Juego coopertaivo de ganacias

15 de Gener de 2015

Autora: Andrea Esteban Fleta15 DE GENER DE 2015

ÍNDEX

Pag.

Enunciat del problema ................................................................................................ 2

Apartat a) Obtenció de la funció característica ........................................................... 3

Apartat b) Demostració de que el joc és de cooperació total ..................................... 4

Apartat c) Solució del joc de cooperació total i anàlisi de .......................................... 8

l’estabilitat i eficiència de la solució.

Apartat d) Canvi d’alguns valors de l’enunciat inicial de ............................................. 14

manera que el joc esdevingui de cooperació parcial: no súperadditivitat

Apartat e) Solució del joc de cooperació parcial i anàlisi ............................................ 19

de l’estabilitat i eficiència de la solució.

Página 1 de 25


PROBLEMA. JOC COOPERATIU DE GUANYS

Amb motiu de la gran demanda de marisc provocada per l’arribada de les festes

nadalenques, una empresa de marisc a decidit que no vol quedar-se sense

subministra en meitat de les festes com li va passar l’any passat i a creat una oferta

per als camioners autònoms, que consisteix en que contra més kg de marisc li sigui

transferit a la vegada, més diners pagarà, i així evitar quedar-se sense subministres.

Normalment aquesta empresa paga 5 euros/kg de marisc, però amb motiu de

l’arribada d’aquestes festes, han fet l’oferta que per cada tona de marisc que portin el

kg vingui després el pagaran a 6 euros/kg, i si porten 2 tones de cop els pagaran 20

mil euros.

5 camioners s’han assabentat d’aquesta oferta: el primer camioner te capacitat per

transporta 400kg de marisc, el segon 350kg, el tercer camioner por transportar 300kg,

el quart camió 500kg, i l’últim camioner pot transportar 450kg.

a) Obtingueu la funció característica del joc associat d’aquest 5 camioners.

b) Determina quina coalició o coalicions es formen entre els camioners

encarregats de transportar el marisc, i demostra que es un joc de cooperació

total de súperadditivitat.

c) Proposeu una assignació dels beneficis que obtindran els 5 camioners, de

manera que garanteixin l’eficiència i l’estabilitat de la coalició o coalicions

formades.

d) Al finalitzar les festes de nadal, l’empresa deixa de fer la oferta de les 2 tones,

ja que no es capaç de vendre tot el material de cop, però segueix tenint una

demanda continuada.

Per això refà una oferta per a tindre material continuat però en menys quantitat.

La nova oferta consisteix en augmentar a 7 euros el kg de marisc, als rebuts

que porten entre 1000 i 1100 kg.

Obtingueu la funció característica del nou joc, i després determina quina

coalició o coalicions es formaran ara.

e) Proposeu una assignació dels beneficis que obtindran els 5 camioners, de

manera que garanteixin l’eficiència i l’estabilitat de la coalició o coalicions

formades.

Página 2 de 25


a) Obtingueu la funció característica del joc associat d’aquest 5 camioners.

Camioner 1 = v(1)

Camioner 2 = v(2)

Camioner 3 = v(3)

Funció característica

v(1) = 400·5 = 2000 €

v(2) = 350·5 = 1750€

v(3) = 300·5 = 1500€

v(4) = 500·5 = 2500€

v(5) = 450·5 = 2250€

b) Determina quina coalició o coalicions es formen entre els camioners encarregats de

transportar el marisc, i demostra que es un joc de total de súdditivitat.

Página 3 de 25

v(12) = (400+350)·5 = 3750€

v(13) = (400+300)·5 = 3500€

v(14) = (400+500)·5 = 4500€

v(15) = (400+450)·5 = 4250€

v(23) = (350+300)·5 = 3250€

v(24) = (350+500)·5 = 4250€

v(25) = (350+450)·5 = 4000€

v(34) = (300+500)·5 = 4000€

v(35) = (300+450)·5 = 3750€

v(45) = (500+450)·5 = 4750€

v(1234) = (400+350+300+500)·6 = 9300€

v(1235) = (400+350+300+450)·6 = 9000€

v(1245) = (400+350+500+450)·6 = 10200€

v(1345) = (400+300+500+450)·6 = 9900€

v(2345) = (350+300+500+450)·6 = 9600€

v(123) = (400+350+300)·6 = 6300€

v(124) = (400+350+500)·6 = 7500€

v(125) = (400+350+450)·6 = 7200€

v(134) = (400+300+500)·6 = 7200€

v(135) = (400+300+450)·6 = 6900€

v(145) = (400+500+450)·6 = 8100€

v(234) = (350+300+500)·6 = 6900€

v(235) = (350+300+450)·6 = 6600€

v(245) = (350+500+450)·6 = 7800€

v(345) = (300+500+450)·6 = 7500€

v(12345) = (400+350+300+500+450) = 20000 €

Camioner 4 = v(4)

Camioner 5 = v(5)


b)Determina quina coalició o coalicions es formen entre els camioners encarregats de

transportar el marisc, i demostra que es un joc de cooperació total de súperadditivitat.

Anàlisis de súperadditivitat

2 jugadors

v(1)+v(2) = 2000+1750 = 3750€ = v(12) = 3750€

v(1)+v(3) = 2000+1500 = 3750€ = v(13) = 3750€

v(1)+v(4) = 2000+2500 = 4500€ = v(14) = 4500€

v(1)+v(5) = 2000+2250 = 4250€ = v(15) = 4250€

v(2)+v(3) = 1750+1500 = 3250€ = v(23) = 3250€

v(2)+v(4) = 1750+2500 = 4250€ = v(24) = 4250€

v(2)+v(5) = 1750+2250 = 4250€ = v(25) = 4250€

v(3)+v(4) = 1500+2500 = 4000€ = v(34) = 4000€

v(3)+v(5) = 1500+2250 = 3750€ = v(35) = 3750€

v(4)+v(5) = 2500+2250 = 4750€ = v(45) = 4750€

3 jugadors

<123>

v(12)+v(3) = 3750+1500 = 5250€ < v(123) = 6300€

v(13)+v(2) = 3500+1750 = 5250€ < v(123) = 6300€

v(23)+v(1) = 3250+2000 = 5250€ < v(123) = 6300€

<124>

v(12)+v(4) = 3750+2500 = 6250€ < v(124) = 7500€

v(14)+v(2) = 4500+1750 = 6250€ < v(124) = 7500€

v(24)+v(1) = 4250+2000 = 6250€ < v(124) = 7500€

<125>

v(12)+v(5) = 3750+2250 = 6000€ < v(125) = 7200€

v(15)+v(2) = 4250+1750 = 6000€ < v(125) = 7200€

v(25)+v(1) = 4000+2000 = 6000€ < v(125) = 7200€

Página 4 de 25

Hi ha súperadditivitat





<134>

v(13)+v(4) = 3500+2500 = 6000€ < v(134) = 7200€

v(14)+v(3) = 4500+1500 = 6000€ < v(134) = 7200€

v(34)+v(1) = 4000+2000 = 6000€ < v(134) = 7200€

<135>

v(13)+v(5) = 3500+2250 = 5750€ < v(135) = 6900€

v(15)+v(3) = 4250+1500 = 5750€ < v(135) = 6900€

v(35)+v(1) = 3750+2000 = 5750€ < v(135) = 6900€

<145>

v(14)+v(5) = 4500+2250 = 6750€ < v(145) = 8100€

v(15)+v(4) = 4250+2500 = 6750€ < v(145) = 8100€

v(45)+v(1) = 4750+2000 = 6750€ < v(145) = 8100€

<234>

v(23)+v(4) = 3250+2500 = 5750€ < v(234) = 6900€

v(24)+v(3) = 4250+1500 = 5750€ < v(234) = 6900€

v(34)+v(2) = 4000+1750 = 5750€ < v(234) = 6900€

<235>

v(23)+v(5) = 3250+2250 = 5500€ < v(235) = 6600€

v(25)+v(3) = 4000+1500 = 5500€ < v(235) = 6600€

v(35)+v(2) = 3750+1750 = 5500€ < v(235) = 6600€

<245>

v(24)+v(5) = 4250+2250 = 6500€ < v(245) = 7800€

v(25)+v(4) = 4000+2500 = 6500€ < v(245) = 7800€

v(45)+v(2) = 4750+1750 = 6500€ < v(245) = 7800€

<345>

v(34)+v(5) = 4000+2250 = 6250€ < v(345) = 7500€

v(35)+v(4) = 3750+2500 = 6250€ < v(345) = 7500€

v(45)+v(3) = 4750+1500 = 6250€ < v(345) = 7500€

Página 5 de 25









4 jugadores

<1234>

v(1)+v(234) = 2000+6900 = 8900€ < v(1234) = 9300€

v(2)+v(134) = 1750+7200 = 8950€ < v(1234) = 9300€

v(3)+v(124) = 1500+7500 = 9000€ < v(1234) = 9300€

v(4)+v(123) = 2500+6300 = 6550€ < v(1234) = 9300€

v(12)+v(34) = 3750+4000 = 7750€ < v(1234) = 9300€

v(13)+v(24) = 3500+4250 = 7750€ < v(1234) = 9300€

v(14)+v(23) = 4500+3250 = 7750€ < v(1234) = 9300€

<1235>

v(1)+v(235) = 2000+6600 = 8600€ < v(1235) = 9000€

v(2)+v(135) = 1750+6900 = 8650€ < v(1235) = 9000€

v(3)+v(125) = 1500+7200 = 8700€ < v(1235) = 9000€

v(5)+v(123) = 2250+6300 = 5850€ < v(1235) = 9000€

v(12)+v(35) = 3750+3750 = 7500€ < v(1235) = 9000€

v(13)+v(25) = 3500+4000 = 7500€ < v(1235) = 9000€

v(15)+v(23) = 4250+3250 = 7500€ < v(1235) = 9000€

<1245>

v(1)+ v(245) = 2000+7800 = 9800€ < v(1245) = 10200€

v(2)+ v(145) = 1750+8100 = 9850€ < v(1245) = 10200€

v(4)+ v(125) = 2500+7200 = 9700€ < v(1245) = 10200€

v(5)+ v(124) = 2250+7500 = 9750€ < v(1245) = 10200€

v(12)+ v(45) = 3500+4750 = 8250€ < v(1245) = 10200€

v(14)+ v(25) = 4500+4000 = 8500€ < v(1245) = 10200€

v(15)+ v(24) = 4250+4250 = 8500€ < v(1245) = 10200€

Página 6 de 25





<1345>

v(1)+ v(345) = 2000+7500 = 9500€ < v(1345) = 9900€

v(3)+ v(145) = 1500+8100 = 9600€ < v(1345) = 9900€

v(4)+ v(135) = 2500+6900 = 9400€ < v(1345) = 9900€

v(5)+ v(134) = 2250+7200 = 9450€ < v(1345) = 9900€

v(13)+ v(45) = 3500+4750 = 8250€ < v(1345) = 9900€

v(14)+ v(35) = 4500+3750 = 8250€ < v(1345) = 9900€

v(15)+ v(34) = 4250+4000 = 8250€ < v(1345) = 9900€

<2345>

v(2)+ v(345) = 1750+7500 = 9250€ < v(2345) = 9600€

v(3)+ v(245) = 1500+7800 = 9300€ < v(2345) = 9600€

v(4)+ v(235) = 2500+6600 = 9100€ < v(2345) = 9600€

v(5)+ v(234) = 2250+6900 = 9150€ < v(2345) = 9600€

v(23)+ v(45) = 3250+4750 = 8000€ < v(2345) = 9600€

v(24)+ v(35) = 4250+3750 = 8000€ < v(2345) = 9600€

v(25)+ v(34) = 4000+4000 = 8000€ < v(2345) = 9600€

5 jugadores

v(1)+ v(2345) = 2000+9600 = 11600€ < v(12345) = 20000€

v(2)+ v(1345) = 1750+9900 = 11650€ < v(12345) = 20000€

v(3)+ v(1245) = 1500+10200 = 11700€ < v(12345) = 20000€

v(4)+ v(1235) = 2500+9000 = 11500€ < v(12345) = 20000€

v(5)+ v(1234) = 2250+9300 = 11550€ < v(12345) = 20000€

v(12)+ v(345) = 3750+7500 = 11250€ < v(12345) = 20000€

v(13)+ v(245) = 3500+7800 = 11300€ < v(12345) = 20000€

v(14)+ v(235) = 4500+6600 = 11100€ < v(12345) = 20000€

v(15)+ v(234) = 4250+6900 = 11150€ < v(12345) = 20000€

v(23)+ v(145) = 3250+8100 = 11350€ < v(12345) = 20000€

Página 7 de 25





v(24)+ v(135) = 4250+6900 = 11150€ < v(12345) = 20000€

v(25)+ v(134) = 4000+7200 = 11200€ < v(12345) = 20000€

v(34)+ v(125) = 4000+7200 = 11200€ < v(12345) = 20000€

v(35)+ v(124) = 3750+7500 = 11250€ < v(12345) = 20000€

v(45)+ v(123) = 6650+7350 = 14000€ < v(12345) = 20000€

c) Proposeu una assignació dels beneficis que obtindran els 5 camioners, de manera

que garanteixin l’eficiència i l’estabilitat de la coalició o coalicions formades.

El joc es un joc súperadditiu total

Valor de Shapley

Ф s=∑i ɇ s

Ɣ ( s )[v (s )−v (s− {i })]

Ɣ (s) en joc de n=5 jugadors

Ɣ (S)=(n−s )! (s−1 ) !

n !

S amb 1 jugador s=1

Ɣ (1 )= (5−1 ) ! (1−1 ) !5 !

=4 ! 0!5!

=15

S amb 2 jugador s=2

Ɣ (2 )= (5−2 )! (2−1 )!5!

=3 !1!5 !

= 120

S amb 3 jugador s=3

Ɣ (3 )= (5−3 )! (3−1 ) !5 !

=2!2 !5 !

= 130

S amb 4 jugador s=4

Ɣ (4 )= (5−4 ) ! (4−1 )!5 !

=1 !3!5 !

= 120

Página 8 de 25


S amb 5 jugador s=5

Ɣ (5 )= (5−5 )! (5−1 )!5 !

=0! 4 !5 !

=15

Valor de Shapley

ɸ1=Ɣ (1 ) [v (1 )−v (Ø ) ]+Ɣ (12 ) [v (12 )−v (2 ) ]+Ɣ (13 ) [v (13 )−v (3 ) ]+Ɣ (14 ) [v (14 )−v (4 ) ]+Ɣ (15 ) [ v (15 )−v (5 ) ]+Ɣ (123 ) [v (123 )−v (23 ) ]+Ɣ (124 ) [v (124 )−v (24 ) ]+Ɣ (125 ) [v (125 )−v (25 ) ]+Ɣ (134 ) [v (134 )−v (34 ) ]+Ɣ (135 ) [v (135 )−v (35 ) ]+Ɣ (145 ) [ v (145 )−v (45 ) ]+Ɣ (1234 ) [v (1234 )−v (234 ) ]+Ɣ (1235 ) [v (1235 )−v (235 ) ]+Ɣ (1245 ) [ v (1245 )−v (245 ) ]+Ɣ (1345 ) [v (1345 )−v (345 ) ]+Ɣ (12345 ) [ v (12345 )−v (2345 ) ]

ɸ1=15

[2000 ]+ 120

[3750−1750 ]+ 120

[3500−1500 ]+ 120

[4500−2500 ]+ 120

[4250−2250 ]+ 130

[6300−3250 ]+ 130

[7500−4250 ]+ 130

[7200−4000 ]+ 130

[7200−4000 ]+ 130

[6900−3750 ]+ 130

[8100−4750 ]+ 120

[9300−6900 ]+ 120

[9000−6600 ]+ 120

[10200−7800 ]+ 120

[9900−7500 ]+ 15

[20000−9600 ]

ɸ1=15

(2000 )+ 120

(2000+2000+2000+2000 )+ 130

(3050+3250+3200+3200+3150+3350 )+ 120

(2400+2400+2400+240 )+ 15(10400)

ɸ1=400+400+640+480+2080=4000€

ɸ2=Ɣ (2 ) [ v (2 )−v (Ø ) ]+Ɣ (21 ) [v (21 )−v (1)]+Ɣ (23 ) [ v (23 )−v (3 ) ]+Ɣ (24 ) [ v (24 )−v (4 ) ]+Ɣ (25 ) [v (25 )−v (5 ) ]+Ɣ (213 ) [v (213 )−v (13 ) ]+Ɣ (214 ) [ v (214 )−v (14 ) ]+Ɣ (215 ) [v (215 )−v (15 ) ]+Ɣ (234 ) [v (234 )−v (34 ) ]+Ɣ (235 ) [ v (235 )−v (35 ) ]+Ɣ (245 ) [v (245 )−v (45 ) ]+Ɣ (2134 ) [ v (2134 )−v (134 ) ]+Ɣ (2135 ) [v (2135 )−v (235 ) ]+Ɣ (2145 ) [ v (2145 )−v (145 ) ]+Ɣ (2345 ) [v (2345 )−v (345 ) ]+Ɣ (21345 ) [ v (21345 )−v (1345 ) ]

ɸ2=15

[1750 ]+ 120

[3750−20000 ]+ 120

[3250−1500 ]+ 120

[4250−2500 ]+ 120

[4000−2250 ]+ 130

[6300−3500 ]+ 130

[7500−4500 ]+ 130

[7200−4250 ]+ 130

[6900−4000 ]+ 130

[6600−3750 ]+ 130

[7800−4750 ]+ 120

[9300−7200 ]+ 120

[9000−6900 ]+ 120

[10200−8100 ]+ 120

[9600−7500 ]+ 15

[20000−9900 ]

ɸ2=15

(1750 )+ 120

(1750+1750+1750+1750 )+ 130

(2800+3000+2950+2900+2850+3050 )+ 120

(2100+2100+2100+2100 )+ 15(10100)

ɸ2=350+350+585+420+2020=3725€

ɸ3=Ɣ (3 ) [ v (3 )−v (Ø ) ]+Ɣ (31 ) [ v (31 )−v (1 ) ]+Ɣ (32 ) [ v (32 )−v (2 ) ]+Ɣ (34 ) [v (34 )−v (4 ) ]+Ɣ (35 ) [ v (35 )−v (5 ) ]+Ɣ (312 ) [v (312 )−v (12 ) ]+Ɣ (314 ) [v (314 )−v (14 ) ]+Ɣ (315 ) [ v (315 )−v (15 ) ]+Ɣ (324 ) [ v (324 )−v (24 ) ]+Ɣ (325 ) [v (325 )−v (25 ) ]+Ɣ (345 ) [ v (345 )−v (45 ) ]+Ɣ (3124 ) [v (3124 )−v (124 ) ]+Ɣ (3125 ) [v (3125 )−v (125 ) ]+Ɣ (3145 ) [ v (3145 )−v (145 ) ]+Ɣ (3245 ) [v (3245 )−v (245 ) ]+Ɣ (31245 ) [ v (31245 )−v (1245 ) ]

ɸ3=15

[1500 ]+ 120

[3500−2000 ]+ 120

[3250−1750 ]+ 120

[4000−2500 ]+ 120

[3750−2250 ]+ 130

[6300−3750 ]+ 130

[7200−4500 ]+ 130

[6900−4250 ]+ 130

[6900−4250 ]+ 130

[6600−4000 ]+ 130

[7500−4750 ]+ 120

[9300−7500 ]+ 120

[9000−7200 ]+ 120

[9900−8100 ]+ 120

[9600−7800 ]+ 15

[20000−10200 ]

Página 9 de 25


ɸ3=15

(1500 )+ 120

(1500+1500+1500+1500 )+ 130

(2550+2700+2650+2650+2600+2750 )+ 120

(1800+1800+1800+1800 )+15(9800)

ɸ3=300+300+530+360+1960=3450 €

ɸ4=Ɣ (4 ) [v (4 )−v (Ø ) ]+Ɣ (41 ) [v (41 )−v (1 ) ]+Ɣ (42 ) [v (42 )−v (2 ) ]+Ɣ (43 ) [v (43 )−v (3 ) ]+Ɣ (45 ) [ v (45 )−v (5 ) ]+Ɣ (412 ) [ v (412 )−v (12 ) ]+Ɣ (413 ) [ v (413 )−v (13 ) ]+Ɣ (415 ) [v (415 )−v (15 ) ]+Ɣ (423 ) [v (423 )−v (23 ) ]+Ɣ (425 ) [v (425 )−v (25 ) ]+Ɣ (435 ) [v (435 )−v (35 ) ]+Ɣ (4123 ) [ v (4123 )−v (123 ) ]+Ɣ (4125 ) [v (4125 )−v (125 ) ]+Ɣ (4135 ) [ v (4135 )−v (135 ) ]+Ɣ (4235 ) [v (4235 )−v (235 ) ]+Ɣ (41235 ) [v (41235 )−v (1235 ) ]

ɸ4=15

[2500 ]+ 120

[4500−2000 ]+ 120

[4250−1750 ]+ 120

[4000−1500 ]+ 120

[4750−2250 ]+ 130

[7500−3750 ]+ 130

[7200−3500 ]+ 130

[8100−4250 ]+ 130

[6900−3250 ]+ 130

[7800−4000 ]+ 130

[7500−3750 ]+ 120

[9300−6300 ]+ 120

[10200−7200 ]+ 120

[9900−6900 ]+ 120

[9600−6600 ]+ 15

[20000−9000 ]

ɸ4=15

(2500 )+ 120

(2500+2500+2500+2500 )+ 130

(3750+3700+3850+3650+3800+3750 )+ 120

(3000+3000+3000+3000 )+ 15(11000)

ɸ4=500+500+750+600+2200=4550 €

ɸ5=Ɣ (5 ) [ v (5 )−v (Ø ) ]+Ɣ (51 ) [ v (51 )−v (1 ) ]+Ɣ (52 ) [v (52 )−v (2 ) ]+Ɣ (53 ) [v (53 )−v (3 ) ]+Ɣ (54 ) [v (54 )−v (4 ) ]+Ɣ (512 ) [ v (512 )−v (12 ) ]+Ɣ (513 ) [ v (513 )−v (13 ) ]+Ɣ (514 ) [v (514 )−v (14 ) ]+Ɣ (523 ) [ v (523 )−v (23 ) ]+Ɣ (524 ) [ v (524 )−v (24 ) ]+Ɣ (534 ) [v (534 )−v (34 ) ]+Ɣ (5123 ) [v (5123 )−v (123 ) ]+Ɣ (5124 ) [v (5124 )−v (124 ) ]+Ɣ (5134 ) [ v (5134 )−v (134 ) ]+Ɣ (5234 ) [v (5234 )−v (234 ) ]+Ɣ (51234 ) [ v (51234 )−v (1234 ) ]

ɸ5=15

[2250 ]+ 120

[4250−2000 ]+ 120

[4000−1750 ]+ 120

[3750−1500 ]+ 120

[4750−2500 ]+ 130

[7200−3750 ]+ 130

[6900−3500 ]+ 130

[8100−4500 ]+ 130

[6600−3250 ]+ 130

[7800−4250 ]+ 130

[7500−4000 ]+ 120

[9000−6300 ]+ 120

[10200−7500 ]+ 120

[9900−7200 ]+ 120

[9600−6900 ]+15

[20000−9300 ]

ɸ5=15

(2250 )+ 120

(2250+2250+2250+2250 )+ 130

(3450+3400+3600+3350+3550+3500 )+ 120

(2700+2700+2700+2700 )+ 15(10700)

ɸ5=450+450+695+540+2140=4275€

Anàlisis d’eficiència i l’estabilitat de la solució del joc.

-Principi d’eficiència de la solució

x (N )=∑i=1

n

xi=v (N )

Página 10 de 25

ɸ1=x1=4000€

ɸ2=x2=3725€

ɸ3=x3=3450€

ɸ4=x4=4550 €

ɸ5=x5=4275 €


x (N )=∑i=1

n

x i=x1+ x2+x3+x4+x5=4000+3725+3450+4550+4270=2000=v (N )

Es eficient

-Principi d’estabilitat de la solució

Principi de racionalitat individual:

x i≥ v (i ) , per a i=1 ,2 ,3 ,4 i5

ɸ1=x1=4000€ >v (1 )=2000 €

ɸ2=x2=3725€>v (2 )=1750€

ɸ3=x3=3450€>v (3 )=1500 €

ɸ4=x4=4550 €>v (4 )=2500€

ɸ5=x5=4275 €>v (5 )=2250€

Compleix el principi de racionalitat individual.

Principi de racionalitat coalicional:

x (S )≥v (S ) ,∀ S⊂N

x (12 )=4000+3725=7725>v (12 )=3750

x (13 )=4000+3450=7450>v (13 )=3500

x (14 )=4000+4550=8550>v (14 )=4500

x (15 )=4000+4275=8275>v (15 )=4250

x (23 )=3725+3450=7175>v (23 )=3250

x (24 )=3725+4550=8275>v (24 )=4250

x (25 )=3725+4275=8000>v (25 )=4000

x (34 )=3450+4550=8000>v (34 )=4000

Página 11 de 25


x (35 )=3450+4275=7725>v (35 )=3750

x (45 )=4550+4275=8825>v (45 )=4750

x (123 )=4000+3725+3450=11175>v (123 )=6300

x (124 )=4000+3725+4550=12275>v (124 )=7500

x (125 )=4000+3725+4275=12000>v (125 )=7200

x (134 )=4000+3450+4550=12000>v (134 )=7200

x (135 )=4000+3450+4275=9725>v (135 )=6900

x (145 )=4000+4550+4275=12825>v (145 )=8100

x (234 )=3725+3450+4550=11725>v (234 )=6900

x (235 )=3725+3450+4275=11450>v (235 )=6600

x (245 )=3725+4550+4275=12550>v (245 )=7800

x (345 )=3450+4550+4275=12275>v (345 )=7500

Compleix el principi de racionalitat coalicional.

d) Al finalitzar les festes de nadal, l’empresa deixa de fer la oferta de les 2 tones, ja

que no es capaç de vendre tot el material de cop, però segueix tenint una demanda

continuada.

Página 12 de 25


Per això refà una oferta per a tindre material continuat però en menys quantitat. La

nova oferta consisteix en augmentar a 7 euros el kg de marisc, als rebuts que porten

entre 1000 i 1100 kg.

Obtingueu la funció característica del nou joc, i després determina quina coalició o

coalicions es formaran ara.

Funció característica

v(1) = 400·5 = 2000 €

v(2) = 350·5 = 1750€

v(3) = 300·5 = 1500€

v(4) = 500·5 = 2500€

v(5) = 450·5 = 2250€

Anàlisis de súperadditivitat

2 jugadors

v(1)+v(2) = 2000+1750 = 3750€ = v(12) = 3750€

Página 13 de 25

v(12) = (400+350)·5 = 3750€

v(13) = (400+300)·5 = 3500€

v(14) = (400+500)·5 = 4500€

v(15) = (400+450)·5 = 4250€

v(23) = (350+300)·5 = 3250€

v(24) = (350+500)·5 = 4250€

v(25) = (350+450)·5 = 4000€

v(34) = (300+500)·5 = 4000€

v(35) = (300+450)·5 = 3750€

v(45) = (500+450)·5 = 4750€

v(123) = (400+350+300)·7 = 7350€

v(124) = (400+350+500)·6 = 7500€

v(125) = (400+350+450)·6 = 7200€

v(134) = (400+300+500)·6 = 7200€

v(135) = (400+300+450)·6 = 6900€

v(145) = (400+500+450)·6 = 8100€

v(234) = (350+300+500)·6 = 6900€

v(235) = (350+300+450)·7 = 7700€

v(245) = (350+500+450)·6 = 7800€

v(345) = (300+500+450)·6 = 7500€

v(1234) = (400+350+300+500)·6 = 9300€

v(1235) = (400+350+300+450)·6 = 9000€

v(1245) = (400+350+500+450)·6 = 10200€

v(1345) = (400+300+500+450)·6 = 9900€

v(2345) = (350+300+500+450)·6 = 9600€

v(12345) = (400+350+300+500+450)·6 = 12000 €


v(1)+v(3) = 2000+1500 = 3750€ = v(13) = 3750€

v(1)+v(4) = 2000+2500 = 4500€ = v(14) = 4500€

v(1)+v(5) = 2000+2250 = 4250€ < v(15) = 6350€

v(2)+v(3) = 1750+1500 = 3250€ = v(23) = 3250€

v(2)+v(4) = 1750+2500 = 4250€ = v(24) = 4250€

v(2)+v(5) = 1750+2250 = 4250€ = v(25) = 4250€

v(3)+v(4) = 1500+2500 = 4000€ = v(34) = 4000€

v(3)+v(5) = 1500+2250 = 3750€ = v(35) = 3750€

v(4)+v(5) = 2500+2250 = 4750€ < v(45) = 6650€

3 jugadors

<123>

v(12)+v(3) = 3750+1500 = 5250€ < v(123) = 7350€

v(13)+v(2) = 3500+1750 = 5250€ < v(123) = 7350€

v(23)+v(1) = 3250+2000 = 5250€ < v(123) = 7350€

<124>

v(12)+v(4) = 3750+2500 = 6250€ < v(124) = 7500€

v(14)+v(2) = 4500+1750 = 6250€ < v(124) = 7500€

v(24)+v(1) = 4250+2000 = 6250€ < v(124) = 7500€

<125>

v(12)+v(5) = 3750+2250 = 6000€ < v(125) = 7200€

v(15)+v(2) = 4250+1750 = 6000€ < v(125) = 7200€

v(25)+v(1) = 4000+2000 = 6000€ < v(125) = 7200€

<134>

v(13)+v(4) = 3500+2500 = 6000€ < v(134) = 7200€

v(14)+v(3) = 4500+1500 = 6000€ < v(134) = 7200€

Página 14 de 25







v(34)+v(1) = 4000+2000 = 6000€ < v(134) = 7200€

<135>

v(13)+v(5) = 3500+2250 = 5750€ < v(135) = 6900€

v(15)+v(3) = 4250+1500 = 5750€ < v(135) = 6900€

v(35)+v(1) = 3750+2000 = 5750€ < v(135) = 6900€

<145>

v(14)+v(5) = 4500+2250 = 6750€ < v(145) = 8100€

v(15)+v(4) = 4250+2500 = 6750€ < v(145) = 8100€

v(45)+v(1) = 4750+2000 = 6750€ < v(145) = 8100€

<234>

v(23)+v(4) = 3250+2500 = 5750€ < v(234) = 6900€

v(24)+v(3) = 4250+1500 = 5750€ < v(234) = 6900€

v(34)+v(2) = 4000+1750 = 5750€ < v(234) = 6900€

<235>

v(23)+v(5) = 3250+2250 = 5500€ < v(235) = 7700€

v(25)+v(3) = 4000+1500 = 5500€ < v(235) = 7700€

v(35)+v(2) = 3750+1750 = 5500€ < v(235) = 7700€

<245>

v(24)+v(5) = 4250+2250 = 6500€ < v(245) = 7800€

v(25)+v(4) = 4000+2500 = 6500€ < v(245) = 7800€

v(45)+v(2) = 4750+1750 = 6500€ < v(245) = 7800€

<345>

v(34)+v(5) = 4000+2250 = 6250€ < v(345) = 7500€

v(35)+v(4) = 3750+2500 = 6250€ < v(345) = 7500€

v(45)+v(3) = 4750+1500 = 6250€ < v(345) = 7500€

4 jugadores

<1234>

v(1)+v(234) = 2000+6900 = 8900€ < v(1234) = 9300€

Página 15 de 25








v(2)+v(134) = 1750+7200 = 8950€ < v(1234) = 9300€

v(3)+v(124) = 1500+7500 = 9000€ < v(1234) = 9300€

v(4)+v(123) = 2500+7350 = 9850€ > v(1234) = 9300€

v(12)+v(34) = 3750+4000 = 7750€ < v(1234) = 9300€

v(13)+v(24) = 3500+4250 = 7750€ < v(1234) = 9300€

v(14)+v(23) = 4500+3250 = 7750€ < v(1234) = 9300€

<1235>

v(1)+v(235) = 2000+7700 = 9700€ > v(1235) = 9000€

v(2)+v(135) = 1750+6900 = 8650€ < v(1235) = 9000€

v(3)+v(125) = 1500+7200 = 8700€ < v(1235) = 9000€

v(5)+v(123) = 2250+7350 = 9600€ > v(1235) = 9000€

v(12)+v(35) = 3750+3750 = 7500€ < v(1235) = 9000€

v(13)+v(25) = 3500+4000 = 7500€ < v(1235) = 9000€

v(15)+v(23) = 4250+3250 = 7500€ < v(1235) = 9000€

<1245>

v(1)+ v(245) = 2000+7800 = 9800€ < v(1245) = 10200€

v(2)+ v(145) = 1750+8100 = 9850€ < v(1245) = 10200€

v(4)+ v(125) = 2500+7200 = 9700€ < v(1245) = 10200€

v(5)+ v(124) = 2250+7500 = 9750€ < v(1245) = 10200€

v(12)+ v(45) = 3500+4750 = 8250€ < v(1245) = 10200€

v(14)+ v(25) = 4500+4000 = 8500€ < v(1245) = 10200€

v(15)+ v(24) = 4250+4250 = 8500€ < v(1245) = 10200€

<1345>

v(1)+ v(345) = 2000+7500 = 9500€ < v(1345) = 9900€

v(3)+ v(145) = 1500+8100 = 9600€ < v(1345) = 9900€

Página 16 de 25

NO Hi ha súperadditivitat




v(4)+ v(135) = 2500+6900 = 9400€ < v(1345) = 9900€

v(5)+ v(134) = 2250+7200 = 9450€ < v(1345) = 9900€

v(13)+ v(45) = 3500+4750 = 8250€ < v(1345) = 9900€

v(14)+ v(35) = 4500+3750 = 8250€ < v(1345) = 9900€

v(15)+ v(34) = 4250+4000 = 8250€ < v(1345) = 9900€

<2345>

v(2)+ v(345) = 1750+7500 = 9250€ < v(2345) = 9600€

v(3)+ v(245) = 1500+7800 = 9300€ < v(2345) = 9600€

v(4)+ v(235) = 2500+7700 = 10200€ > v(2345) = 9600€

v(5)+ v(234) = 2250+6900 = 9150€ < v(2345) = 9600€

v(23)+ v(45) = 3250+4750 = 8000€ < v(2345) = 9600€

v(24)+ v(35) = 4250+3750 = 8000€ < v(2345) = 9600€

v(25)+ v(34) = 4000+4000 = 8000€ < v(2345) = 9600€

5 jugadores

v(1)+ v(2345) = 2000+9600 = 11600€ < v(12345) = 12000€

v(2)+ v(1345) = 1750+9900 = 11650€ < v(12345) = 12000€

v(3)+ v(1245) = 1500+10200 = 11700€ < v(12345) = 12000€

v(4)+ v(1235) = 2500+9000 = 11500€ < v(12345) = 12000€

v(5)+ v(1234) = 2250+9300 = 11550€ < v(12345) = 12000€

v(12)+ v(345) = 3750+7500 = 11250€ < v(12345) = 12000€

v(13)+ v(245) = 3500+7800 = 11300€ < v(12345) = 12000€

v(14)+ v(235) = 4500+7700 = 12200€ > v(12345) = 12000€

v(15)+ v(234) = 4250+6900 = 11150€ < v(12345) = 12000€

v(23)+ v(145) = 3250+8100 = 11350€ < v(12345) = 12000€

v(24)+ v(135) = 4250+6900 = 11150€ < v(12345) = 12000€

v(25)+ v(134) = 4000+7200 = 11200€ < v(12345) = 12000€

v(34)+ v(125) = 4000+7200 = 11200€ < v(12345) = 12000€

Página 17 de 25





v(35)+ v(124) = 3750+7500 = 11250€ < v(12345) = 12000€

v(45)+ v(123) = 4750+7350 = 12100€ > v(12345) = 12000€

e) Proposeu una assignació dels beneficis que obtindran els 5 camioners, de manera

que garanteixin l’eficiència i l’estabilitat de la coalició o coalicions formades en les

coalicions parcial trobades anteriorment.

Hi ha 6 possibilitats:

1) v(4)+v(123) = 2500+7350 = 9850€ > v(1234) = 9300€

(4)

ɸ4=2500

(123)

ɸ1=Ɣ (1 ) [v (1 )−v (Ø ) ]+Ɣ (12 ) [v (12 )−v (2 ) ]+Ɣ (13 ) [v (13 )−v (3 ) ]++Ɣ (123 ) [ v (123 )−v (23 ) ]

ɸ1=13

[2000 ]+ 16

[3750−1750 ]+ 16

[3500−1500 ]+ 13

[7350−3250 ]

ɸ1=20003

+ 20006

+ 20006

+ 41003

=2700

ɸ2=Ɣ (2 ) [ v (2 )−v (Ø ) ]+Ɣ (21 ) [v (21 )−v (1)]+Ɣ (23 ) [ v (23 )−v (3 ) ]+Ɣ (213 ) [v (213 )−v (13 ) ]

ɸ2=13

[1750 ]+ 16

[3750−2000 ]+ 16

[3250−1500 ]+ 13

[7350−3500 ]

ɸ2=17503

+ 17506

+ 17506

+ 38503

=2450

ɸ3=Ɣ (3 ) [ v (3 )−v (Ø ) ]+Ɣ (31 ) [ v (31 )−v (1 ) ]+Ɣ (32 ) [ v (32 )−v (2 ) ]+Ɣ (312 ) [v (312 )−v (12 ) ]

ɸ3=13

[1500 ]+ 16

[3500−2000 ]+ 16

[3250−1750 ]+13

[7350−3750 ]

ɸ3=15003

+ 15006

+ 15006

+36003

=2200

2) v(1)+v(235) = 2000+7700 = 9700€ > v(1235) = 9000€

(1)

Página 18 de 25


ɸ1=2000

(235)

ɸ2=Ɣ (2 ) [ v (2 )−v (Ø ) ]+Ɣ (23 ) [ v (23 )−v (3 ) ]+Ɣ (25 ) [v (25 )−v (5 ) ]+Ɣ (235 ) [ v (235 )−v (35 ) ]

ɸ2=13

[1750 ]+ 16

[3250−1500 ]+ 16

[4000−2250 ]+ 13

[7700−3750 ]

ɸ2=17503

+ 17506

+ 17506

+ 39503

=74503

ɸ3=Ɣ (3 ) [ v (3 )−v (Ø ) ]+Ɣ (32 ) [ v (32 )−v (2 ) ]+Ɣ (35 ) [ v (35 )−v (5 ) ]+Ɣ (325 ) [v (325 )−v (25 ) ]

ɸ3=13

[1500 ]+ 16

[3250−1750 ]+ 16

[3750−2250 ]+13

[7700−4000 ]

ɸ3=15003

+ 15006

+ 15006

+37003

=67003

ɸ5=Ɣ (5 ) [ v (5 )−v (Ø ) ]+Ɣ (52 ) [ v (52 )−v (2 ) ]+Ɣ (53 ) [ v (53 )−v (3 ) ]+Ɣ (523 ) [v (523 )−v (23 ) ]

ɸ5=13

[2250 ]+ 16

[4000−1750 ]+ 16

[3750−1500 ]+ 13

[7700−3250 ]

ɸ3=22503

+ 22506

+ 22506

+ 44503

=89503

3) v(5)+v(123) = 2250+7350 = 9600€ > v(1235) = 9000€

(5)

ɸ5=15

[2250 ]=450

(123)

ɸ1=Ɣ (1 ) [v (1 )−v (Ø ) ]+Ɣ (12 ) [v (12 )−v (2 ) ]+Ɣ (13 ) [v (13 )−v (3 ) ]++Ɣ (123 ) [ v (123 )−v (23 ) ]

ɸ1=13

[2000 ]+ 16

[3750−1750 ]+ 16

[3500−1500 ]+ 13

[7350−3250 ]

ɸ1=20003

+ 20006

+ 20006

+ 41003

=2700

ɸ2=Ɣ (2 ) [ v (2 )−v (Ø ) ]+Ɣ (21 ) [v (21 )−v (1)]+Ɣ (23 ) [ v (23 )−v (3 ) ]+Ɣ (213 ) [v (213 )−v (13 ) ]

ɸ2=13

[1750 ]+ 16

[3750−2000 ]+ 16

[3250−1500 ]+ 13

[7350−3500 ]

Página 19 de 25


ɸ2=17503

+ 17506

+ 17506

+ 38503

=2450

ɸ3=Ɣ (3 ) [ v (3 )−v (Ø ) ]+Ɣ (31 ) [ v (31 )−v (1 ) ]+Ɣ (32 ) [ v (32 )−v (2 ) ]+Ɣ (312 ) [v (312 )−v (12 ) ]

ɸ3=13

[1500 ]+ 16

[3500−2000 ]+ 16

[3250−1750 ]+13

[7350−3750 ]

ɸ3=15003

+ 15006

+ 15006

+36003

=2200

4) v(4)+ v(235) = 2500+7700 = 10200€ > v(2345) = 9600€

(4)

ɸ4=15

[2500 ]=500

(235)

ɸ2=Ɣ (2 ) [ v (2 )−v (Ø ) ]+Ɣ (23 ) [ v (23 )−v (3 ) ]+Ɣ (25 ) [v (25 )−v (5 ) ]+Ɣ (235 ) [ v (235 )−v (35 ) ]

ɸ2=13

[1750 ]+ 16

[3250−1500 ]+ 16

[4000−2250 ]+ 13

[7700−3750 ]

ɸ2=17503

+ 17506

+ 17506

+ 39503

=74503

ɸ3=Ɣ (3 ) [ v (3 )−v (Ø ) ]+Ɣ (32 ) [ v (32 )−v (2 ) ]+Ɣ (35 ) [ v (35 )−v (5 ) ]+Ɣ (325 ) [v (325 )−v (25 ) ]

ɸ3=13

[1500 ]+ 16

[3250−1750 ]+ 16

[3750−2250 ]+13

[7700−4000 ]

ɸ3=15003

+ 15006

+ 15006

+37003

=67003

ɸ5=Ɣ (5 ) [ v (5 )−v (Ø ) ]+Ɣ (52 ) [ v (52 )−v (2 ) ]+Ɣ (53 ) [ v (53 )−v (3 ) ]+Ɣ (523 ) [v (523 )−v (23 ) ]

ɸ5=13

[2250 ]+ 16

[4000−1750 ]+ 16

[3750−1500 ]+ 13

[7700−3250 ]

ɸ3=22503

+ 22506

+ 22506

+ 44503

=89503

5) v(14)+ v(235) = 4500+7700 = 12200€ > v(12345) = 12000€

Página 20 de 25


(14)

ɸ1=Ɣ (1 ) [v (1 )−v (Ø ) ]+Ɣ (14 ) [ v (14 )−v (4 ) ]

ɸ1=12

[2000 ]+ 12

[4500−2500 ]=2000

ɸ4=Ɣ (4 ) [v (4 )−v (Ø ) ]+Ɣ (14 ) [v (14 )−v (1 ) ]

ɸ4=12

[2500 ]+ 12

[4500−2000 ]=2500

(235)

ɸ2=Ɣ (2 ) [ v (2 )−v (Ø ) ]+Ɣ (23 ) [ v (23 )−v (3 ) ]+Ɣ (25 ) [v (25 )−v (5 ) ]+Ɣ (235 ) [ v (235 )−v (35 ) ]

ɸ2=13

[1750 ]+ 16

[3250−1500 ]+ 16

[4000−2250 ]+ 13

[7700−3750 ]

ɸ2=17503

+ 17506

+ 17506

+ 39503

=74503

ɸ3=Ɣ (3 ) [ v (3 )−v (Ø ) ]+Ɣ (32 ) [ v (32 )−v (2 ) ]+Ɣ (35 ) [ v (35 )−v (5 ) ]+Ɣ (325 ) [v (325 )−v (25 ) ]

ɸ3=13

[1500 ]+ 16

[3250−1750 ]+ 16

[3750−2250 ]+13

[7700−4000 ]

ɸ3=15003

+ 15006

+ 15006

+37003

=67003

ɸ5=Ɣ (5 ) [ v (5 )−v (Ø ) ]+Ɣ (52 ) [ v (52 )−v (2 ) ]+Ɣ (53 ) [ v (53 )−v (3 ) ]+Ɣ (523 ) [v (523 )−v (23 ) ]

ɸ5=13

[2250 ]+ 16

[4000−1750 ]+ 16

[3750−1500 ]+ 13

[7700−3250 ]

ɸ3=22503

+ 22506

+ 22506

+ 44503

=89503

6) v(45)+ v(123) = 4750+7350 = 12100€ > v(12345) = 12000€

(45)

ɸ4=Ɣ (4 ) [v (4 )−v (Ø ) ]+Ɣ (45 ) [ v (45 )−v (5 ) ]

ɸ5=12

[2500 ]+ 12

[ 4750−2250 ]=2500

ɸ5=Ɣ (5 ) [ v (5 )−v (Ø ) ]+Ɣ (54 ) [v (54 )−v (4 ) ]

Página 21 de 25


ɸ5=12

[2250 ]+ 12

[ 4750−2500 ]=2250

(123)

ɸ1=Ɣ (1 ) [v (1 )−v (Ø ) ]+Ɣ (12 ) [v (12 )−v (2 ) ]+Ɣ (13 ) [v (13 )−v (3 ) ]++Ɣ (123 ) [ v (123 )−v (23 ) ]

ɸ1=13

[2000 ]+ 16

[3750−1750 ]+ 16

[3500−1500 ]+ 13

[7350−3250 ]

ɸ1=20003

+ 20006

+ 20006

+ 41003

=2700

ɸ2=Ɣ (2 ) [ v (2 )−v (Ø ) ]+Ɣ (21 ) [v (21 )−v (1)]+Ɣ (23 ) [ v (23 )−v (3 ) ]+Ɣ (213 ) [v (213 )−v (13 ) ]

ɸ2=13

[1750 ]+ 16

[3750−2000 ]+ 16

[3250−1500 ]+ 13

[7350−3500 ]

ɸ2=17503

+ 17506

+ 17506

+ 38503

=2450

ɸ3=Ɣ (3 ) [ v (3 )−v (Ø ) ]+Ɣ (31 ) [ v (31 )−v (1 ) ]+Ɣ (32 ) [ v (32 )−v (2 ) ]+Ɣ (312 ) [v (312 )−v (12 ) ]

ɸ3=13

[1500 ]+ 16

[3500−2000 ]+ 16

[3250−1750 ]+13

[7350−3750 ]

ɸ3=15003

+ 15006

+ 15006

+36003

=2200

Trien el jugador dos i tres, ja que son els dos jugadors que es repeteixen en les dos

opcions per a una coalició.

Jug 2: (123) = 2450 i (235) = 74503

≈ 2483,3

Jug 3: (123) = 2200 i (235) = 67003

≈ 2233,3

Amb aquest resultat observem que tant el jugador 2, com el jugador 3 guanyen més

diners escollin la coalició (235), per lo tant, esta clar que serà aquesta la coalició que

agafaran per avarca la demanda.

La resta de jugadors que no es troben en les coalicions, no els importa quina sigui la

decisió final, ja que el seus beneficis no es veuen afectats per aquestes decisions.

Página 22 de 25


Anàlisis d’eficiència i l’estabilitat de la solució del joc.

N = (235)

-Principi d’eficiència de la solució

x (N )=∑i=1

n

xi=v (N )

x (N )=∑i=1

n

x i=x2+ x3+x5=74503

+67003

+ 89503

=7700=v (N )

Es eficient

-Principi d’estabilitat de la solució

Principi de racionalitat individual:

x i≥ v (i ) , per a i=2 ,3 i5

ɸ2=x2=74503€>v (2 )=1750 €

ɸ3=x3=67003€ >v (3 )=1500€

ɸ5=x5=89503

€>v (5 )=2250€

Compleix el principi de racionalitat individual.

Principi de racionalitat coalicional:

x (S )≥v (S ) ,∀ S⊂N

x (23 )=74503

+ 67003

=141503

>v (23 )=3250

x (25 )=74503

+ 89503

=164003

>v (25 )=4000

Página 23 de 25


x (35 )=67003

+ 89503

=156503

>v (35 )=3750

Compleix el principi de racionalitat coalicional.

Página 24 de 25

Juego coopertaivo de ganacias

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