pagos factibles de un juego

Definición 7.3

?'4, *" 2*#3, 7/"/%, )% _* ]5 >>>5 * !g +]5 >>>5 +!`5 4#&/.,$ 6*# #+ -#&%,1 4# ('3,$

I , ]5 , 5 >>>5 , !J #$ ,/.)*40# , /0./-2/40# $/ #$ *"' &,.</"'&/@" +/"#'+ &,"-#N' 4# +,$-#&%,1#$ 4# ('3,$ &,11#$(,"4/#"%#$ ' (#17/+#$ 4# #$%1'%#3/'$ (*1'$ 4# )>

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Pagos a J2

Pagos a J1S(1, 1)

R(5, 0)

P(0, 5)

Q(4, 4)

Pagos a J2

Pagos a J1

S(1, 1)

R(2, 0)

P(0, 2)

Q(5, 5)

Pagos a J2

Pagos a J1S(0, 0)

R(0, 0)

P(3, 7)

Q(5, 5)

Pagos a J2

Pagos a J1S(0, 0)

R(0, 0)

P(1, 2)

Q(2, 1)

(a) Dilema del prisionero (b) Caza del ciervo

(c) Juego de disuasión 1 (d) Batalla de los sexos

Figura 7.1 Vectores de pagos factibles.

Algunos ejemplos introductorios

Ejemplo 7.2

F,"$/4#1#.,$ #+ &'$, #" #+ 6*# #+ 2*#3, 4# +' &'0' 4#+ &/#1-,5 4#7/"/4, #" #+ !2#.(+,^>W5 $# 1#(/%# 4,$ -#&#$> L &,"%/"*'&/@" .,$%1'.,$5 ('1' #+ &'$, ('1%/&*+'1 #" 6*#a% W = h% ]5 +' 1#(1#$#"%'&/@" #" 7,1.' #$%1'%:3/&' 4#+ 2*#3, 4# #%'(' = +' 1#(1#A$#"%'&/@" #" 7,1.' #N%#"$/-' 4#+ 2*#3, 3+,<'+ I#" :$%' $# 9' '<1#-/'4, .//01#2# .#A

4/'"%# FffB = 3+452# 6-17#1 .#4/'"%# KT5 = $# 9'" ,<%#"/4, +,$ ('3,$ 4#+ 2*#3,3+,<'+ $*.'"4, +,$ 4#+ 2*#3, 4# #%'('J>

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J1 J1J1 J1

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COO BL BLCOO

COO COO COO COOBL BL BL BL

COO BL BL BL BL BL BL BL BLCOOCOO

COO COO COOCOO

COO

10 5 7 6 5 0 2 1 7 2 4 3 6 1 3 210 7 5 6 7 4 2 3 5 2 0 1 6 3 1 2

Figura 7.2 Caza del ciervo repetido dos veces, con d%1.

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Ejemplo 7.3

F,"$/4#1#.,$ '9,1' #+ &'$, #" #+ 6*# #+ 2*#3, 4# #%'(' #$ #+ 4/+#.' 4#+ (1/$/,"#1,54#7/"/4, #" #+ !2#.(+, ^>]> L &,"%/"*'&/@" .,$%1'.,$ 4# "*#-, +' 7,1.' #$%1'%:3/&'4#+ 2*#3, 4# #%'('>

C"#$% +# #)/(/D +*0#3/ +#0 ('*&*%-#'%

B'#&% <

F'++'1 F,"7#$'1

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F,"7#$'1 W5 X ]5 ]

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45 F,"$/4:1#$# '9,1' 6*# #+ ";.#1, 4# #%'('$ $# 4#&/4' +'"0'"4, *" 4'4, #6*/+/A<1'4, %1'$ &'4' #%'('5 = &,"&+*=#"4, #+ 2*#3, $@+, $/ $# ,<%/#"# *" H> D*(@"3'$# %'.A</:" 6*# +,$ ('3,$ 3+,<'+#$ $# ,<%/#"#" $*.'"4, +,$ ('3,$ 4# +'$ #%'('$ #7#&%/-'.#"%#

2*3'4'$ I#$ 4#&/15 6*# +' (1#7#1#"&/' %#.(,1'+ (*1' $# &'1'&%#1/0' (,1 *"' ('&/#"&/'#N%1#.' #" +,$ 2*3'4,1#$J> !$%# #$ *" 2*#3, 1#(#%/4, /"7/"/%'.#"%#> T'$ $/3*/#"%#$ $,"'+3*"'$ #$%1'%#3/'$ 4#+ 2*3'4,1 ] 6*# (,41)'.,$ ++'.'1 /"&,"4/&/,"'4'$ I(,16*# ", 9'A&#" 4#(#"4#1 4#+ 4#$'11,++, (1#-/, 4#+ 2*#3, +'$ '&&/,"#$ 4#&/4/4'$ #" &'4' #%'('JM

./!;142# #" &*'+6*/#1 #%'('>

.2992# #" &*'+6*/#1 #%'('>

./!;142# &," (1,<'</+/4'4 ]/ V = .2992# &," (1,<'</+/4'4 ^/ V5 #" &*'+6*/#1 #%'('>

./!;142# #" +'$ #%'('$ ('1#$ = .2992# #" +'$ /.('1#$>

B,1 #+ &,"%1'1/,5 +'$ $/3*/#"%#$ $," '+3*"'$ #$%1'%#3/'$ &,"4/&/,"'4'$ 4#+ 2*3'4,1 ]M

./!;142# #" +' (1/.#1' #%'('5 .2992# #" +'$ (,$%#1/,1#$ $/ #+ 2*3'4,1 ^ &'++@ #" +'(1/.#1'5 = ./!;142# #" +'$ (,$%#1/,1#$ $/ #+ 2*3'4,1 ^ &,"7#$@ #" +' (1/.#1'>

.2992# #" +' (1/.#1' #%'(' = 1#(#%/1 #" +' #%'(' I" !]JA<4-$2 +, 6*# #+ 2*3'4,1 ^9'=' 9#&9, #" +' " A<4-$2>

B'1#&# #-/4#"%# 6*#5 %'.</:" '9,1'5 #+ (#17/+ #$%1'%:3/&, #" #+ 6*# &'4' 2*3'4,1 $#(1,(,"# ./!;142# #" %,4,$ $*$ &,"2*"%,$ 4# /"7,1.'&/@" #$ *" !EBD> D/" #.<'13,5

P$#3*/18 $/#"4, #+ ;"/&, !EBDQ !" +'$ (1@N/.'$ $#&&/,"#$ $# &,"%#$%'18 ' #$%' (1#3*"%'>

Ejemplo 7.4

D*(,"3'.,$ '9,1' 6*# #+ 2*#3, 4# #%'(' #$ #+ 2*#3, 4# 4/$*'$/@" ]5 4#7/"/4, #" #+!2#.(+, G>]> L &,"%/"*'&/@" $# .*#$%1'5 ('1' 4,$ #%'('$5 +' 1#(1#$#"%'&/@" #" 7,1.'#N%#"$/-' 4#+ 2*#3, 3+,<'+ I#" :$%' $# 9' '<1#-/'4, =/ 1!"#2# (,1 =15 >!"#2# (,1 > 5./$01"-# ?+#/ (,1 .? = ./$01"-# 4+2v1 (,1 .45 = '4#.8$ $# 9'" ,<%#"/4, +,$ ('3,$4#+ 2*#3, 3+,<'+ $*.'"4, +,$ 4#+ 2*#3, 4# #%'('J>

!+ 2*#3, 4# #%'(' %/#"# '6*) 4,$ !E5 6*# $," I> 5 .4J = I =15 .? J5 (#1, $@+, #+ (1/.#A

1, 4# #++,$ #$ !EBD5 %'+ &,., $# (*$, 4# .'"/7/#$%, #" #+ !2#.(+, G>]W> ?#</4, ' 6*##+ 2*#3, 4# #%'(' #$ #" #$%# &'$, 4/"8./&, 4# /"7,1.'&/@" (#17#&%'5 %'.</:" +, #$ #+ 2*#3, 1#(#%/4, , 3+,<'+5 '+ &,"%1'1/, 4# +, 6*# 9' ,&*11/4, #" +,$ 4,$ #2#.(+,$ '"%#1/,A1#$5 &*=, 2*#3, 4# #%'(' #$ #$%8%/&,> !" #$%# &'$, '.<,$ 2*3'4,1#$ %/#"#" G &,"2*"%,$4# /"7,1.'&/@"5 I*", &,11#$(,"4/#"%# ' +' (1/.#1' #%'(' = V ' +' $#3*"4'J>

B,1 $#1 4# /"7,1.'&/@" (#17#&%'5 #$%# 2*#3, #$ 1#$,+*<+# (,1 /"4*&&/@" 9'&/' '%18$5+, 6*# ",$ (#1./%# /4#"%/7/&'1 #+ ;"/&, !EBD 4#+ 2*#3,5 6*# #$ 4% I4]5 $^J5 4,"4# 4] #$+' #$%1'%#3/' 4# !E[dfE &,"$/$%#"%# #" >!"#2# #" &*'+6*/#1' 4# $*$ &,"2*"%,$ 4# /"A7,1.'&/@"5 = 4^ #$ +' #$%1'%#3/' 4# UEFOCKdfE &,"$/$%#"%# #" ./$01"-# 4+2v1 #"&*'+6*/#1' 4# $*$ &,"2*"%,$ 4# /"7,1.'&/@"> D# 9' $#Z'+'4, .#4/'"%# +)"#'$ 31*#$'$ #"

+' \/3*1' Y>V>

1.ª etapa

2.ª etapa

ENTRON

ENTRONENTRONENTRON

INCUMBRON

INCUMBRON

INCUMBRON INCUMBRON

Ne

Ne Ne Ne

E

EEE

Cd

Cd Cd CdCs Cs Cs

Cs

x1

x2

x3 x4 x5

x7 x8 x6

614

37

812

37

00

55

812

55

1010

Figura 7.3 Juego de disuasión 1 repetido dos veces, con d%1.

L+ /"%#1(1#%'1 #$%# 2*#3, 1#(#%/4,5 #$ .8$ 1#'+/$%' &,"$/4#1'1 6*# $# %1'%' 4# 4,$.#1&'4,$ 4/$%/"%,$5 '*"6*# /4:"%/&,$ #" ('3,$ I#" +*3'1 4# *" ;"/&, .#1&'4,J5 4# *"';"/&' #.(1#$' =' /"$%'+'4'5 6*# #$ UEFOCKdfE5 = 4# *"' (,%#"&/'+ #"%1'"%# !EA[dfE I, -'1/'$ /4:"%/&'$J 6*# %,.' $#&*#"&/'+.#"%# 4#&/$/,"#$ 4# #"%1'4' #" +,$ 4/$A

%/"%,$ .#1&'4,$>

7.2. JUEGOS REPETIDOS EN UN NÚMERO FINITO DE ETAPAS

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7,1.' #$%1'%:3/&'5 =' 6*# #++, $/.(+/7/&' +'$ (1@N/.'$ 4#7/"/&/,"#$> B'1' #-/%'1 &,"7*$/,A"#$ 4# ",%'&/@" #"%1# +,$ #+#.#"%,$ 6*# 4#7/"#" *" 2*#3, 4# #%'(' = +,$ 6*# &,11#$(,"A4#" '+ 2*#3, 1#(#%/4,5 '4,(%'1#.,$ +' $/3*/#"%# &,"-#"&/@"M

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Definición 7.4

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/5 L"%#$ 4# 6*# #.(/#&# #+ 2*#3, #$ 4# 4,./"/, (;<+/&,M

/;5 j*# #+ 7'&%,1 4# 4#$&*#"%, 4# &'4' 2*3'4,1 - #$ d->/<5 j*# ) $# -' ' 2*3'1 ( -#&#$>/=5 j*# +,$ ('3,$ 4# )( IJ $,"5 ('1' &'4' 2*3'4,1 -5 #+ -'+,1 (1#$#"%# I('1'

#+ 7'&%,1 4# 4#$&*#"%, d-J 4# +' $#&*#"&/' 7/"/%' 4# $*$ ('3,$ 4# #%'('>

45 L"%#$ 4# 6*# #.(/#&# &*'+6*/#1 #%'('5 $," 4# 4,./"/, (;<+/&, +'$ 2*3'4'$1#'+/0'4'$ #" %,4'$ +'$ #%'('$ '"%#1/,1#$>

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<: f<$:1-#$# 6*# '6*) ", $# &,"$/4#1' #+ 7'&%,1 4# /"&#1%/4*.<1# I(1,<'</+/4'4 4#6*# ", $# 9'3' #7#&%/-, *" ('3,J 6*# (*#4# /"%#1-#"/1 #" ,%1'$ ,&'$/,"#$ #" +' 4#%#1./A"'&/@" 4#+ 7'&%,1 4# 4#$&*#"%,> !$ 4#&/15 d -/#"# 4#%#1./"'4, ;"/&'.#"%# (,1 +' (1#7#A1#"&/' 4# +/6*/4#0>

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=' $," (+'"#$ &,.(+#%,$ 4# '&&/@"5 = '4#.8$ )( IdJ (*#4# %#"#1 ,%1,$ $*<2*#3,$ '('1%#4# +,$ '"%#1/,1.#"%# 4#$&1/%,$> B,1 #2#.(+,5 #" #+ 2*#3, 4# 4/$*'$/@" ] 1#(#%/4, 4,$ -#A&#$5 1#(1#$#"%'4, #" +' \/3*1' Y>V5 UEFOCKdfE /"/&/' *" $*<2*#3, #" #+ ",4, 4# 4#&/A$/@" , 5 ' (#$'1 4# 6*# ", #$ /"/&/, 4# #%'('>

Historias, estrategias y subjuegos

B'1' (,4#1 4#7/"/1 &," 1/3,1 +,$ &,"&#(%,$ 4# #$%1'%#3/' = 4# $*<2*#3, #" *" 2*#3, 1#(#%/A4,5 #$ ;%/+ (1#&/$'1 #" (1/.#1 +*3'1 #+ &,"&#(%, 4# 9/$%,1/' 4#+ 2*#3, #" *" (#1/,4, ,#%'(' &,"&1#%,>

Definición 7.5

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B " %__2: ]5 2:

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: - à @-`

4,"4# 2: - 9' 4# /"%#1(1#%'1$# &,., +' '&&/@" 1#'+/0'4' (,1 #+ 2*3'4,1 - #" +' #%'(' : >

L$) (*#$5 $/ ++'.'.,$ @% @]#ñ# @! '+ &,"2*"%, 4# +,$ (#17/+#$ #" #$%1'%#3/'$

(*1'$ 4#+ 2*#3, 4# #%'(' )5 &'4' " A9/$%,1/' #$ *" #+#.#"%, 4# @" .]> F,"-#"3'.,$ #" 6*##N/$%# *"' ;"/&' ]A9/$%,1/'5 6*# /4#"%/7/&'.,$ &," #+ ",4, /"/&/'+5 = ++'.#.,$ B '+ &,"A 2*"%, 4# %,4'$ +'$ 9/$%,1/'$ 4#+ 2*#3, 1#(#%/4, )( IdJ5 #$ 4#&/15

B %

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B "

f<$:1-#$# 6*# +,$ 1#$*+%'4,$ 4#+ 2*#3, 1#(#%/4,5 #" #+ $#"%/4, 4# 4#$'11,++,$ #7#&%/A-,$ , %1'=#&%,1/'$ &,.(+#%'$5 $," +'$ ( !]A9/$%,1/'$ _I2:

]5 2: ^5 >>>5 2:

!J`: %]5^5>>>5( > !$ 4#&/15

+,$ 1#$*+%'4,$ 4#+ 2*#3, 1#(#%/4, #$(#&/7/&'" *" (#17/+ 4# '&&/,"#$ 1#'+/0'4'$ (,1 &'4'#%'(' 4#+ 2*#3,>

Ejemplo 7.5

D/ #+ 2*#3, 4# #%'(' ) #$ #+ 4/+#.' 4#+ (1/$/,"#1,5 &,., #" #+ !2#.(+, Y>V5 $* &,"2*"%,4# (#17/+#$ #$%1'%:3/&,$ %/#"# GI%^#^J #+#.#"%,$> B,1 %'"%,5 #" #+ 2*#3, )( IdJM

k'= GI%G]J 9/$%,1/'$ 9'$%' #+ .,.#"%, ^5 %'"%'$ &,., (#17/+#$ #" #$%1'%#3/'$

(*1'$ I#$ 4#&/15 .,4,$ 4# 2*3'1 (,$/<+#$ &," #$%1'%#3/'$ (*1'$J> !"*.#1:.,$+'$M B % _I./!;142# 5 ./!;142# J5 I./!;142# 5 .2992# J5 I.2992# 5 ./!;142# J5 I.2992# 5 .2992# J`>

k'= GW HA9/$%,1/'$> O"' 4# #++'$ #$ AH% II./!;142# 5 .2992# J5 I.2992# 5 ./!;142# J5I./!;142# 5 .2992# J5 I.2992# 5 ./!;142# J5 I./!;142# 5 .2992# JJ>

L(+/6*#.,$ '+ &,"%#N%, 4# 2*#3,$ 1#(#%/4,$ +'$ 4#7/"/&/,"#$ 3#"#1'+#$ 4# #$%1'%#3/' =4# $*<2*#3,M

?'4, #+ 2*#3, 4# #%'(' )% _ @]5 >>>5 @!g ']5 >>>5 '!` #" 7,1.' #$%1'%:3/&' = #+ 2*#3,1#(#%/4, )( IdJM

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- I>J5 *"' (,1 &'4'#%'(' : 5 4# .,4, 6*# 2:

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p-% I2]- IA]J5 2^

- IA^J5 >>>5 2( - IA( JJ

D/ +' '(+/&'&/@" #$ &,"$%'"%# #" &'4' #%'(' 4#&/.,$ 6*# +' #$%1'%#3/' #$ /"&,"4/&/,"'A4' , /"4#(#"4/#"%# 4# +' 9/$%,1/'> B,1 #2#.(+,5 +'$ #$%1'%#3/'$ R 3+452# 9-17#1 #" '.<'$

#%'('$S = R 3+452# 9-17#1 #" +' (1/.#1' #%'(' = .//01#2# #" +' $#3*"4'S5 $," '.<'$/"&,"4/&/,"'4'$ #" )^IdJ5 $/#"4, ) #+ 2*#3, 4# +' &'0' 4#+ &/#1-,5 ./#"%1'$ 6*# +' #$%1'A%#3/'$ R 3+452# 9-17#1 #" +' (1/.#1' #%'(' = &,(/'1 #" +' $#3*"4' #%'(' +' 2*3'4' 6*# #+,%1, 2*3'4,1 9/0, #" +' (1/.#1'S ", +, #$>

f<$:1-#$# 6*# #" +,$ 2*#3,$ 1#(#%/4,$ 9'= *"' #6*/-'+#"&/' &,.(+#%' #"%1# +'$ #$%1'A%#3/'$ ./N%'$ (1,(/'.#"%# 4/&9'$ I+,%#1)'$ 4# #$%'%#3/'$ (*1'$ 4#+ 2*#3, 1#(#%/4,J = +'$#$%1'%#3/'$ 4# &,.(,1%'./#"%, I+,%#1)'$ 4# '&&/,"#$ 4# ) #" &'4' #%'('J>

O" (#17/+ 4# #$%1'%#3/'$ (*1'$ , ./N%'$ p% Ip]5 >>>5 p-5 >>>5 p!J 4#+ 2*#3, 1#(#%/4,)( IdJ 4#%#1./"' I2*"%, &," +'$ &,11#$(,"4/#"%#$ 1#'+/0'&/,"#$ 4#+ '0'15 #" #+ &'$, 4##$%1'%#3/'$ ./N%'$5 = ' %1'-:$ 4# *" (1,&#$, /%#1'%/-,J +' $#&*#"&/' 4# &,.</"'&/,"#$ 4#'&&/,"#$

_I2: ]IA: IpJJ5 2:

^IA: IpJJ5 >>>5 2: !IA: IpJJJ`: %]5^5>>>5(

*"' #" &'4' #%'(' : 5 6*# $# (,"418" #" (18&%/&' 4# .'"#1' #7#&%/-'> D# 9' 4#",./"'4,A: IpJ ' +' : A9/$%,1/' '$) 4#%#1./"'4' (,1 p> L #$%' $#&*#"&/' 4# &,.</"'&/,"#$ $# +'++'.' %1'=#&%,1/' 4#+ 2*#3, 4#%#1./"'4' (,1 p> B,1 $* ('1%#5 #$%'$ &,.</"'&/,"#$ 4#%#1A./"'" +,$ ('3,$ 4# #%'(' = :$%,$ 4#%#1./"'"5 4# '&*#14, &," #+ 7'&%,1 4# 4#$&*#"%,5 +,$('3,$ 3+,<'+#$ ('1' &'4' 2*3'4,1 4#+ 2*#3, 1#(#%/4,>

T,$ &"4!"#$%& 4# )( IdJ &,./#"0'" '+ /"/&/'1$# &'4' #%'(' = %/#"#" +' 7,1.'

)( ." IdJ5 4,"4# Xm " a( > !N/$%#" %'"%,$ $*<2*#3,$ 6*# &,./#"&#" &," +' #%'(' " &,.," A9/$%,1/'$ $#'" (,$/<+#$>

Ejemplo 7.6

D#' #+ 2*#3, 4# #%'(' ) #+ 2*#3, 4# +' &'0' 4#+ &/#1-,> !$%*4/#.,$ &," 4#%#"/./#"%, #"#+ 2*#3, 1#(#%/4, ^ -#&#$ )ÎdJ #+ (#17/+ #$%1'%:3/&, p% Ip]5 p^J 4,"4# p] #$ +' #$%1'%#A3/' R.//01#2# #" +' (1/.#1' #%'(' = #" &'4' #%'(' (,$%#1/,1 : &,(/'1 +, 6*# 9'=' 9#&9,

#+ 2*3'4,1 ^ #" +' #%'(' : .]S5 = p^ #$ +' #$%1'%#3/' R 3+452# 9-17#1 #" +' (1/.#1' #%'(' =#" &'4' #%'(' (,$%#1/,1 : &,(/'1 +, 6*# 9'=' 9#&9, #+ 2*3'4,1 ] #" +' #%'(' : .]S>

T' &,.</"'&/@" 4# '&&/,"#$ 6*# 4/&9, (#17/+ 4#%#1./"' #" +' #%'(' ] #$ I2]]IA]IpJJ5

2]ÎA]IpJJJ% I.//01#2# 5 3+452# 9-17#1J> !$%' &,.</"'&/@" #$ +' Â9/$%,1/' 4#+ 2*#3,

4#%#1./"'4' (,1 #+ (#17/+ p5 = (,1 %'"%, +' ++'.'1#.,$ AÎpJ> F,",&/4' =' +' Â9/$%,1/'#7#&%/-'5 (*#4# (1,$#3*/1 #+ &8+&*+,> T' &,.</"'&/@" 4# '&&/,"#$ 6*# p 4#%#1./"' #" +'#%'(' ^ #$ I2^

]IAÎpJJ5 2^ÎAÎpJJJ%I 3+452# 9-17#15 .//01#2# J>

L$) (*#$5 +' $*&#$/@" &,.(+#%' 4# &,.</"'&/,"#$ 4# '&&/,"#$ 6*# &,"$%/%*=# #+1#$*+%'4, 4#+ 2*#3, $/ $# 2*#3' #+ (#17/+ #$%1'%:3/&, p #$ II.//01#2# 5 3+452# 9-17#1J5I 3+452# 9-17#15 .//01#2# JJ5 6*# 4#%#1./"' #+ -#&%,1 4# ('3,$ IX5 ^J #" +' (1/.#1' #%'('

= #+ -#&%,1 4# ('3,$ I^5 XJ #" +' $#3*"4'> !" &,"&+*$/@"5 #+ 1#$*+%'4, 4#+ 2*#3, #"%:1./",$ 4# ('3,$ #$ #+ -#&%,1 4# ('3,$ 3+,<'+#$ I^d5 ^J>

P!$ *" !E #+ (#17/+ #$%1'%:3/&, pQ D# &,.(1*#<' /".#4/'%'.#"%# 6*# ", +, #$ ('1'"/"3;" -'+,1 4# d> !" #7#&%,5 $/ db X5 +' #$%1'%#3/' p^ ", #$ 1#$(*#$%' @(%/.' 4#+ 2*3'A4,1 ^ ' p]5 4#</4, ' 6*# +' #$%1'%#3/' /"&,"4/&/,"'4' R 3+452# 9-17#1 #" &*'+6*/#1 #%'A('S 9'<1)' /.(+/&'4, +,$ 1#$*+%'4,$ 4# #%'(' I.//01#2# 5 3+452# 9-17#1J = I 3+452#

9-17#15 3+452# 9-17#1J5 6*# +# 9'<1)'" (#1./%/4, ,<%#"#1 *" ('3, 3+,<'+ ^! db ^> B,1,%1' ('1%#5 $/ d% X5 +' #$%1'%#3/' p] ", #$ 1#$(*#$%' @(%/.' 4#+ 2*3'4,1 ] ' p^5 4#</4, '6*# +' #$%1'%#3/' /"&,"4/&/,"'4' R 3+452# 9-17#1 #" &*'+6*/#1 #%'('S 9'<1)' /.(+/&'4,+,$ 1#$*+%'4,$ 4# #%'(' I 3+452# 9-17#15 3+452# 9-17#1J = I 3+452# 9-17#15 3+452# 9-17#1J5

6*# +# 9'<1)'" (#1./%/4, ,<%#"#1 *" ('3, 3+,<'+ ]!db ^d>

E,$ (1,(,"#.,$ '9,1' #$%*4/'1 +'$ $/3*/#"%#$ &*#$%/,"#$ 1#7#1#"%#$ ' +,$ #6*/+/<1/,$4# *" 2*#3, 1#(#%/4, )( IdJM

Pd#6*/#1#" +,$ !EBD 4#+ 2*#3, 1#(#%/4, 6*# $# 2*#3*# #" &'4' #%'(' '+3;" !E 4#+ 2*#3, 4# #%'(' )Q

P?#(#"4# +' &,"%#$%'&/@" ' +' (1#3*"%' '"%#1/,1 4#+ -'+,1 4# ( , 4#+ -'+,1 4# dQP?#(#"4# 4#+ ";.#1, 4# !E 4# )Q P?#(#"4# 4# 6*# #+ 2*#3, $# 1#(/%' 7/"/%' , /"7/"/%'A.#"%#Q

!" '6*#++,$ 2*#3,$ 4# #%'(' 6*#5 &,., #+ 4/+#.' 4#+ (1/$/,"#1,5 %/#"#" !E ;"/&,$B'1#%,A4,./"'4,$ (,1 (#17/+#$ 6*# (,41)'.,$ ++'.'1 4# R&,,(#1'&/@"S 6*# ", $," !E5P(*#4# #$%'1 +' &,,(#1'&/@" $,(,1%'4' (,1 #6*/+/<1/,$ 4#+ 2*#3, 1#(#%/4,Q5 #$ 4#&/15 P#N/$%#'+3;" !EBD 4#+ 2*#3, 1#(#%/4, 6*# /.(+/6*# 2*3'1 #" &'4' #%'(' #+ (#17/+ 4# &,,(#1'&/@"Q

C#1#&# +' (#"' '-'"0'1 /"%*/%/-'.#"%# '+3*"'$ 4# +'$ 1#$(*#$%'$> !" +,$ 2*#3,$ 1#(#A%/4,$ 7/"/%'.#"%#5 $#'" &*'+#$ $#'" +,$ -'+,1#$ 4# ( = 4# d5 $@+, #$ (,$/<+# 1#'+/0'1 2*3'A4'$ 4# #%'(' 6*# 7,1.' ('1%# 4# *" #6*/+/<1/, 4#+ 2*#3, 1#(#%/4, = 6*# ", $#'" 4##6*/+/<1/, 4# ) $/ #N/$%#" -'1/,$ !E #" )> B,1 %'"%,5 +' &,,(#1'&/@" #$ /"'+&'"0'<+# #"#6*/+/<1/, ('1' #+ 4/+#.' 4#+ (1/$/,"#1, 1#(#%/4, 7/"/%'.#"%#> D/" #.<'13,5 ('1' 2*#3,$1#(#%/4,$ /"7/"/%'.#"%# I= #" ('1%/&*+'1 ('1' #+ 4/+#.' 4#+ (1/$/,"#1,J $) #$ (,$/<+# #"A

&,"%1'1 #6*/+/<1/,$ 6*# $,(,1%#" +' &,,(#1'&/@"5 $/#.(1# 6*# d %#"3' *" -'+,1 $*7/&/#"%#A.#"%# &#1&'", ' ]>

Cálculo de los ENPS de un juego finitamente repetido

Teorema 7.1

D#' ) *" 2*#3, #" 7,1.' #$%1'%:3/&'>

/5 D/ ) %/#"# *" ;"/&, #6*/+/<1/, 4# E'$95 #"%,"&#$ ('1' &*'+6*/#1 ( 7/"/%, =&*'+6*/#1 7'&%,1 4# 4#$&*#"%, d5 #+ 2*#3, 1#(#%/4, )( IdJ %/#"# *" L-*.% !EBD5 6*#&,"$/$%# #" 6*# &'4' 2*3'4,1 2*#3*# 4# .'"#1' /"&,"4/&/,"'4' #" &'4' #%'(' $* ;"/&'#$%1'%#3/' 4# #6*/+/<1/,>

45 D/ %,4,$ +,$ #6*/+/<1/,$ 4# E'$9 4# ) 1#$*+%'" #" +,$ ./$.,$ ('3,$5 #"%,"&#$('1' &*'+6*/#1 ( 7/"/%, = &*'+6*/#1 7'&%,1 4# 4#$&*#"%, d5 &*'+6*/#1 !EBD 4#+ 2*#3,1#(#%/4, )( IdJ (1#$&1/<# 6*# #" &'4' #%'(' $# 2*#3*# *" !E 4#+ 2*#3, 4# #%'(' )>

M#3%&)'/.*8-D

K'$%' &," 4#.,$%1'1 I<J5 (*#$ I'J #$ *" &'$, ('1%/&*+'1> D#' p%

Ip]5 p^5 >>>5 p!J *"!EBD> d'0,"#.,$ (,1 /"4*&&/@" 9'&/' '%18$ 3#"#1'+/0'4'> !" &*'+6*/#1 $*<2*#3, &,A11#$(,"4/#"%# ' +' ;+%/.' #%'('5 #$ 4#&/15 %1'$ &*'+6*/#1 ( A9/$%,1/'5 p 9' 4# (1#$&1/</1*" !E 4#+ 2*#3, 4# #%'('> L$) (*#$5 #+ ('3, 4# #%'(' 6*# ' &'4' 2*3'4,1 +# -' '&,11#$(,"4#1 #" +' ;+%/.' #%'(' #$ /"4#(#"4/#"%# I$* ('3, #" #$# !E 2*3'4, #" +';+%/.' #%'('J 4# +' ( A9/$%,1/' 6*# 9' &,"4*&/4, 9'$%' #$' #%'('5 = (,1 %'"%,5 +' ,(%/.'A+/4'4 4# 1#$(*#$%'$ #" #+ 2*#3, 3+,<'+ $#18 #6*/-'+#"%# ' +' ,(%/.'+/4'4 #" #+ 2*#3,6*# '&'<' #" +' (#";+%/.' #%'('> d'0,"'"4, 4# /3*'+ .,4,5 (*#4# &,"&+*/1$# 6*# #"+' #%'(' (#";+%/.' 9' 4# 2*3'1$# %'.</:" *" !E 4# #%'('> d'0,"'"4, 1#(#%/4'.#"%#4# .,4, '"8+,3, I/"4*&%/-'.#"%#J $# &,"&+*=# 6*# #" +' (1/.#1' #%'(' 9' 4# 2*3'1$#

%'.</:" *" !E 4# #%'('> !" &,"&+*$/@"5 p 9' 4# (1#$&1/</1 #" &*'+6*/#1 #%'(' 6*# $# 2*#3*# *" #6*/+/<1/, 4# E'$9 4# #%'('>

Ejemplo 7.7

D#' #+ 4/+#.' 4#+ (1/$/,"#1, 1#(#%/4, ^ -#&#$5 &," 7'&%,1 4# 4#$&*#"%, /3*'+ ' ]> D#3;"+' /"4*&&/@" 9'&/' '%18$ 3#"#1'+/0'4'5 #" +' $#3*"4' #%'(' $# -' ' 2*3'1 #+ #6*/+/<1/, 4#E'$9 ;"/&, I./!;142# 5 ./!;142# J> !" &,"$#&*#"&/'5 #" +' (1/.#1' #%'(' +,$ 2*3'4,1#$'71,"%'" #+ 2*#3, $/3*/#"%#M

B'#&% <

F'++'1 F,"7#$'1

B'#&% ;F'++'1 W5 W ]5 H

F,"7#$'1 H5 ] ^5

!$%# 2*#3, #$ #6*/-'+#"%# #" ('3,$ '+ 4/+#.' 4#+ (1/$/,"#1, I$@+, $# 4/7#1#"&/'" #"#+ &'.</, 4# #$&'+'5 (,1 %1'$+'&/@"5 4# +'$ 7*"&/,"#$ 4# ('3,$5 (*#$ +,$ ('3,$ 4# #$%# 2*#3, 1#$*+%'" 4# $*.'1 *"' *"/4'4 ' +,$ 4#+ 4/+#.' 4#+ (1/$/,"#1,J> B,1 %'"%,5 $* ;"/&,

!E #$ I./!;142# 5 ./!;142# J> !" &,"$#&*#"&/'5 #+ ;"/&, 1#$*+%'4, (#17#&%, #" $*<2*#3,$#$ II./!;142# 5 ./!;142# J5 I./!;142# 5 ./!;142# JJ> !+ ;"/&, !EBD #$ '6*#+ #" 6*# &'4'

2*3'4,1 *$' +' #$%1'%#3/' /"&,"4/&/,"'+ R./!;142# #" +' (1/.#1' #%'('5 = ./!;142# #" +'$#3*"4'5 /"4#(#"4/#"%#.#"%# 4#+ 1#$*+%'4, 4# +' (1/.#1'S>

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

42# J> L$) (*#$5 #+ 1#$*+%'4, (#17#&%, #" $*<2*#3,$ I6*# #$ ;"/&,J #$ '6*#+ #" #+ 6*# &'4' 2*3'4,1 2*#3' #" &'4' #%'(' +' #$%1'%#3/' 6*# +# &,11#$(,"4# #" #+ !E 4#+ 2*#3, 4##%'('>

B,1 ,%1' ('1%#5 #+ !EBD I6*# %'.</:" #$ ;"/&,J #$ '6*#+ #" 6*# &'4' 2*3'4,1 #2#&*%'+' #$%1'%#3/' , (+'" &,.(+#%, $/3*/#"%#M Ri*3'1 ./!;142# $/#.(1#S>

74&#'6/.*8- 9:>

/5 d#&*:14#$# 6*# ", #$ +, ./$., 1#$*+%'4, (#17#&%, #" $*<2*#3,$ 6*# #6*/+/<1/,I4# E'$9J (#17#&%, #" $*<2*#3,$> !+ !EBD #$ *" (#17/+ #$%1'%:3/&, ./#"%1'$ 6*# #+ dBD #$*" 4#$'11,++, &,"&1#%, 4#+ 2*#3,> C/#"%1'$ 6*# *" (#17/+ #$%1'%:3/&, 4#%#1./"' #+ 4#$'11,A++, , %1'=#&%,1/' 4#+ 2*#3, 4#$4# #+ ",4, /"/&/'+ '+ ",4, %#1./"'+5 ", $# 1#4*&# ' 4/&9'%1'=#&%,1/'5 (*#$ %'.</:" #$(#&/7/&' 6*# 9'1)' &'4' 2*3'4,1 $/ $# #"&,"%1'1' #" *" ",4,4# 4#&/$/@" ", /"&+*/4, #" 4/&9' %1'=#&%,1/'> B*#4# ,&*11/1 6*# 4,$ (#17/+#$ 4/$%/"%,$ 4#A%#1./"#" +' ./$.' %1'=#&%,1/'> a#'.,$ +, 6*# ,&*11#5 #" #+ !2#.(+, Y>Y5 &," +,$ 4,$(#17/+#$ $/3*/#"%#$M

p% Ip]5 p^J 4,"4# p]%p^%R./!;142# $/#.(1#S

pñ% Ipñ]5 p ñ J 4,"4# pñ]% pñ% R./!;142# #" +' (1/.#1' #%'('5 = #" +' $#3*"4' #%'('./!;142# $/ #+ ,%1, &,"7#$@ #" +' (1/.#1' = .2992# $/ #+ ,%1, &'++@ #" +' (1/.#1'S

B*#$ </#"5 '.<,$ (#17/+#$ 4#%#1./"'" #+ ./$., 1#$*+%'4, , %1'=#&%,1/'5 #+ '11/<'4#$&1/%,5 &,., ;"/&, dBD> l $/" #.<'13,5 #+ (#17/+ p #$ !EBD5 ./#"%1'$ 6*# #+ (#17/+pñ ", +, #$>

45 d#&*:14#$# %'.</:" 6*# ", #$ +, ./$., *" $*<2*#3, 6*# *" 2*#3, 4# #%'('> T,$ 2*#3,$ 4# #%'(' $@+, %/#"#" #" &*#"%' +, 6*# $*&#4# #" +' #%'(' I/"4/&'" '6*#++' ('1%# 4#+ 2*#3, 6*# $# 1#(/%#J5 +,$ $*<2*#3,$ %/#"#" #" &*#"%' ('1' &'4' #%'(' ", $@+, #+ 2*#3, 4##%'(' $/", %'.</:" %,4, #+ 4#$'11,++, (,$%#1/,1 4#+ 2*#3, ' ('1%/1 4# #$' #%'(' I= $/ #+

2*#3, 4# #%'(' #$ 4/"8./&, = %/#"# /"7,1.'&/@" (#17#&%'5 9'= $*<2*#3,$ 6*# ", &,./#"A0'" '+ /"/&/, 4# *"' #%'('J>

Cálculo de los ENPS en el caso general en que cada juego de etapapuede tener más de un EN

!" #+ $/3*/#"%# %#,1#.' $# #$%'<+#&#" '+3*"'$ &,"4/&/,"#$ $#"&/++'$ I"#&#$'1/'$ #" #+ (1/A.#1 '('1%'4, = $*7/&/#"%#$ #" #+ $#3*"4,J ('1' 6*# *" (#17/+ #$%1'%:3/&, $#' !EBD #" *"

2*#3, 1#(#%/4, 7/"/%'.#"%#>

Teorema 7.2

D#' ) *" 2*#3, #" 7,1.' #$%1'%:3/&'5 = $#' #+ 2*#3, 1#(#%/4, )( IdJ 4,"4# ( #$ 7/"/%,>

/5 F*'+6*/#1 !EBD 4#+ 2*#3, 1#(#%/4, %/#"# 6*# 4'1 +*3'1 #" +' ;+%/.' #%'(' ' *"!E 4# )>

45 F*'+6*/#1' 4# +,$ 1#$*+%'4,$ Ia&I]J5 a&I^J5 >>>5 a&I( JJ5 4,"4# &'4' aeI: J #$ *"(#17/+ #$%1'%:3/&, 6*# #$ *" !E 4# )5 #$ *" 1#$*+%'4, (#17#&%, #" $*<2*#3,$ 4# )( IdJ>

L4#.8$5 #+ (#17/+ 4# #$%1'%#3/'$ p%

Ip]5 p^5 >>>5 p!J &,"$/$%#"%# #" 6*# &'4' 2*3'4,1 2*#3*# 4# .'"#1' /"&,"4/&/,"'4' #" +' #%'(' : A:$/.' +' #$%1'%#3/' 6*# +# &,11#$(,"4##" a&I: J #$ *" !EBD>

M#3%&)'/.*8-D

/5 D*(,"3'.,$ 6*# #+ (#17/+ #$%1'%:3/&, p% Ip]5 p^5 >>>5 p!J #$ *" !EBD 4#)( IdJ> !+ (#17/+ p 3#"#1' *" !E #" &*'+6*/#1' 4# +,$ $*<2*#3,$ 6*# &,./#"0'" #" +';+%/.' #%'('5 (#1, #$# $*<2*#3, ", #$ ,%1, 6*# #+ 2*#3, 4# #%'(' )5 = (,1 %'"%, p 9' 4#4'1 +*3'1 #" #$' #%'(' ' *" !E 4# )>

45 D#' Ia&I]J5 a&I^J5 >>>5 a&I( JJ *"' $#&*#"&/' 4# #6*/+/<1/,$ 4# E'$9 I/3*'+#$ ,

4/$%/"%,$J 4# )> D#' p% Ip]5 p^5 >>>5 p!J #+ (#17/+ 4# #$%1'%#3/'$ &,"$/$%#"%# #" 6*# &'4' 2*3'4,1 2*#3*# 4# .'"#1' /"&,"4/&/,"'4' #" +' #%'(' : A:$/.' +' #$%1'%#3/' 6*# +# &,A11#$(,"4# #" #+ (#17/+ : A:$/.,5 a&I: J5 4# +' $#&*#"&/' '"%#1/,1>

!$ 78&/+ &,.(1,<'1 6*#5 $/ +,$ 4#.8$ 2*3'4,1#$ $# &,"7,1.'" ' #$# (#17/+ p5 #$1#$(*#$%' @(%/.' 4# &*'+6*/#1 2*3'4,1 - 1#$(,"4#1 &," +' #$%1'%#3/' 6*# +# &,11#$(,"4##" 4/&9, (#17/+5 (*#$ #" &'4' #%'(' 2*3'18 *"' 1#$(*#$%' @(%/.' ' +'$ 2*3'4'$ 6*# +,$4#.8$ 1#'+/0'" #" #$' #%'('5 ,<%#"/#"4, '$) *" 7+*2, 4# ('3,$ &*=,$ ('3,$ 4# #%'('$," .8N/.,$> B,1 %'"%,5 p #$ *" !E 4# 4# )( IdJ>

B,1 ,%1' ('1%#5 = 4'4, *" $*<2*#3, &*'+6*/#1' 4# )( IdJ &,.#"0'"4, #" +' #%'(' : 5&,11#$(,"4/#"%# ' &*'+6*/#1 9/$%,1/' (1#-/' I/"&+*$, $/ $# 9'" (1,4*&/4, 4#$-/'&/,"#$

4#+ #6*/+/<1/, pJ5 (*#$%, 6*# ' ('1%/1 4# #$# .,.#"%, +,$ 4#.8$ 2*3'4,1#$ -'" ' 2*3'14# .'"#1' /"&,"4/&/,"'4' #" +'$ #%'('$ (,$%#1/,1#$ ' : +' #$%1'%#3/' 6*# +#$ &,11#$(,"A4# #" +' $#&*#"&/' 4# (#17/+#$ '"%#1/,15 +' 1#$(*#$%' @(%/.' 4# - #" #$'$ #%'('$ $#3*/18$/#"4, +' #$%1'%#3/' 6*# ' - +# &,11#$(,"4# #" 4/&9,$ (#17/+#$5 %'+ &,., (1#$&1/<# #+#6*/+/<1/, p> B,1 %'"%,5 p #$ *" !E 4# 4/&9, $*<2*#3,5 = #" &,"$#&*#"&/' *" !EBD 4#)( IdJ>

f<$:1-#$# 6*# #$%# %#,1#.' ", /.(/4# 6*# #N/$%'" !EBD '4/&/,"'+#$ ' +,$ .#"&/,"'A4,$ #" I<J> B,1 *"' ('1%#5 (,41)'" &,"$%1*/1$# !EBD 4#+ %/(, $/3*/#"%#M R\/2'4,$ 4,$ !E

4# #%'(' 4/$%/"%,$ ++'.'4,$ aeI]J = aeI^J5 2*3'1 aeI]J #" +' (1/.#1' #%'('5 = #" +'$ $/A3*/#"%#$ 2*3'1 $/#.(1# aeI]J5 $'+-, 6*# #$%#.,$ #" *"' 9/$%,1/' #" 6*# ", $# 9' 2*3'4,

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

Ejemplo 7.8

!" #+ 2*#3, 1#(#%/4, )( 5 4,"4# ) #$ +' <'%'++' 4# +,$ $#N,$5 &*'+6*/#1' 4# +,$ ^( 1#$*+%'A

4,$ bI2]5]5 2^5]J5 I2]5^5 2^5^J5 >>>5 I2]5( 5 2^5( Jc5 4,"4# I2]5: 5 2^5: J #$ IF/"#5 F/"#J , I\;%<,+5\;%<,+J5 #$ (#17#&%, #" $*<2*#3,$5 = &*'+6*/#1' 4# +,$ (#17/+#$ #$%1'%:3/&,$ 6*# #" &'4'#%'(' : A:$/.' O : % ]5 >>>5 ( 5 4' +*3'1 ' '+3*", 4# 4/&9,$ (#17/+#$ 4# '&&/,"#$ #$ *"!EBD>

Ejemplo 7.9

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&,"%1'1/, $# 2*#3' ,%1, (#17/+ 4# '&&/,"#$ #" +' (1/.#1' #%'('5 $# ++#3' ' IK5 ?J 6*#%'.</:" #$ *" !E 4# %,4,$ +,$ $*<2*#3,$ 1#$%'"%#$ I+,$ V 1#$%'"%#$J>

<: T'$ #$%1'%#3/'$ (1,(*#$%'$ $," *" !E 4#+ 2*#3, &,.(+#%,M 4'4' +' #$%1'%#3/' p]

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74&#'6/.*8- 9:F

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# I^GJG%^# ^G#^]H

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7.3. JUEGOS REPETIDOS EN UN NÚMERO INFINITO DE ETAPAS

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Definición 7.6

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Historias, estrategias y subjuegos de un juego repetido infinitamente

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ENPS en el dilema del prisionero repetido infinitamente

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Oligopolio de Cournot repetido infinitamente. Modelo simplificado

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7.7 F,"$/4#1# #+ $/3*/#"%# 2*#3, #" 7,1.' #$%1'%:3/&'M

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45 D*(,"3' 6*# #" #+ 2*#3, 1#(#%/4, /"7/"/%,$ (#1),4,$5 &'4' 2*3'4,1 $/3*# +'

$/3*/#"%# #$%1'%#3/'M &,./#"0' #+/3/#"4, B5 = +, $/3*# 9'&/#"4, ./#"%1'$ 6*##+ ,%1, 2*3'4,1 %'.</:" #+/2' Bg (#1, $/ $* 1/-'+ #+/3# k5 #+#3/18 k 4*1'"%# ^(#1),4,$5 = ' &,"%/"*'&/@" #+#3/18 Bg .'"%#"418 #$%# &,.(,1%'./#"%, ' +,+'13, 4#+ 2*#3, I#$ 4#&/15 6*# #+#3/18 B ./#"%1'$ 6*# #+ ,%1, +, 9'3'5 = $/ $*1/-'+ $# 4#$-)' R+, &'$%/3'18S 4*1'"%# 4,$ (#1),4,$ = +*#3, -,+-#18 ' #+#3/1BJ> ?#%#1./"# ('1' 6*: -'+,1#$ 4#+ 7'&%,1 4# 4#$&*#"%, p +'$ #$%1'%#3/'$ (1,A(*#$%'$ $," *" #6*/+/<1/, 4# E'$9>

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" %G5 #+ &'$%/3, $@+, 4*1' *" (#1),4,> PD," +'$ #$%1'%#3/'$ (1,(*#$%'$ *"#6*/+/<1/, 4# E'$9Q

7.8 F,"$/4#1# #+ $/3*/#"%# 2*#3, #" 7,1.' #$%1'%:3/&'M

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D*(,"3' 6*# #+ 2*#3, $# 1#(/%# /"7/"/%,$ (#1),4,$>

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45 ?#%#1./"# $/ +'$ #$%1'%#3/'$ (1,(*#$%'$ $," *" #6*/+/<1/, 4# E'$9 (#17#&%,#" $*<2*#3,$> d'0,"# $* 1#$(*#$%'>

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7.9 F'4' *"' 4# +'$ #.(1#$'$ ] = ^ %/#"# 4,$ ,(,1%*"/4'4#$ 4# .#1&'4, L = K> D/'.<'$ $# '(1,-#&9'" 4# +' ,(,1%*"/4'4 4# .#1&'4, L5 ,<%/#"#" *"' 3'"'"&/' 4#

V &'4' *"' 4# #++'$> D/" #.<'13,5 $/ &*'+6*/#1' 4# #++'$ '<'"4,"' +' ,(,1%*"/4'4L = '(1,-#&9' +' K ,<%/#"# *"' 3'"'"&/' 4# G5 %#"/#"4, *"' 3'"'"&/' 4# +' 6*#$# 6*#4' #" L> \/"'+.#"%#5 #" +' ,(,1%*"/4'4 4# .#1&'4, K $@+, 9'= $/%/, ('1'*"' #.(1#$'g $/ '.<'$ #.(1#$'$ #"%1'" #" K5 '.<'$ ,<%/#"#" X> D*(,"3'.,$6*# #+ 2*#3, $# 1#(/%# /"7/"/%'$ -#&#$> F'+&*+# #+ -'+,1 4# +' %'$' 4# 4#$&*#"%, '('1%/1 4# +' &*'+ $# (*#4#" ,<%#"#1 +,$ ('3,$ IV5VJ #" &'4' 1#(#%/&/@"5 4#7/"/#"4,7,1.'+.#"%# +'$ #$%1'%#3/'$ 6*# 9'&#" (,$/<+# ,<%#"#1 %'+ #6*/+/<1/,>

7.10 F,"$/4#1# #+ $/3*/#"%# .,4#+, 4# K#1%1'"4 4/$&1#%,> ?,$ #.(1#$'$ /4:"%/&'$ !] =

!^ &,.(/%#" #" (1#&/,$5 '.<'$ &," &,$%#$ .'13/"'+#$ &,"$%'"%#$ # /4:"%/&,$5%^ = $/" &,$%#$ 7/2,$> T' 4#.'"4' 4# .#1&'4, ('1' #+ </#" 6*# (1,4*&#" -/#A"# 4'4' (,1M JI 0J% ]^. 05 = +,$ (1#&/,$ 4# .#1&'4, #$%8" 1#3*+'4,$ (,1 +#=5$/#"4, 0- à _X5 ]5 ^5 V5 G5 W`5 O -%]5 ^> B,1 %1'%'1$# 4# *" (1,4*&%, 9,.,3:"#,5+,$ &,"$*./4,1#$ +, &,.(1'18" 4# '6*#++' 6*# +, -#"4' '+ (1#&/, .8$ <'2,5 = $/+'$ 4,$ #.(1#$'$ ,71#&#" #+ ./$., (1#&/, +' 4#.'"4' $# 4/$%1/<*=# #" ('1%#$/3*'+#$>

D*(,"3' 6*# +' #&,",.)' 4*1' *" %,%'+ 4# ( %^ (#1),4,$5 = 6*# +'$ #.A(1#$'$ %/#"#" *" 7'&%,1 4# 4#$&*#"%, d%]M

/5 ?#%#1./"# '+3;" (#17/+ 4# #$%1'%#3/'$ 6*# $#' !EBD = (#1./%' ' +'$ #.(1#A

$'$ (,"#1 *" (1#&/, /3*'+ ' G #" +' (1/.#1' #%'('5 I 0]5]5 0^5]J% IG5 GJ> P!N/$A%# '+3;" (#17/+ !EBD 6*# &,"$/3' I 0]5]5 0^5]J% IW5 WJQ

45 D*(,"3' 6*# +,$ &,$%#$ .'13/"'+#$ 4# +'$ #.(1#$'$ $," 4/7#1#"%#$5 4# %'+.'"#1' 6*# 5]% ] = 5^% ^> P!N/$%# '+3;" !EBD 6*# (#1./%' ' +' #.(1#$'!^ &,"$#3*/1 *"' 3'"'"&/' %,%'+ .'=,1 6*# XQ Pl .'=,1 6*# GQ

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7.11 F,"$/4#1# *" ,+/3,(,+/, 4# K#1%1'"4 &," ! #.(1#$'$ /4:"%/&'$5 6*# &,.(/%#" #"(1#&/,$5 %,4'$ #++'$ &," &,$%#$ .'13/"'+#$ &,"$%'"%#$ # /3*'+#$ ' 5b X5 = $/"&,$%#$ 7/2,$> T' 4#.'"4' 4# .#1&'4, ('1' #+ </#" 6*# (1,4*&#" -/#"# 4'4' (,1M

%-I 0J%

E X $/ P 0 I %'+ 6*# 0-b 0 I

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: $/ O 0 I5 #$ 0-m 0 I5 = ('1' : #.(1#$'$ 0-% 0A

4,"4# %I 0-J%2. 70-5 25 7b X5 2b 55 = 0- à C!

5 O ->D*(,"3' 6*# +' #&,",.)' 4*1' ( (#1),4,$5 ( aä5 = 6*# +'$ #.(1#$'$

'(+/&'" *" 7'&%,1 4# 4#$&*#"%, d5 Xa da ] ('1' ,<%#"#1 #+ -'+,1 (1#$#"%# 4#$*$ <#"#7/&/,$M

/5 ?#%#1./"# #+ #6*/+/<1/, 4# E'$9 (#17#&%, #" $*<2*#3,$ 1'0,"'"4, +' 1#$A(*#$%'>

D*(,"3' '9,1' 6*# +' #&,",.)' 4*1' *" ";.#1, /"7/"/%, 4# (#1),4,$M

45 B1,(,1&/,"# *" #6*/+/<1/, 4# E'$9 (#17#&%, #" $*<2*#3,$ #" 6*# %,4'$ +'$#.(1#$'$ %/#"#" <#"#7/&/,$ "*+,$>

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7.12 F,"$/4#1# #+ .,4#+, 4# 4*,(,+/, 4# K#1%1'"4 &," 4,$ #.(1#$'$ = (1,4*&%,$ 4/A7#1#"&/'4,$ 4#$'11,++'4, #" +' D#&&/@" ^>H 4#+ F'()%*+, ^5 #" #+ 6*# '.<'$ #.A(1#$'$ 4#<#" 4#%#1./"'1 *" (1#&/, 0] = 0^ 6*# (#1%#"#&# '+ /"%#1-'+, bX5 äJ54,"4# +'$ 7*"&/,"#$ 4# 4#.'"4' 4# '.<,$ (1,4*&%,$ 4#(#"4#" 4#+ -#&%,1 4#(1#&/,$ I 0]5 0^J 4#+ $/3*/#"%# .,4,M

%]I 0]5 0^J%2. 0]! 70^ = %^I 0]5 0^J%2. 0^! 70]

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7.13 F,"$/4#1# #+ $/3*/#"%# 2*#3, 1#(#%/4, ( %^ -#&#$ = &," 7'&%,1 4# 4#$&*#"%,d%]M

C"$/+%' <

U F ?

C"$/+%' ;

L V5 ] ^5 X V5

C H5 , V5 q ]5

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7.14 F,"$/4#1# #+ $/3*/#"%# 2*#3, 4# #%'(' r &," 4,$ 2*3'4,1#$M

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C V5 X5 V ^5

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CAPÍTULO

8Juegos cooperativos

!"#$ %&'(#)*+ #,&#& "+-,$ .)$/+" %++'$,&#01+"2 3)$ 4& 5)$,+6 06#,+7)%07+" $6 $* 8&'(#)9*+ :; <$ #,&#& 7$ $"#)70&, %=>+ ')$7$6 &%#)&, /,)'+" 7$ .)/&7+,$"2 "06 7$#$6$,6+" $6 *&&%%0+6$" 0670107)&*$" 7$ *+" >0">+"; !* ',+-*$>& 5)67&>$6#&* 3)$ 1&>+" & $"#)70&, $"

$* 7$ %=>+ ')$7$ ?&%$,"$ )6& 70"#,0-)%0=6 7$ '&/+" $6#,$ *+" .)/&7+,$" 3)$ 5+,>&6 )6&%+&*0%0=6 4 ?&6 +-#$607+ )6& /&6&6%0& &%#)&67+ %++,706&7&>$6#$2 7$ >&6$,& %++'$,&#091&; @&,& ,$"+*1$, 70%?+ ',+-*$>&2 4 &* 0/)&* 3)$ $6 $* %&"+ 7$ *+" .)$/+" 6+ %++'$,&#01+"2"$ ?&6 ',+')$"#+ 701$,"+" %+6%$'#+" 7$ "+*)%0=6; !6 $"#$ %&'(#)*+ "$ $"#)70&6 #,$" 7$$**+"A $* %+,$2 $* !"#$%&$"' 4 $* 1&*+, 7$ <?&'*$4;

8.1. INTRODUCCIÓN

!6 $* 8&'(#)*+ : 7$* *0-,+ 4& ?0%0>+" )6& -,$1$ 06#,+7)%%0=6 & *+" .)$/+" %++'$,&#01+"2

7$5060$67+ *& 5+,>& %+&*0%0+6&* + 5+,>& 5)6%0=6 %&,&%#$,("#0%& 7$ )6 .)$/+ %+6 )#0*07&7$"#,&6"5$,0-*$" 4 ',$"$6#&67+ #,$" $.$>'*+";

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

E %+6#06)&%0=62 #,&" ,$%+,7&, *& 7$5060%0=6 7$ .)$/+ $6 5+,>& %+&*0%0+6&* B+ .)$/+ $6

5+,>& 7$ 5)6%0=6 %&,&%#$,("#0%&C2 "$ 06#,+7)%$6 +#,&" 7$5060%0+6$" 3)$ "$ 1&6 & )#0*0F&, &*+ *&,/+ 7$* %&'(#)*+;

Definición 8.1

G6 .)$/+ $6 5+,>& %+&*0%0+6&* + $6 5+,>& 7$ 5)6%0=6 %&,&%#$,("#0%& %+6 )#0*07&7$"#,&6"5$,0-*$" %+6"0"#$ $6A

G6 %+6.)6#+ 5060#+ 7$ .)/&7+,$" ( %

H:2 I2 ;;;2 !J;G6& 5)6%0=6 %&,&%#$,("#0%&2 3)$ &"+%0& & %&7& ")-%+6.)6#+ ) 7$ ( B+ %+&*0%0=6C)6 6K>$,+ ,$&* vB) C B1&*+, 7$ *& %+&*0%0=6C2 "0$67+ vBC% L;

@+, #&6#+2 *% B ( 2 vC $" )6 .)$/+ $6 5+,>& %+&*0%0+6&* + $6 5+,>& 5)6%0=6 %&,&%#$9,("#0%& %+6 )#0*07&7$" #,&6"5$,0-*$" "0 ( 4 v $"#D6 $"'$%050%&7+";

<0 &* %,$%$, $* 6K>$,+ 7$ .)/&7+,$" 3)$ 5+,>&6 )6& %+&*0%0=6 "$ %)>'*$ 3)$ $* -$6$950%0+ + /&6&6%0& 3)$ +-#0$6$ *& %+&*0%0=6 6+ 70">06)4$ $"#&>+" &6#$ )6 .)$/+ %++'$,&#091+ >+6=#+6+2 #&* %+>+ "$ 7$506$ 5+,>&*>$6#$ & %+6#06)&%0=6;

Definición 8.2

<$ 70%$ 3)$ )6 .)$/+ *% B ( 2 vC $" !"#$%"#" "0 O ) 2 + Ñ ( 2 %+6 ) Ñ+ 2 "$ 1$,050%& 3)$

vB) Cm vB+ C

E "0>'*$ 10"#& "$ %+>',)$-& 3)$ *+" #,$" $.$>'*+" 7$ .)$/+" %++'$,&#01+" 06#,+7)%097+" $6 $* 8&'(#)*+ : B!.$>'*+" :;II2 :;IM 4 :;INC "+6 .)$/+" >+6=#+6+";

E %+6#06)&%0=6 "$ 7$506$ $* %+6%$'#+ 7$ .)$/+ ")'$,&70#01+2 $6 $* %)&* %)&67+ 7+"

%+&*0%0+6$" %+6 06#$,"$%%0=6 1&%(& "$ )6$6 $* -$6$50%0+ + /&6&6%0& 7$ *& 6)$1& %+&*0%0=6$" &* >$6+" 0/)&* & *& ")>& 7$ *+" -$6$50%0+" 7$ *&" %+&*0%0+6$" 3)$ "$ )6$6;

Definición 8.3

<$ 70%$ 3)$ )6 .)$/+ *% B ( 2 vC $" &'()*+,-%-." "0 O ) 2 + Ñ ( 2 %+6 ) ç+ %2 "$1$,050%& 3)$

vB) C! vB+ Cm vB) é + C

<0 *& 7$"0/)&*7&7 7$ *& 7$5060%0=6 &6#$,0+, "$ 7& $6 "$6#07+ +')$"#+ "$ 70%$ 3)$ $* .)$/+ $" ")-&70#01+;

E"(2 $6 $* %&"+ 7$ #,$" .)/&7+,$" *& ")'$,&70#0107&7 "$ %+6%,$#& $6 *& 1$,050%&%0=6 7$*&" "0/)0$6#$" 7$"0/)&*7&7$"A

vBH:JC! vBHIJCm vBH:2 IJC2 vBH:JC! vBHMJCm vBH:2 MJC2

vBHIJC! vBHMJCm vBHI2 MJC2 vBH:JC! vBHI2 MJCm vBH:2 I2 MJC2

vBHIJC! vBH:2 MJCm vBH:2 I2 MJC2 vBHMJC! vBH:2 IJCm vBH:2 I2 MJC

<$ %+>',)$-& 06>$70&#&>$6#$ 3)$ $"&" 7$"0/)&*7&7$" "$ %)>'*$6 '&,& %&7& )6+ 7$

*+" .)$/+" 3)$ "$ ',$"$6#&6 $6 *+" !.$>'*+" :;II2 :;IM 4 :;IN2 '+, *+ 3)$ 70%?+" .)$/+""+6 ")'$,&70#01+";

E %+6#06)&%0=6 "$ 06#,+7)%$ )6& ',+'0$7&7 >D" 5)$,#$ 3)$ *& &6#$,0+,; <0 7+" %+&*09%0+6$" B%+6 06#$,"$%%0=6 6+ 6$%$"&,0&>$6#$ 1&%(&C "$ )6$62 $6#+6%$" *& ")>& 7$ *+" -$6$950%0+" 7$ *& )60=6 $ 06#$,"$%%0=6 $" &* >$6+" 0/)&* & *& ")>& 7$ -$6$50%0+" 7$ *&"%+&*0%0+6$" 3)$ "$ )6$6;

Definición 8.4

<$ 70%$ 3)$ )6 .)$/+ *% B ( 2 vC $" /"#.)0" "0 O ) 2 + Ñ ( 2 "$ 1$,050%& 3)$

vB) C! vB+ Cm vB) é+ C! vB) ç+ C

<0 *& 7$"0/)&*7&7 7$ *& 7$5060%0=6 &6#$,0+, "$ 7& $6 "$6#07+ +')$"#+ "$ 70%$ 3)$ $* .)$/+ $" %=6%&1+;

Ejemplo 8.1

!* .)$/+ 7$ *&" #,$" $>',$"&" 7$* !.$>'*+ :;II $" %+61$O+2 4& 3)$ '&,& %)&*3)0$, '&,7$ %+&*0%0+6$" 3)$ "$ #+>$6 "$ %)>'*$ *& 7$"0/)&*7&7 7$ *& 7$5060%0=6; E"( '+,$.$>'*+A

vBH:2 IJC! vBH:2 MJC%:PI! :PI%MQNa vBH:2 I2 MJC! vBH:JC% MRL!SI%NMI

vBH:2 IJC! vBHI2 MJC%:PI! :NN%MMRa vBH:2 I2 MJC! vBHIJC% MRL!MR%MPR

!* .)$/+ 7$* !.$>'*+ :;IM #&>-0T6 $" %+61$O+;!* .)$/+ 7$* !.$>'*+ :;IN 6+ $" %+61$O+ 4& 3)$

vBH:2 MJC! vBHI2 MJC%:! :%Ib vBH:2 I2 MJC! vBHMJC% :!L%:

E %+6#06)&%0=6 "$ 7$506$6 *+" %+6%$'#+" 7$ .)$/+ L96+,>&*0F&7+ 4 7$ .)$/+ BL2:C96+,>&*0F&7+;

Definición 8.5

<$ 70%$ 3)$ )6 .)$/+ *% B ( 2 vC $" 12#"*!+3-4+," "0 "$ 1$,050%& 3)$

vBH,JC%L2 O , à (

!* .)$/+ 7$* !.$>'*+ :;II 6+ $" L96+,>&*0F&7+U $6 %&>-0+ *+" .)$/+" 7$ *+" !.$>9'*+" :;IM 4 :;IN "( *+ "+6;

<$ ')$7$ +-#$6$, *& L96+,>&*0F&%0=6 7$ )6 .)$/+2 7$5060$67+ *& "0/)0$6#$ 5)6%0=6%&,&%#$,("#0%&A

vLB) C% vB) C. ;,à)

vBH,JC2 O ) Ñ (

Juegos cooperativos 453

Definición 8.6

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Definición 8.7

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8.2. EJEMPLOS DE JUEGOS COOPERATIVOS

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Ejemplo 8.2

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Ejemplo 8.3

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Ejemplo 8.4

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Ejemplo 8.5

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Ejemplo 8.6

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Ejemplo 8.7

B$ 3"%4& 5% $7 D7!#7EE&87?

!* '*&6#$&>0$6#+ 7$ )6& "0#)&%0=6 7$ -&6%&,,+#& %+>+ )6 .)$/+ %++'$,&#01+ "$ 7$-$

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8.3. EL CONJUNTO DE IMPUTACIONES

<$& *% B ( 2 vC )6 .)$/+ $6 ") 5+,>& 5)6%0=6 %&,&%#$,("#0%& B#&>-0T6 **&>&7& 5+,>& %+&*09%0+6&*C2 $6 7+67$ ( % H:2 I2 ;;;2 !J $" $* %+6.)6#+ 7$ .)/&7+,$" 4 v $" *& 5)6%0=6 %&,&%#$9,("#0%&; <0 $6 )6 .)$/+ *+" .)/&7+,$" 7$%07$6 #,&-&.&, %+6.)6#&>$6#$2 $" 7$%0, %++'$,&,2 $*

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Definición 8.8

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_& %+670%0=6 3)$ %)>'*$6 *+" 1$%#+,$" 7$ 70"#,0-)%0=6 7$ '&/+" 3)$ '$,#$6$%$6 &*%+6.)6#+ 7$ ',$0>')#&%0+6$" 7$* .)$/+ "$/K6 *& %)&* *& ")>& 7$ *+" '&/+" 3)$ ,$%0-$6*+" .)/&7+,$" "$& 0/)&* &* 1&*+, 7$ *& %+&*0%0=6 #+#&*2 ,$%0-$ $* 6+>-,$ 7$ ',06%0'0+ 7$$50%0$6%0&;

Z$',$"$6#&%0=6 /,D50%&A "0 !%M2 $* %+6.)6#+ 7$ ',$0>')#&%0+6$" $" )6 '*&6+ $6 $*$"'&%0+ 7$ #,$" 70>$6"0+6$" 3)$ %+,#& & *+" $.$" $6 *+" ')6#+" BvB ( C2 L2 LC2 BL2 vB ( C2 LC 4BL2 L2 vB ( CC;

Ejemplo 8.8

<$ %+6"07$,& $* "0/)0$6#$ .)$/+ %+6 #,$" .)/&7+,$"A

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_& ]0/),& Q;: ,$%+/$ *& ,$',$"$6#&%0=6 /,D50%& 7$* %+6.)6#+ 7$ ',$0>')#&%0+6$"; <$#,&#& 7$ )6 '*&6+2 3)$ %+,#& & *+" $.$" $6 *+" ')6#+" BY2 L2 LC2 BL2 Y2 LC 4 BL2 L2 YC;

Figura 8.1 Representación gráfica del conjunto de preimputaciones.

8&-$ '$6"&, 3)$ 606/K6 .)/&7+, &%$'#&,D )6 '&/+ 065$,0+, &* 3)$ +-#$67,(& '+, "( >0">+ "06 '&,#0%0'&, $6 606/)6& %+&*0%0=6; <),/$ $6#+6%$" $* %+6%$'#+ 3)$ "$ 7$506$ &%+6#06)&%0=6;

Definición 8.9

!* %+6.)6#+ 7$ -!('%+/-"#)& 7$ )6 .)$/+ *% B ( 2 vC $" $* "0/)0$6#$ %+6.)6#+ 7$ 1$%9#+,$" 7$ '&/+"A

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_& %+670%0=6 7$ 3)$ '&,& %&7& .)/&7+, , #0$6$ 3)$ %)>'*0,"$ 3)$ N ,n vBH,JC ,$%0-$ $*6+>-,$ 7$ ',06%0'0+ 7$ ,&%0+6&*07&7 0670107)&*;

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3)$ %$,+C; <$ #,&#&2 '+, #&6#+2 7$ )6 #,0D6/)*+ $6 $* $"'&%0+ 7$ #,$" 70>$6"0+6$"2 %)4+"1T,#0%$" "$ $6%)$6#,&6 $6 *+" ')6#+" BvB ( C2 L2 LC2 BL2 vB ( C2 LC 4 BL2 L2 vB ( CC;

Ejemplo 8.9

<$ %+6"07$,& $* .)$/+ 7$* !.$>'*+ Q;Q;<$ #0$6$ 3)$

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!" )6 .)$/+ L96+,>&*0F&7+; _& ]0/),& Q;I ,$%+/$ *& ,$',$"$6#&%0=6 /,D50%& 7$* %+69

.)6#+ 7$ 0>')#&%0+6$"; <$ #,&#& 7$* #,0D6/)*+ 7$ 1T,#0%$" BY2 L2 LC2 BL2 Y2 LC 4 BL2 L2 YC $6$* $"'&%0+ 7$ #,$" 70>$6"0+6$";

Figura 8.2 Representación gráfica del conjunto de imputaciones.

c+,>&*>$6#$2 $6 $* %&"+ 7$ #,$" .)/&7+,$"2 "$ &%+"#)>-,& & ,$',$"$6#&, $* %+6.)6#+7$ 0>')#&%0+6$" 70,$%#&>$6#$ $6 $* '*&6+; E"(2 '&,& $* .)$/+ 7$* !.$>'*+ Q;P "$ &%+"9#)>-,& & ,$',$"$6#&, $* %+6.)6#+ 7$ 0>')#&%0+6$" 70,$%#&>$6#$ $6 $* '*&6+2 #&* %+>+&'&,$%$ $6 *& ]0/),& Q;M;

Figura 8.3 Representación gráfica del conjunto de imputaciones, en el plano.

!6 $* "0/)0$6#$ $.$>'*+ "$ ,$',$"$6#& /,D50%&>$6#$ $6 $* '*&6+ $* %+6.)6#+ 7$ 0>')9#&%0+6$" '&,& )6 .)$/+ 3)$ 6+ $" L96+,>&*0F&7+; E #,&1T" 7$* $.$>'*+ "$ 0670%& %=>+?&4 3)$ ',+%$7$, $6 $"#+" %&"+";

Ejemplo 8.10

<$ %+6"07$,& $* "0/)0$6#$ .)$/+ %+6 #,$" .)/&7+,$"A

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@&,& ,$',$"$6#&, 70%?+ %+6.)6#+2 '&,#0>+" 7$ *& 06#$,"$%%0=6 7$* '*&6+ N :! N I! N M%Y %+6 $* +,#&6#$ 6+ 6$/&#01+ B$" 7$%0, %+6 *+" B N :2 N I2 N MC #&*$" 3)$ N :nL2 N In L2 N Mn LC 4 & %+6#06)&%0=6 &a&70>+" *&" ,$"#,0%%0+6$" N :n I2 N In L2 N Mn :2 #&*%+>+ &'&,$%$ $6 *& ]0/),& Q;N;

Figura 8.4 Conjunto de imputaciones del juego.

]06&*>$6#$ 3)$7& %+>+ &'&,$%$ $6 *& ]0/),& Q;Y;

Figura 8.5 Conjunto de imputaciones del juego.

!6 $* %&"+ 7$ N .)/&7+,$"2 $* %+6.)6#+ 7$ 0>')#&%0+6$" "$ &%+"#)>-,& & ,$',$"$6#&,%+>+ )6 #$#,&$7,+ $6 $* '*&6+2 /)&,7&67+ &6&*+/(& %+6 *& ,$',$"$6#&%0=6 3)$ "$ ?& 10"#+'&,& #,$" .)/&7+,$"; !6 *& ]0/),& Q;R "$ ,$',$"$6#& /,D50%&>$6#$ $* %+6.)6#+ 7$ 0>')#&%0+96$" %+,,$"'+670$6#$ &* .)$/+ 7$ '&#$6#$" 7$* !.$>'*+ Q;N; @&,& 70%?+ .)$/+ "$ #0$6$ 3)$

L B ( 2 vC%BH N :2 N I2 N M2 N NJC à .NA N :! N I! N M! N N%:M2 N :n L2 N In L2 N Mn L2 N Nn LJ

Figura 8.6 Conjunto de imputaciones para 4 jugadores.


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Proposición 8.1

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Definición 8.10

<$ 70%$ 3)$ $* .)$/+ *% B ( 2 vC $" )&)#/-+3 "0 1$,050%& 3)$ L B ( 2 vCÇ;

8.4. EL CORE

!* ',06%0'0+ 7$ ,&%0+6&*07&7 0670107)&* 3)$ "$ ,$%+/$ $6 $* %+6.)6#+ 7$ 0>')#&%0+6$"')$7$ $O#$67$,"$ & #+7&" *&" %+&*0%0+6$" >$70&6#$ $* ',06%0'0+ 7$ ,&%0+6&*07&7 %+&*0%0+96&*; _*$/&>+" $6#+6%$" &* %+6%$'#+ 7$ #&E% 7$ )6 .)$/+ %++'$,&#01+;

Definición. 8.11

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Proposición 8.2

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<$ #,&#& '+, #&6#+ 7$ *& 06#$,"$%%0=6 5060#& 7$ %+6.)6#+" %$,,&7+" 3)$ $" %$,,&7+; b$?$%?+ *&" ,$"#,0%%0+6$" 3)$ 7$506$6 &* %+,$ "+6 7$ #0'+ 0/)&*2 + >&4+, + 0/)&*2 *+ %)&*0>'*0%& 3)$ $* %+6.)6#+ "$& %$,,&7+;

f$&>+" &?+,& 3)$ O B ( 2 vC $" )6 %+6.)6#+ &%+#&7+; !6 $5$%#+A<$& N %B N :2 N I2 ;;;2 N !C à O B ( 2 vC; <$ %)>'*$ 3)$ N ,nvBH,JC2 '&,& %&7& ,%:2 I2 ;;;2 !;

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vBH,JCm N ,m vB ( C. vB ( . H,JC

42 '+, #&6#+ $* %+6.)6#+ O B ( 2 vC $"#D &%+#&7+;]06&*>$6#$2 1$&>+" 3)$ $* %+6.)6#+ $" %+61$O+; !6 $5$%#+A

<$&6 N % B N :2 N I2 ;;;2 N !C à O B ( 2 vC2 P% B P:2 PI2 ;;;2 P!C à O B ( 2 vC2 j à VL2 :W; f$&>+"3)$ j N ! B:. jC P à O B ( 2 vC; !6 $5$%#+A%+>+

j N ! B:. jC P% jB N :2 N I2 ;;;2 N !C! B:. jCB P:2 PI2 ;;;2 P!C%

% Bj N :! B:. jC P:2 j N I! B:. jC PI2 ;;;2 j N !! B:. jC P!C

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@+, #&6#+2 O B ( 2 vC $" %+61$O+ 4 *& ',+'+"0%0=6 3)$7& 7$>+"#,&7&;

E %+6#06)&%0=6 1&>+" & +-#$6$, $* %+,$ '&,& &*/)6+" .)$/+";

Ejemplo 8.11

<$ #,&#& 7$ +-#$6$, $* %+,$ 7$* .)$/+ 7$ *& 506%& ,K"#0%& 7$* !.$>'*+ Q;I;@$,#$6$%$,D6 &* %+,$ *+" ')6#+" B N :2 N I2 N MC 3)$ "&#0"5&/&6 *&" "0/)0$6#$" ,$"#,0%%0+6$"A

N :! N I! N M%SSY B',06%0'0+ 7$ $50%0$6%0&C2

N :n L2 N In L2 N Mn MYL B,&%0+6&*07&7 0670107)&*C2

N :! N In L2 N :! N Mn SLL2 N I! N Mn SSYB,&%0+6&*07&7 '&,& *&" %+&*0%0+6$" 5+,>&7&" '+, 7+" .)/&7+,$"C

!" 7$%0,2 *+" $*$>$6#+" 7$* %+,$ "+6 *+" ')6#+" 3)$ '$,#$6$%$6 &* %+6.)6#+ 7$ 0>')9#&%0+6$" 7$* .)$/+ 42 &7$>D"2 1$,050%&6 *&" ,$"#,0%%0+6$" %+,,$"'+670$6#$" & *& ,&%0+6&*097&7 '&,& *&" %+&*0%0+6$" 7$ 7+" .)/&7+,$";

\$60$67+ $6 %)$6#& *& ,$"#,0%%0=6 N :! N I! N M%SSY B',06%0'0+ 7$ $50%0$6%0&C2 "$#0$6$ 3)$

N :! N In L á SSY. N Mn L á N Mm SSY

N :! N Mn SLL á SSY. N In SLL á N Im SY

N I! N Mn SSY á SSY. N :n SSY á N :m L

@+, #&6#+2

O B ( 2 vC%HB N :2 N I2 N MC à .MA N :! N I! N M%SSY2 N :%L2 Lm N ImSY2 MYLm N MmSSYJ%

%HB N :2 N I2 N MC à .MA N :%L2 Lm N ImSY2 MYLmSSY. N ImSSY2 N M%SSY. N IJ%

% HB N :2 N I2 N MC à .MA N :%L2 Lm N Im SY2 Lm N Im NIY2 N M% SSY. N IJ%

% HBL2 N I2 SSY. N ICA Lm N Im SYJ

<$ ')$7$ ,$',$"$6#&, /,D50%&>$6#$ $* %+,$2 '&,#0$67+ 7$* %+6.)6#+ 7$ 0>')#&%0+6$"7$* .)$/+ 4 &a&70$67+ *&" ,$"#,0%%0+6$" N Mm SSY2 N Im SY 4 N :m L2 #&* %+>+ &'&,$%$ $6*& ]0/),& Q;S;

(0, 0, 775)

(0, 435, 350)(425, 0, 350)

x1 05

x3 3505

x2 05

x2 755

Core

Figura 8.7 Core del juego.

_&" 70"#,0-)%0+6$" 7$ '&/+" B N :2 N I2 N MC 3)$ %)>'*$6 $* ',06%0'0+ 7$ $50%0$6%0& 4 6+'$,#$6$%$6 &* %+,$ "+6 06&%$'#&-*$" '&,& &*/)6& %+&*0%0=6 3)$ "$ ')$7& 5+,>&, 4 %+6"09/)$ >$.+,$" ,$")*#&7+" 7$ *+" 3)$ +-#0$6$6 %+6 B N :2 N I2 N MC; E"(2 '+, $.$>'*+A

BYL2 SY2 RYLC 6+ 06#$,$"&,(& & *& %+&*0%0=6 HI2 MJ2 3)$ ')$7$ +-#$6$, '+, "( >0">&SSY2 3)$ $" >&4+, 3)$ SY! RYL%SIY2 %&6#07&7 3)$ +-#0$6$ %+6 *& 70"#,0-)%0=6 7$'&/+" BYL2 SY2 RYLC;

BL2 :LL2 RSYC 6+ 06#$,$"&,(& & *& %+&*0%0=6 H:2 MJ2 3)$ ')$7$ +-#$6$, '+, "( >0">&

SLL2 3)$ $" >&4+, 3)$ L!RSY% RSY2 3)$ +-#0$6$ 70%?& %+&*0%0=6 %+6 *& 70"#,0-)%0=67$ '&/+" 7&7&;

ASSY

M2

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M2

SSY

M B 6+ 06#$,$"&,(& & *& %+&*0%0=6 HI2 MJ2 3)$ ')$7$ +-#$6$, '+, "(

>0">& SSY2 3)$ $" )6& %&6#07&7 >&4+, 3)$SSY

M!

SSY

M2 %&6#07&7 3)$ +-#0$6$ 70%?&

%+&*0%0=6 %+6 *& 70"#,0-)%0=6 7$ '&/+" 7&7&;

Ejemplo 8.12

<$ #,&#& 7$ +-#$6$, $* %+,$ 7$* .)$/+ 7$ *+" %&6#&6#$" 7$* !.$>'*+ Q;M; E7$>D"2 ?&4 3)$$6%+6#,&, *&" %+670%0+6$" 3)$ 7$-$6 %)>'*0, *+" 1&*+,$" 0 4 1 '&,& 3)$ $* %+,$ "$&1&%(+2 )60#&,0+2 + #$6/& >D" 7$ )6 $*$>$6#+;

@$,#$6$%$,D6 &* %+,$ *+" ')6#+" B N :2 N I2 N MC 3)$ "&#0"5&/&6 *&" "0/)0$6#$" ,$"#,0%%0+6$"A

N :! N I! N M% 0 B',06%0'0+ 7$ $50%0$6%0&C2

N :n L2 N In L2 N Mn L B,&%0+6&*07&7 0670107)&*C2

N :! N In 12 N :! N Mn 12 N I! N Mn 1B,&%0+6&*07&7 '&,& *&" %+&*0%0+6$" 5+,>&7&" '+, 7+" .)/&7+,$"C

8+>+ ?$>+" 10"#+ $6 $* $.$>'*+ &6#$,0+,2 *+" $*$>$6#+" 7$* %+,$ "+6 *+" ')6#+" 3)$'$,#$6$%$6 &* %+6.)6#+ 7$ 0>')#&%0+6$" 7$* .)$/+ 42 &7$>D"2 1$,050%&6 *&" ,$"#,0%%0+6$"%+,,$"'+670$6#$" & *& ,&%0+6&*07&7 '&,& *&" %+&*0%0+6$" 7$ 7+" .)/&7+,$";

\$60$67+ $6 %)$6#& *& ,$"#,0%%0=6 N :! N I! N M% 0 B',06%0'0+ 7$ $50%0$6%0&C2 "$ #0$96$ 3)$

N :

! N I

n 1 á 0. N M

n 1 á N M

m 0. 1

N :! N Mn 1 á 0. N In 1 á N Im 0. 1

N I! N Mn 1 á 0. N :n 1 á N :m 0. 1

@+, #&6#+2

O B ( 2 vC%HB N :2 N I2 N MC à .MA N :! N I! N M% 02 Lm N ,m 0. 12 O ,%:2 I2 MJ%

%HB N :2 N I2 0. N :. N IC à .MA Lm N :m 0. 12 Lm N Im 0. 12 1m N :! N Im 0J

E *& 10"#& 7$ *&" $O',$"0+6$" &6#$,0+,$" 3)$ 7$506$6 $* %+,$ "$ **$/& & 3)$A

<0 B 0. 1C! B 0. 1C% 1 B$" 7$%0,2 I 0%M 1C2 $6#+6%$" $* %+,$ $" )60#&,0+2 +-#$960T67+"$ 3)$ O B ( 2 vC% HB 0. 12 0. 12 I 1. 0CJ; _& ,$',$"$6#&%0=6 /,D50%& $"#D $6 *&]0/),& Q;Q;

Figura 8.8 El core es unitario si 2A%3B .

<0 B 0. 1C! B 0. 1Ca 1 B$" 7$%0,2 I 0a M 1C2 $6#+6%$" $* %+,$ $" 1&%(+ B]0/)9,& Q;PC;

Figura 8.9 El core es vacío si 2Aa3B .

<0 B 0. 1C! B 0. 1Cb 1 B$" 7$%0,2 I 0b M 1C2 $6#+6%$" $* %+,$ $" 6+ 1&%(+2 6+)60#&,0+ B]0/),& Q;:LC;

(0, 0, ) A

( 0, 0) A, (0, , 0) A

x1 5 A B x2 5 A B

x2 05 x150

x350

x3 5 A B

Figura 8.10 El core es no vacío, no unitario, si 2Ab3B .

@+, $.$>'*+2 ")'+6/&>+" 3)$ 0%:YL; !6#+6%$"2 "0 1%:LL2 $* %+,$ $" )60#&,0+2"0$67+ BYL2 YL2 YLC ") K60%+ ')6#+; <0 1b :LL $* %+,$ $" 1&%(+2 >0$6#,&" 3)$ "0

1a :LL2 $* %+,$ $" 6+ 1&%(+2 6+ )60#&,0+;

Ejemplo 8.13

E?+,& 1&>+" & +-#$6$, $* %+,$ 7$* .)$/+ 7$ '&#$6#$"2 %+,,$"'+670$6#$ &* !.$>'*+ Q;N;<$ #,&#& 7$ )6 .)$/+ %+6 N .)/&7+,$"; _&" ,$"#,0%%0+6$" 3)$ 7$-$6 %)>'*0,2 $6 $"#$

%&"+2 *+" $*$>$6#+" 7$* %+,$ "+6 *&" "0/)0$6#$"A

@,06%0'0+ 7$ $50%0$6%0&A

N :! N I! N M! N N% :M

Z&%0+6&*07&7 0670107)&*A

N :n L2 N In L2 N Mn L2 N Nn L

Z&%0+6&*07&7 7$ *&" %+&*0%0+6$" 5+,>&7&" '+, 7+" .)/&7+,$"A

N :!

N In L2 N :!

N Mn R2Y2 N :!

N Nn R2Y2 N I! N Mn R2Y2 N I! N Nn R2Y2 N M! N Nn L

Z&%0+6&*07&7 7$ *&" %+&*0%0+6$" 5+,>&7&" '+, #,$" .)/&7+,$"A

N :! N I! N Mn :L2 N :! N I! N Nn :L2

N :! N M! N Nn :L2 N I! N M! N Nn :L

G#0*0F&67+ *& 0/)&*7&7 7$* ',06%0'0+ 7$ $50%0$6%0& $6 *&" 7$"0/)&*7&7$" %+,,$"'+6970$6#$" & *& ,&%0+6&*07&7 7$ %+&*0%0+6$" 7$ #,$" .)/&7+,$"2 3)$7&A

N :! N I! N Mn :L á :M. N Nn :L á N Nm M

N :! N I! N Nn :L á :M. N Mn :L á N Mm M

N :! N M! N Nn :L á :M. N In :L á N Im M

N I! N M! N Nn :L á :M. N :n :L á N :m M

!" 0>'+"0-*$ 3)$ N :m M2 N Mm M 4 & *& 1$F N :! N Mn R2Y; @+, #&6#+2 $* %+,$ 7$ $"#$ .)$/+ $" $* %+6.)6#+ 1&%(+;

E?+,& 6+" ?&%$>+" *& "0/)0$6#$ ',$/)6#&A g$O0"#$ &*/)6& %*&"$ 7$ .)$/+" 3)$ #$6/&6$* %+,$ 6+ 1&%(+h @&,& ,$"'+67$, & $"#& ',$/)6#& "$ ',$"$6#&6 & %+6#06)&%0=6 7+" 7$50609%0+6$" 4 )6 #$+,$>&;

Definición 8.12

G6& 5&>0*0& H) :2 ) I2 ;;;2 ) QJ 7$ ")-%+6.)6#+" 7$ ( 2 70"#06#+" 4 6+ 1&%(+"2 $" );'-3-<*+2,+ "+-,$ ( "0 $O0"#$6 6K>$,+" '+"0#01+" a:2 aI2 ;;;2 aQ2 7$6+>06&7+" '$"+"2 #&*$" 3)$'&,& #+7+ ,%:2 I2 ;;;2 ! 1$,050%&6A

;H 3A ,à)

jJ

a 3% :

Ejemplo 8.14

<$& ( % H:2 I2 MJ; f$&>+" &*/)6&" 5&>0*0&" $3)0*0-,&7&" "+-,$ ( ;<$& :% HH:J2 HIJ2 HMJJ;\$6$>+" 3)$ ) :% H:J2 ) I% HIJ2 ) M% HMJ;<$ #,&#& 7$ )6& 5&>0*0& $3)0*0-,&7& "+-,$ ( 4& 3)$ $O0"#$6 a:%aI% aM%:2 1$,0509

%D67+"$ 3)$A

@&,& ,%:2 ;H 3A:à)

jJ

a 3%a:%:

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jJ

a 3%aI%:

@&,& ,%M2 ;H 3AMà) jJ a 3%aM%:

<$& I% HH:J2 HI2 MJJ;!6#+6%$" ) :% H:J2 ) I% HI2 MJ;<$ #,&#& 7$ )6& 5&>0*0& $3)0*0-,&7& "+-,$ ( 4& 3)$ $O0"#$6 a:%aI%:2 1$,050%D67+9

"$ 3)$A

@&,& ,%:2 ;H 3A:à) jJ a 3%a:%:

@&,& ,%I2 ;H 3AIà)

jJ

a 3%aI%:

@&,& ,%M2 ;H 3AMà)

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a 3%aI%:

<$& M% HH:2 IJ2 H:2 MJ2 HI2 MJJ;\$6$>+" 3)$ ) :% H:2 IJ2 ) I%H:2 MJ2 ) M%HI2 MJ;<$ #,&#& 7$ )6& 5&>0*0& $3)0*0-,&7& "+-,$ ( 4& 3)$ $O0"#$6 a:%aI% aM%:/ I2 1$,09

50%D67+"$ 3)$A

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a 3%aI! aM%:

<$& N% HH:J2 H:2 IJ2 H:2 I2 MJJ;

\$6$>+" 3)$ ) :% H:J2 ) I% H:2 IJ2 ) M% H:2 I2 MJ; <$ %+>',)$-& 06>$70&#&>$6#$3)$ $" )6& 5&>0*0& $3)0*0-,&7&2 %+6 "=*+ #+>&, a:%L2 aI%L2 aM%:;

<$& Y% HH:J2 HIJ2 HMJ2 H:2 IJ2 H:2 MJ2 HI2 MJ2 H:2 I2 MJJ;!6 $"$ %&"+ ) :% H:J2 ) I% HIJ2 ) M% HMJ2 ) N% H:2 IJ2 ) Y% H:2 MJ2 ) R% HI2 MJ2

) S% H:2 I2 MJ;<$ %+>',)$-& 06>$70&#&>$6#$ 3)$ $" )6& 5&>0*0& $3)0*0-,&7&2 #+>&67+A

a:%:2 aI%L2 aM%L2 aN%L2 aY%L2 aR%:2 aS%L

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a:%L2 aI%L2 aM%L2 aN%L2 aY%L2 aR%L2 aS%:

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Ejemplo 8.15

<$& H) :2 ) I2 ;;;2 ) QJ )6& '&,#0%0=6 "+-,$ ( ; <$ #,&#& 7$ )6& 5&>0*0& $3)0*0-,&7& "+-,$ ( 2 *+

%)&* "$ %+>',)$-& 06>$70&#&>$6#$ &* #+>&, a:%aI%ñ%a>%:;

E %+6#06)&%0=6 "$ 7$506$ $* %+6%$'#+ 7$ .)$/+ $3)0*0-,&7+;

Definición 8.13

<$ 70%$ 3)$ $* .)$/+ B ( 2 v

C $" );'-3-<*+," "0 '&,& %)&*3)0$, 5&>0*0& $3)0*0-,&7& H) ,2 ;;;2) QJ "+-,$ ( 2 %+6 '$"+" a:2 ;;;2 aQ2 "$ 1$,050%& 3)$

Q

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Teorema 8.1

G6 .)$/+ B ( 2 vC #0$6$ %+,$ 6+ 1&%(+ "0 4 "=*+ "0 B ( 2 vC $" )6 .)$/+ $3)0*0-,&7+;

9)!"&%*+/-$#:

úC <)'+6/&>+" 3)$ B ( 2 vC #0$6$ %+,$ 6+ 1&%(+; f$&>+" 3)$ B ( 2 vC $" )6 .)$/+$3)0*0-,&7+; !6 $5$%#+2 '+, 7$5060%0=6 7$ %+,$2

O B ( 2 vCÇ ú P N % B N :2 N I2 ;;;2 N !C à O B ( 2 vC

4 '+, #&6#+ 1$,050%& 3)$2 '&,& #+7& %+&*0%0=6 ) "$ 1$,050%& 3)$

N B) C%

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B) C

<$& % H) :2 ) I2 ;;;2 ) QJ )6& 5&>0*0& $3)0*0-,&7& "+-,$ ( 2 %+6 '$"+" &"+%0&7+" a:2aI2 ;;;2 aQ;

b&7+ N à O B ( 2 vC2 "$ #0$6$ 3)$

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8+>+ $* %+6.)6#+ 1 $" %$,,&7+2 &%+#&7+ 065$,0+,>$6#$ 4 %+61$O+2 $* ',+/,&>&

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;) à/B ( C

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PR, D%;,à (

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'+, *+ 3)$ $* .)$/+ B ( 2 vC 6+ $"#D $3)0*0-,&7+; b$ $"#& 5+,>&2 "$ **$/& & )6& %+6#,&70%9

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Ejemplo 8.16

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d$>+" 10"#+ $6 *& "$%%0=6 &6#$,0+, 3)$ $* #&E% $" )6 %+6%$'#+ 7$ "+*)%0=6 3)$ #0$6$ )6&7050%)*#&7 0>'+,#&6#$A $6 &*/)6&" +%&"0+6$" $" )6 %+6.)6#+ >)4 /,&67$ 4 $6 +#,&" $" )6%+6.)6#+ 1&%(+; !* %+6%$'#+ 7$ !"#$%&$"' ',+'+6$ )6& "+*)%0=6 3)$2 "0$>',$ 3)$ $* %+69 .)6#+ 7$ 0>')#&%0+6$" "$& 6+ 1&%(+2 ")'$,& *& 7050%)*#&7 &6#$,0+,2 ')$" $" 6+ 1&%(+ 4 K60%+;E7$>D"2 '$,#$6$%$ &* #&E% "0 T"#$ $" 6+ 1&%(+; <$ %+6"07$,& )6 .)$/+ %++'$,&#01+ B ( 2 vC;

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Definición 8.14

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Ejemplo 8.17

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!6 *& "0/)0$6#$ #&-*& "$ ,$%+/$6 *+" 1&*+,$"2 *+" $O%$"+" + 3)$.&" 7$ %&7& %+&*0%0=6'&,& %&7& )6+ 7$ *+" 1$%#+,$" 7&7+";

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Definición 8.15

@&,& %&7& N à L B ( 2 vC2 "$ 7$506$ $* 1$%#+, 7$ $O%$"+" %+>+ $* "0/)0$6#$ 1$%#+, hB N C2 %+6I! %+>'+6$6#$"A

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Definición 8.16

<$&6 N 2 N ñ à L B ( 2 N C;

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á h:B N Ca h:B N ñC2 + -0$62 '&,& 3b :2 h 3B N Ca h 3B N ñC 4 h,B N C%h,B N ñC2 '&,& ,%:2 ;;;2 3.:

<8 hB N C% V hB N ñC á h 3B N C% h 3B N ñC2 O 3

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@+, #&6#+2 7&7+" 7+" 1$%#+,$" 7$ $O%$"+"2 '&,& %+>'&,&,*+" "$/K6 $* +,7$6 *$O0%+/,D950%+2 "$ +-"$,1&6 "=*+ *&" ',0>$,&" %+>'+6$6#$"U "0 *& ',0>$,& %+>'+6$6#$ 7$ )6 1$%#+,

$" >$6+, 3)$ *& ',0>$,& %+>'+6$6#$ 7$* +#,+ 1$%#+, $* ',0>$, 1$%#+, $" >$6+, 3)$ $*"$/)67+ "$/K6 $* +,7$6 *$O0%+/,D50%+ 7$50607+; <0 *+" 7+" 1$%#+,$" #0$6$6 0/)&*$" ")"',0>$,&" %+>'+6$6#$" "$ %+>'&,&6 ")" "$/)67&" %+>'+6$6#$" "0$67+ >$6+, "$/K6 $*+,7$6 *$O0%+/,D50%+ &3)$* 1$%#+, %)4& "$/)67& %+>'+6$6#$ "$& >$6+,; <0 ")" "$/)67&"%+>'+6$6#$" "+6 0/)&*$" "$ %+>'&,&6 ")" #$,%$,&" %+>'+6$6#$" 4 &"( ")%$"01&>$6#$;

Ejemplo 8.18

@&,& *+" 1$%#+,$" N 2 P2 U 7$50607+" $6 $* !.$>'*+ Q;:S "$ #0$6$ 3)$A

hB N Ca V hB PC2 4& 3)$ *& ',0>$,& %+>'+6$6#$ 7$* 1$%#+, hB N C2 3)$ $" L2 $" >$6+,

3)$ *& ',0>$,& %+>'+6$6#$ 7$* 1$%#+, hB PC2 3)$ $" I;hB UCa V hB PC2 4& 3)$ *& ',0>$,& %+>'+6$6#$ 7$* 1$%#+, hB UC2 3)$ $" L2 $" >$6+,3)$ *& ',0>$,& %+>'+6$6#$ 7$* 1$%#+, hB PC2 3)$ $" I;

hB N Ca V hB UC2 4& 3)$ *&" %)&#,+ ',0>$,&" %+>'+6$6#$" 7$ &>-+" 1$%#+,$" "+60/)&*$" 4 *& 3)06#& %+>'+6$6#$ 7$* 1$%#+, hB N C2 3)$ $" .Y2 $" >$6+, 3)$ *& 3)06#&%+>'+6$6#$ 7$* 1$%#+, hB UC2 3)$ $" .N;

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Definición 8.17

!* !"#$%&$"' 7$ )6 .)$/+ B ( 2 vC $" $* %+6.)6#+ J B ( 2 vC 7$50607+ 7$ *& "0/)0$6#$ 5+,>&A

J B ( 2 vC% H N à L B ( 2 vCA hB N Cm V hB PC2 O P à L B ( 2 vCJ

@+, #&6#+2 "$ ')$7$ 7$%0, 3)$ $* !"#$%&$"' %+6#0$6$ &3)$**&" 70"#,0-)%0+6$" 7$ '&/+"3)$ "+6 0>')#&%0+6$"2 4 '&,& *&" %)&*$" "$ >060>0F& $* >&4+, 7$ *+" /,&7+" 7$ 06"&#0"5&%%0=6;

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Teorema 8.2

<$& B ( 2 vC )6 .)$/+ $"$6%0&* B*+ %)&* 3)0$,$ 7$%0, 3)$ ") %+6.)6#+ 7$ 0>')#&%0+6$" $"6+ 1&%(+C; !6#+6%$" "$ 1$,050%& 3)$ $* !"#$%&$"' $O0"#$ 4 $" K60%+;

_& 7$>+"#,&%0=6 7$* #$+,$>& "$ $6%)$6#,& $6 b,0$""$6 B:PQQC;

Proposición 8.3

G6& %+670%0=6 ")50%0$6#$ '&,& 3)$ $* !"#$%&$"' $O0"#& 4 "$& K60%+ $" 3)$

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E %+6#06)&%0=6 "$ 7&6 7+" 7$5060%0+6$" 3)$ "$ )#0*0F&,D6 '+"#$,0+,>$6#$ $6 )6& ',+9

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Definición 8.18

<$ 70%$ 3)$ 7+" .)/&7+,$" ,2 3 "+6 &-!=%*-/"& $6 )6 .)$/+ B ( 2 vC "0 ,$&*0F&6 &'+,#&%0+96$" $3)01&*$6#$" '&,& %&7& %+&*0%0=6; !" 7$%0,2 "0 "$ %)>'*$ 3)$

vB) é H,JC% vB) é H,JC2 '&,& #+7+ ) à /B ( C2 %+6 ,2 3 â )

Definición 8.19

<$ 70%$ 3)$ $* .)/&7+, , $" )6 >'?+,"* (+&-." $6 $* .)$/+ B ( 2 vC "0 6+ &'+,#& 606/K6-$6$50%0+ &70%0+6&* &* ,$"#+ 7$ *+" .)/&7+,$"; !" 7$%0,2 "0 "$ %)>'*$ 3)$

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Ejemplo 8.19

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Proposición 8.4

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Ejemplo 8.21

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Figura 8.11 Resolución gráfica del problema minimax.



Ejemplo 8.23

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+#,& '&,#$2 "$ %)>'*$ *& %+670%0=6 ")50%0$6#$ $6 *& @,+'+"0%0=6 Q;M 3)$ 6+" '$,>0#$

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Figura 8.12 Resolución gráfica del problema minimax en el juego de patentes.

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Ejemplo 8.24

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Método para calcular el nucleolus utilizando programación lineal

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Ejemplo 8.25

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Bibliografía

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506 Bibliografía

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Bibliografía 507



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CERDÁ PEREZ & JIMENO - Teoría de Juegos - 1a

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