Introducci ón a la Teoría No Cooperativ a de
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Introducción a la Teoría No Cooperativa
de Juegos
Juegos en forma estratégica
El Dilema del Prisionero
Delatar No Delatar
Delatar -10 -10 0 -15
No Delatar
-15 0 -1 -1
El equilibrio de Nash
M. Davis (1986). “Introducción a la teoría de juegos”. Alianza Editorial.R. Gibbons (1992). “Un primer curso de teoría de juegos”. Antoni Bosch. (www.antonibosch.com).M.J. Osborne and A. Rubinstein (1994). “A course in game theory”. The MIT Press.
“Much of the modern literature in economics (and related disciplines) takes the following form: A social situation is modelled, as a non-cooperative game, the Nash equilibria of the game are computed, and their properties are translated into insights into the original problem.”
H.W. Kuhn, J.C. Harsanyi, R. Selten, J.W. Weibull, E. van Damme, J.F. Nash and P. Hammerstein (1996). “The work of John Nash in game theory”. Journal of Economic Theory 69, 153-185.
“During the past two decades non-cooperative game theory has become a central topic in economic theory. Many scholars have contributed to this revolution, none more than John Nash.”
A. Rubinstein (1995). “John Nash: The master of economic modeling”. Scandinavian Journal of Economics 97, 9-13.
La Batalla de los Sexos
Cine Teatro
Cine 4 3 2 2
Teatro 1 1 3 4
No tiene equilibrios de
Nash
(L,L) es un equilibrio de
NashEdgar Allan Poe
“La carta robada”
P I
P -1 1 1 -1
I 1 -1 -1 1
Matching Pennies
P I L
P -1 1 1 -1 0 0
I 1 -1 -1 1 0 0
L 0 0 0 0 0 0
Teorema de NashTodo juego finito en forma estratégica en el que los jugadores pueden elegir loterías sobre sus conjuntos de estrategias tiene al menos un equilibrio de Nash.
LADRÓN (L)
RobaV Duerme B
V No Duerme -C
No Roba 0
VIGILANTE (V)
DuermeL No Roba S
L Roba -P
No Duerme
L No Roba 0
L Roba M
B=Botín
C=Cárcel
La Paradoja de la Instigación
S=SiestaP=ParoM=Medalla
D ND
R B -P -C M
NR 0 S 0 0
El único equilibrio de Nash de este juego es:
(S/(S+P+M),(P+M) /(S+P+M))(C/(C+B),B/(C+B))
INSTIGACIÓN CONSECUENCIA
C más grande
L roba lo mismo
V duerme más
P, M más grandes
L roba menos
V duerme lo mismo
(S/(S+P+M),C/(C+B)) es el único equilibrio de Nash.
((2/5,3/5),(1/5,4/5))
V=86
El lanzamiento del sistema exafónico
86 80 95
M A
M 60 20 20 60
A 70 10
50 30
M A
M 160 40 140 60
A 70 130 130 70
PEQUEÑO 1/2 GRANDE 1/2
M A
M 110 30 80 60
A 70 70 90 50
MM MA AM AA
M 110 30 110 30 80 60 80 60
A 70 70 90 50 70 70 90 50
MM MA AM AA
MM 110 30 110 30 80 60 80 60
MA 75 65 95 45 45 95 75 65
AM 115 25 105 35 105 35 95 45
AA 70 70 90 50 70 70 90 50
• R. Selten (1975). “Reexamination of the perfectness concept for equilibrium points in extensive games”. International Journal of Game Theory 4, 25-55.
• E. Van Damme (1991). “Stability and Perfection of Nash Equilibria”. Springer Verlag.
Refinamientos del equilibrio de Nash
10 10 0 10
10 0 1 1
A
a
I D (2,2)
(3,1)
(0,0)
A
a
a
A1
2
(a,D) es el único equilibrio de Nash razonable. (a,A) es el único equilibrio
de Nash razonable.
El modelo de duopolio de Cournot
En un monopolio...
El Modelo de Duopolio de Stackelberg
Al contrario que en el lanzamiento del sistema exafónico, aquí es bueno ser el líder.
R. Gibbons (1992). “Un primer curso de teoría de juegos”. Antoni Bosch. (www.antonibosch.com).
MONOPOLIO
COURNOT
STACKELBERG
21(a-c)
4
1(a-c)
2
1(a-c)
3
2(a-c)
321
(a-c)9
21(a-c)
922
(a-c)9
1(a-c)
2
1(a-c)
4
3(a-c)
421
(a-c)8
21(a-c)
1623
(a-c)16
1x 2x 1 2x x 1 1 2( , )H x x 2 1 2( , )H x x 1 2H H
1(a-c)
3
F. Patrone, I. García-Jurado and S. Tijs (2000). “Game Practice: Contributions from Applied Game Theory”. Kluwer Academic Publishers.
• P. Borm and B. van der Genutgen. “On the exploitation of casino games: how to distinguish between games of chance and games of skill?”
• A. Roth. “Game theory as a tool for market design.”
• E. van Damme. “The Dutch DCS-1800 auction.”
...it is not allowed to: exploit games with monetary prizes if the participants in general do not have a dominant influence on the probability to win, unless in compliance to this act, a license is granted...
Dutch Gaming Act
• In practice the Dutch state only grants such a license to its own Holland Casino’s foundation.
• The formulation of the Gaming Act clearly implies that skill should be considered relatively with respect to chance.
• If it would be possible to rank a broad class of games with chance elements by means of an operational and objective criterium which quantifies the level of skill relatively to chance, e.g. on a scale from zero to one, the legislator would be able to decide on a certain bound on the level of skill, below which a game should be considered as a game of chance.
• Game of chance: it needs a license according to the Gaming Act. Game of skill: all the other games.
P. Borm and B. van der Genutgen. “On the exploitation of casino games: how to distinguish between games of chance and games of skill?”
• Jugador principiante: juega el juego de un modo ingenuo, del modo en que lo jugaría alguien que acaba de conocer las reglas.
• Jugador avanzado real: juega el juego de un modo óptimo.• Jugador avanzado virtual: juega el juego de un modo óptimo
y conoce de antemano el resultado de todos los elementos del juego en los que interviene el azar.
• La legislación se refiere únicamente a la explotación de juegos con premios en metálico. • Efecto del aprendizaje: Es el resultado esperado de un jugador avanzado real (AR), menos el resultado esperado de un jugador principiante (P).• Efecto del azar: Es el resultado esperado de un jugador avanzado virtual (AV), menos el resultado esperado de un jugador avanzado real (AR).
• Market design concerns the creation of a venue for buyers and sellers, and a format for transactions.
• Game theorists have taken the lead in designing a number of different kinds of markets. Perhaps the three best known of these are auction markets for radio spectrum licenses, spot markets for electric power, and labor market clearinghouses.
• My own experience in market design has been with entry-level professional labor markets. Since 1998, the vast majority of jobs for new physicians in the US (about 20,000 per year) are filled by a clearinghouse whose design I directed.
A. Roth. “Game theory as a tool for market design.”
Problema Básico de Asignación
En un problema básico de asignación siempre existe una asignación estable. Algoritmo de Gale y Shapley (1962).
Complejidades del mercado médico:1. Plazas con prerrequisitos.2. Asignación en problemas con parejas.
E. van Damme. “The Dutch DCS-1800 auction.”
“In the case of radio spectrum auctions in the United States, the federal government used to give away licenses, but was ordered by Congress to sell them, both to raise revenue and to promote efficient use.” A. Roth.
“In February 1998 the Dutch government auctioned licenses to operate mobile telecommunications networks according to the DCS-1800 technology. Two “national” licenses and sixteen “regional” ones were auctioned by using a variant of the simultaneous, multiple round auction that was proposed by US-economists and that had been tested in the US. This paper describes how the decision to auction came about, it details the auction rules, and it analizes the resulting outcomes.” E. van Damme.
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Introducción a la Teoría No Cooperativa
de Juegos
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