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1 Evolution & Economics No. 4

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2 Evolutionary Stability in Repeated Games Played by Finite Automata K. Binmore & L. Samuelson J.E.T Automata

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3 Finite Automata playing the Prisoners’ Dilemma C D D C,D C Grim C D D D C Tit For Tat (TFT) C D C D C C Tat For Tit (TAFT) D D C D C C Tweedledum D states (& actions) transitions

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4 Automata playing the Prisoners’ Dilemma D C D C,D C CA D C C,D C Tweedledee D D C,D D C C

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5 Two Automata playing together, eventually follow a cycle (handshake) The payoff is the limit of the means. The cost of an automaton is the number of his states. The cost enters the payoffs lexicographically.

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6 CD C 2, 2-1, 3 D 3, -10, 0 The Structure of Nash Equilibrium in Repeated Games with Finite Automata Dilip Abreu & Ariel Rubinstein Econometrica,1988

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7 CD C 2, 2-1, 3 D 3, -10, 0 The Structure of Nash Equilibrium in Repeated Games with Finite Automata Dilip Abreu & Ariel Rubinstein Econometrica,1988 (0,0) (3,-1) (-1,3) (2,2) N.E. of repeated Game N.E in Repeated Games with Finite Automata (Abreu Rubinstein)

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8 Binmore Samuelson:

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9 x ? x a y x ? x a y If then: x ? x b y

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10 x ? x a y If then: x ? x b y x ? x a y

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11 x ? x a y Q.E.D.

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12 D C,D D C C C is not an ESS, it can be invaded by D. D is not an ESS, it can be invaded by Tit For Tat. C D D D C Tit For Tat (TFT) C

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14 Q.E.D.

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15 Q.E.D. In the P.D. Tit For Tat and Grim are not MESS (they do not use one state against themselves) C D D C,D C Grim C D D D C Tit For Tat (TFT) C

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16 For a general, possibly non symmetric game G. Define the symmetrized version of G: G # #. A player is player 1 with probability 0.5 and player 2 with probability 0.5 The previous lemmas apply to (a 1,a 2 ) 1.An ESS has a single state │a 1 │=│a 2 │=1 2.If (a 1,a 2 ) is a MESS it uses all its states when playing against itself, i.e. a 1,a 2 use all their states when playing against the other.

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18 Q.E.D.

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20 D C D C C Tat For Tit (TAFT) D It can be invaded by: D C D C,D C CA D C C,D C AC D C D D C C CC D C D D C CD C,D D C AA

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21 D C D C C Tat For Tit (TAFT) D It can be invaded by: D C C,D C AC D

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22 D C D C C Tat For Tit (TAFT) D It can be invaded by: D C D C,D C CA

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23 D C D C C Tat For Tit (TAFT) D It can be invaded by: C D D C C CC D

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24 D C D C C Tat For Tit (TAFT) D No other (longer and more sophisticated) automaton can invade. Any exploitation of TAFT (playing D against his C ) makes TAFT play D, so the average of these two periods is (3+0)/2 = 1.5 < 2, the average of cooperating. CD C 2, 2-1, 3 D 3, -10, 0

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25 A population consisting of: D C D C C Tat For Tit (TAFT) D C D D C C CC D C D D C CD C,D can be invaded only by: D C D C CA D C C,D C AC D If AC invaded, it does not do well against CD D C D C ……. C D C D …….

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26 A population consisting of: D C D C C Tat For Tit (TAFT) D C D D C C CC D C D D C CD C,D can be invaded only by CA D C D C,D C CA D C C,D AA If AA invaded, it does not do well against CC D C C C C……. C D D D D…….

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27 A population consisting of: D C D C C Tat For Tit (TAFT) D C D D C C CC D C D D C CD C,D can be invaded only by CA D C D C,D C CA but if CA i nvaded then a sophisticated automaton S c an invade and exploit CA. S starts with C. i f it saw C it continues with C forever (the opponent must be CD or CC ). If it saw D, it plays D again, if the other then plays D it must be TAFT. S plays another D and then C forever. If, however, after 2x D, the other played C, then it must be CA, and S should play D forever.

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28 A population consisting of: D C D C C Tat For Tit (TAFT) D C D D C C CC D C D D C CD C,D can be invaded only by CA D C D C,D C CA When S invades, CA will vanish, and then S which is a complex automaton will die out. Evolution - 5

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