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Game Theory Angel Sánchez Grupo Interdisciplinar de Sistemas Complejos (GISC) Departamento de Matemáticas Universidad Carlos III de Madrid Laredo 2005 Game Theory / Angel Sánchez Contents Introduction Normal and extensive forms Dominated strategies and Nash equilibria Mixed strategies and refinements Some important games and game iteration Evolutionary game theory Dynamics and basic results Spatial effects Discussion and conclusions Basic reference: H. Gintis, Game Theory Evolving (Princeton, 2000) Laredo 2005 Game Theory / Angel Sánchez Introduction ¿What the ΨØ@Ø¥¡¶Ĭ is this Game Theory thing? A universal language to treat with behavioral sciences in a unified manner A toolbox to solve complicated problems… … without a lot of math A way to research the world An study of emergency, transformation, stabilization and diffusion of “strategies” Adventure and fantasy! Laredo 2005 Game Theory / Angel Sánchez Introduction History Ernst Zermelo (1913): Chess John von Neumann y Oskar Morgenstern (1944): The theory of games and economic behavior Laredo 2005 Game Theory / Angel Sánchez Introduction History Ernst Zermelo (1913): Chess John von Neumann y Oskar Morgenstern (1944): The theory of games and economic behavior John Nash (1950)†: “Solution” †Nobel Prize in Economics Laredo 2005 Game Theory / Angel Sánchez Introduction History Ernst Zermelo (1913): Chess John von Neumann y Oskar Morgenstern (1944): The theory of games and economic behavior John Nash (1950)†: “Solution” John Maynard Smith y George Price (1973): Evolution (biological) †Nobel Prize in Economics Laredo 2005 Game Theory / Angel Sánchez Introduction History Ernst Zermelo (1913): Chess John von Neumann y Oskar Morgenstern (1944): The theory of games and economic behavior John Nash (1950)†: “Solution” John Maynard Smith y George Price (1973): Evolution (biological) William Hamilton y Robert Axelrod (1981): Human cooperation John Harsanyi† y Reinhard Selten† (1988): Equilibrium problem †Nobel Prize in Economics Laredo 2005 Game Theory / Angel Sánchez Extensive and normal forms Big Monkey vs Little Monkey Big Monkey Action C W Little Monkey C W W C Terminal node (result) (5,3) (4,4) (9,1) (0,0) Payoffs Extensive form Laredo 2005 Game Theory / Angel Sánchez Extensive and normal forms Big Monkey vs Little Monkey Strategies (Big Monkey plays first) Big Monkey Wait Pure strategies Climb Little Monkey CC (Climb no matter what Big Monkey does) WW (Wait no matter what Big Monkey does) WC (Do exactly as Big Monkey) CW (Do the opposite to Big Monkey) Laredo 2005 Game Theory / Angel Sánchez Extensive and normal forms Big Monkey vs Little Monkey Normal Form Little Monkey CC WW WC CW Big Monkey W 9,1 9,1 0,0 0,0 C 5,3 4,4 5,3 4,4 Laredo 2005 Game Theory / Angel Sánchez Extensive and normal forms Big Monkey vs Little Monkey Backward induction Big Monkey S E Little Monkey S E E S (5,3) (4,4) (9,1) (0,0) Nash equilibrium Laredo 2005 Game Theory / Angel Sánchez Extensive and normal forms Big Monkey vs Little Monkey Dominated strategies Weakly dominated Little Monkey Incredible threat CC WW WC CW Big Monkey W 9,1 9,1 0,0 0,0 C 5,3 4,4 5,3 4,4 Laredo 2005 Game Theory / Angel Sánchez Extensive and normal forms Big Monkey vs Little Monkey Big Monkey Action S E Little Monkey S E Information E S Terminal node set (result) (5,3) (4,4) (9,1) (0,0) Payoffs Simultaneous moves Laredo 2005 Game Theory / Angel Sánchez Extensive and normal forms Big Monkey vs Little Monkey Normal Form Little Monkey S E Big Monkey E 9,1 0,0 S 5,3 4,4 Laredo 2005 Game Theory / Angel Sánchez Dominated strategies and Concepts Nash equilibria Perfect information Perfect rationality N players, symmetry/asymmetry Dominated strategies For player i, si dominates s’i if, for any choice of the rest of players, the payoff obtained with si is larger than that gained from s’i Weakly dominated strategies If elimination of dominated strategies leaves a unique one for each player, the result is a Nash equilibrium Laredo 2005 Game Theory / Angel Sánchez Dominated strategies and Concepts Nash equilibria Nash equilibrium A set of strategies (one per player) from which no player benefits by changing unilaterally A set of strategies such that each one of them is a best response to the joint strategies of the rest Some times weakly dominated strategies Pure strategy equilibria Mixed strategy equilibria Randomization (Populations) Laredo 2005 Game Theory / Angel Sánchez Dominated strategies and Nash Theorem Nash equilibria “Every finite game, with N players, with a finite number of strategies per player, has at least one Nash equilibrium, possibly in mixed strategies” [PNAS 36, 48 (1950)] σ:=(p1,…,pn) are the probabilities of the strategies of one player (σ1,…, σN) is a set of strategies If pi y pj are nonzero in σ, then the payoffs for si y sj played against the rest is the same Way to find equilibria in mixed strategies Laredo 2005 Game Theory / Angel Sánchez Mixed strategies and refinements Big Monkey vs Little Monkey Mixed strategies equilibrium Little Monkey S E Big Monkey E 9,1 0,0 S 5,3 4,4 Laredo 2005 Game Theory / Angel Sánchez Mixed strategies and refinements Big Monkey vs Little Monkey Mixed strategy equilibrium Little Monkey: xC+(1-x)W If Big Monkey uses in a mixed strategy both C and W, their payoff against x has to be equal p(W,x)=9x; p(C,x)=5x+4(1-x) p(W,x)=p(C,x) —› 8x=4 —› x=1/2 ((1/2,1/2), For Little Monkey (1/2,1/2)) is p(y,W)=y+3(1-y); p(y,C)=4(1-y) a Nash eq. p(y,W)=p(y,C) —› 2y=1 —› y=1/2 Laredo 2005 Game Theory / Angel Sánchez Mixed strategies and refinements Refinements A unique Nash equilibrium is only guaranteed in zero-sum games In general we expect more than one Nash equilbrium. ¿How should we decide which one is the solution? Refinements of Nash equlibria: criteria to choose among the possible ones Incredible threats: Subgame perfection Pareto-dominance vs risk-dominance “Trembling hands” Motivation for evolutionary game theory Laredo 2005 Game Theory / Angel Sánchez Mixed strategies and refinements Subgame perfection Big Monkey S E Little Monkey S E E S (5,3) (4,4) (9,1) (0,0) Incredible threat: (W,WW) is not subgame perfect Laredo 2005 Game Theory / Angel Sánchez Important games and iteration Some important games Prisoner’s dilemma Prisoner 2 Unique Nash equilibrium C D Dilemma: the best thing is not to cooperate Prisoner 1 C 3,3 0,5 Communication among players D 5,0 1,1 Paradigm in the study of human cooperation (mainly in iterated form) Symmetrical Laredo 2005 Game Theory / Angel Sánchez Important games and iteration Some important games Stag Hunt Hunter 2 Two Nash equlibria C D (C,C) is Pareto dominant (D,D) is risk dominant Hunter 1 C 6,6 0,5 Experiments: different behaviors D 5,0 1,1 Laredo 2005 Game Theory / Angel Sánchez Important games and iteration Some important games Battle of the sexes She Two undecidable F S Nash equilibria Choice of payoffs to He F 2,1 0,0 represent real S 0,0 1,2 situations is arbitrary A mixed equilibria Coordination game Laredo 2005 Game Theory / Angel Sánchez Important games and iteration Some important games Matching pennies Player 2 No Nash equilibria in pure strategies C X Mixed strategy equilibria Player 1 C 1,-1 -1,1 are difficult to justify in applications X -1,1 1,-1 Laredo 2005 Game Theory / Angel Sánchez Important games and iteration 2x2 game classification C X C 1,1 S,T X T,S 0,0 1 Prisoner’s dilemma 2 Chicken/Hawk-Dove/Snowdrift 3 Leader 4 Battle of the sexes 5 Stag hunt 6 Harmony 12 Deadlock Laredo 2005 Game Theory / Angel Sánchez Important games and iteration Some important games Minority W. Brian Arthur, 1992 A.K.A. “Bar El Farol” N people decide whether to go or not Win (positive payoff) those in the minority Ultrasimplified model of economy Laredo 2005 Game Theory / Angel Sánchez Important games and iteration Some important games Public goods The tragedy of the commons, G. Hardin (1965) N people decide wheter to contribute to a common pool The bank doubles the pool and divides it equally among all players Model of cooperative behavior Laredo 2005 Game Theory / Angel Sánchez Important games and iteration Some important games Ultimatum Laredo 2005 Game Theory / Angel Sánchez Important games and iteration Some important games Ultimatum M euros experimenter OK NO M-u u proponent M-u 0 u 0 respondent (Güth, Schmittberger & Schwarze, 1982) Laredo 2005 Game Theory / Angel Sánchez Important games and iteration Some important games Ultimatum Widely used worldwide in experiments on human behavior The subgame perfect Nash equlibrium is (offer the minimun, accept) The observed behavior does not resemble Nash equilibrium in a vast majority of cases and is qualitatively different Interesting open problem Laredo 2005 Game Theory / Angel Sánchez Important games and iteration Iterated games Iterated prisoner’s dilemma Two prisoners play the prisoner’s dilemma an unknown number of times R. Axelrod and W. D. Hamilton, Science 211, 1390 (1981) Basic strategies: All C All D Tit-for-Tat (TFT) Laredo 2005 Game Theory / Angel Sánchez Important games and iteration Iterated games Iterated prisoner’s dilemma Strategy tournament: TFT winner In general, the best strategies are “nice”, “punishing” and “forgiving” Other important strategies: TF2T Pavlov Applications: trench war during WWI Laredo 2005 Game Theory / Angel Sánchez Important games and iteration Iterated games Generalities If the game is played a predetermined number of times, the predicted equilibria are problematic Usually, discount factors are introduced With an infinite number of iterations, the “Folk theorem” guarantees a continuum of Nash equlibria Laredo 2005 Game Theory / Angel Sánchez Evolutionary game theory Evolutionary game theory John Maynard Smith (1982): “Evolution and the theory of games” Biology meets economics Three main concepts shift as compared to classical game theory: Strategy Equilibrium Interaction among players Laredo 2005 Game Theory / Angel Sánchez Evolutionary game theory Strategy Classical theory: players have strategy sets from where to choose their actions Biology: species have strategy sets from which every individual inherits one Society: the set of alternative cultures can be identified with the strategy set, which individuals inherit or choose Laredo 2005 Game Theory / Angel Sánchez Evolutionary game theory Equilibrium Classical theory: Nash equilibrium Biology: evolutively stable strategy (ESS) Society: similar concept We move from trying to explain the actions of individuals to model the changes and diffusion of behaviors in biology or in the society Laredo 2005 Game Theory / Angel Sánchez Evolutionary game theory Interactions Classical theory: one-shot games and iterated games Biology: random and repeated pairing of individuals, with strategies based on their genome and not on the past Society: applied better as the world becomes more and more interconnected Laredo 2005 Game Theory / Angel Sánchez Evolutionary game theory ESS Strategies {s1,…,sn} Payoff for the player using si vs another one using sj: πij (and πji for its opponent) Game does not depend on being player 1 o 2: symmetrical Game matrix A=(πij) At every time t=1,2,…, agents in a large population are paired and play the game. There are as many types of agents as strategies Laredo 2005 Game Theory / Angel Sánchez Evolutionary game theory ESS At a given time the population state is With this population the payoff for a type i player is Evolutionary game: payoff depends on population Laredo 2005 Game Theory / Angel Sánchez Evolutionary game theory ESS For mixed strategy τ For a randomly chosen individual, the expected payoff is Laredo 2005 Game Theory / Angel Sánchez Evolutionary game theory ESS We replace a fraction Є of the population for a mutant of type j. The new state is Payoffs for incumbent (wild) and mutant are Mutant can invade the incumbent population if Laredo 2005 Game Theory / Angel Sánchez Evolutionary game theory ESS σ is an ESS if it cannot be invaded by any mutant (introduced in small quantities) In terms of Nash equilibria: we can see it as a population of equal agents all playing the mixed strategy σ In biological terms, we have a population of agents, each one with a pure strategy, in the proportion given by σ Laredo 2005 Game Theory / Angel Sánchez Evolutionary game theory ESS Theorem: σ is a ESS if and only if, for any mutant τ, we have This means that mutant does worse against incumbent than the incumbent itself, and if they do equally well, incumbent does better against mutant than the mutant itself Theorem: every ESS is a Nash equilibrium. Every strict Nash equilibrium is an ESS Laredo 2005 Game Theory / Angel Sánchez Dynamics and basic results Dynamics We have discussed “invasion” or “displacement” of some strategies by others We have not specified the dynamics of such process In game theory there are no Newton’s laws or Hamilton´s equation: we have to pose a dynamics depending on the process we want to model There are many possible dynamics Laredo 2005 Game Theory / Angel Sánchez Dynamics and basic results Equilibrium selection We have also seen that Nash equilibrium is considered in classical game theory to be the solution of a game, but there are problems if it is not unique and refinements are needed One reason to introduce dynamics in game theory is precisely as an attempt to obtain equilibrium selection through the dynamics, without artificial refinements (Unfortunately, the problem is not fixed) Laredo 2005 Game Theory / Angel Sánchez Dynamics and basic results Replicator dynamics (Evolutionary game dynamics) Payoff xi xi [ f i ( x ) f ] matrix xi ... abundanceof type i f i ... fitness of type i , f i j aij x j f ... average fitness, f i xi f i Taylor and Jonker, Hofbauer and Sigmund Lotka-Volterra equation of ecology Laredo 2005 Game Theory / Angel Sánchez Dynamics and basic results Replicator-mutant xi j x j f j ( x )Q ji fxi Constant fitness No mutation xi j x j f j Q ji fxi xi xi [ f i ( x ) f ] Quasi-species equation Replicator equation Laredo 2005 Game Theory / Angel Sánchez Dynamics and basic results General Framework Quasi-species equation Replicator-mutant Replicator-mutant- Price equation Price equation equation Lotka-Volterra Game dynamical Replicator-Price equation equation equation Adaptive dynamics Laredo 2005 Game Theory / Angel Sánchez Dynamics and basic results Basic assumptions Everybody begins with a randomly chosen strategy Everybody plays against everybody else Infinite population Payoffs add up Total payoff determines the number of copies: Selection Copies inherit approximately their parent’s strategy: Mutation (Notice the relationship with genetic algorithms) Laredo 2005 Game Theory / Angel Sánchez Dynamics and basic results Replicator dynamics Strategies {s1,…,sn} pi(t): proportion of players with strategy i at time t Payoff for si: πi[P(t)]:= πit, P(t)=(p1,…,pn) At every t, we order π1t≤ π2t ≤…≤ πnt In dt, an agent using si changes to sj with probability (learning) Laredo 2005 Game Theory / Angel Sánchez Dynamics and basic results Replicator dynamics The evolution of pi is given by: Finally, Laredo 2005 Game Theory / Angel Sánchez Dynamics and basic results Properties Replicator dynamics is not a best response dynamics The sum of pit is always 1 Equation can be derived in other contexts Fundamental theorem of natural selection (Fisher, 1930): Laredo 2005 Game Theory / Angel Sánchez Dynamics and basic results Equilibria Theorems: In general, dominated strategies do not survive Every Nash equilibrium is a fixed point Every stable fixed point is a Nash equilibrium Every ESS is an asymptotically stable fixed point. If it uses all the strategies, stability becomes global Laredo 2005 Game Theory / Angel Sánchez Spatial effects Spatial effects So far, space has not played any role, and every player interacts with every other one If agents are spatially distributed, interactions go local In spatial models, there is nothing similar to the replicator equation Numerical simulations (agent based modelling) Equilibria change drastically Laredo 2005 Game Theory / Angel Sánchez Spatial effects Spatial effects Example: M. Nowak and R. May, Nature 359, 826 (1992) Players: pure C or D, without memory or strategy, playing on a square lattice Laredo 2005 Game Theory / Angel Sánchez Spatial effects Spatial effects Example: M. Nowak and R. May, Nature 359, 826 (1992) C->C D->D C->D 1.75<T<1.8 D->C Laredo 2005 Game Theory / Angel Sánchez Spatial effects Spatial effects Example: M. Nowak y R. May, Nature 359, 826 (1992) C->C 1.8<T D->D C->D D->C Laredo 2005 Game Theory / Angel Sánchez Spatial effects Spatial effects Example: M. Nowak y R. May, Nature 359, 826 (1992) Temporal evolution beginning with a D in the middle Laredo 2005 Game Theory / Angel Sánchez Spatial effects Spatial effects Game-lattice coupling M. Zimmerman, V. M. Eguíluz and M. San Miguel, Phys. Rev. E 69, 065102 (2004) Laredo 2005 Game Theory / Angel Sánchez Spatial effects Spatial effects Game-lattice coupling M. Zimmerman, V. M. Eguíluz and M. San Miguel, Phys. Rev. E 69, 065102 (2004) Imitation network Laredo 2005 Game Theory / Angel Sánchez Discussion and conclusions Discussion and conclusions We have done a quick tour over the very basic concepts of Game Theory Equilibrium, in its different flavors, is the fundamental concept: Nash equilibrium Evolutionary stable strategy (ESS) Stable fixed point of the dynamics The equilibrium selection problem remains open (and possibly it has no solution) Laredo 2005 Game Theory / Angel Sánchez Discussion and conclusions Discussion and conclusions We have not analyzed in depth very many concepts: Equilibrium refinements Asymmetric games n player games Continuous strategies Different dynamics Discrete dynamics Finite size/population effects Games and networks Laredo 2005 Game Theory / Angel Sánchez Discussion and conclusions Discussion and conclusions Game Theory as a tool to model any kind of systems where we have interacting agents: Biological Economical Social Modelling can be done at different levels Key: a clear identification/specification of the game and its dynamics Laredo 2005