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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

				
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