ECS1050.06.post

# ECS1050.06.post - Unit III: The Evolution of Cooperation...

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Unit III: The Evolution of Cooperation Can Selfishness Save the Environment? Repeated Games: the Folk Theorem Evolutionary Games A Tournament How to Promote Cooperation 4/14 7/28 7/9

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Repeated Games Some Questions: What happens when a game is repeated? Can threats and promises about the future influence behavior in the present? Cheap talk Finitely repeated games: Backward induction Indefinitely repeated games: Trigger strategies
Can threats and promises about future actions influence behavior in the present? Consider the following game, played 2X : C 3,3 0,5 D 5,0 1,1 Repeated Games C D See Gibbons: 82-104.

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Repeated Games Draw the extensive form game: (3,3) (0,5) (5,0) (1,1) 6,6) (3,8) (8,3) (4,4) (3,8)(0,10)(5,5)(1,6)(8,3) (5,5)(10,0) (6,1) (4,4) (1,6) (6,1) (2
Repeated Games Now, consider three repeated game strategies: D (ALWAYS DEFECT): Defect on every move. C (ALWAYS COOPERATE): Cooperate on every move. T (TRIGGER): Cooperate on the first move, then cooperate after the other cooperates. If the other defects, then defect forever .

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Repeated Games If the game is played twice, the V(alue) to a player using ALWAYS DEFECT (D) against an opponent using ALWAYS DEFECT(D) is: V (D/D) = 1 + 1 = 2, and so on. . . V (C/C) = 3 + 3 = 6 V (T/T) = 3 + 3 = 6 V (D/C) = 5 + 5 = 10 V (D/T) = 5 + 1 = 6 V (C/D) = 0 + 0 = 0 V (C/T) = 3 + 3 = 6 V (T/D) = 0 + 1 = 1 V (T/C) = 3 + 3 = 6
Repeated Games Time average payoffs: n V (D/D) = 1 + 1 + 1 + ... /n = 1 V (C/C) = 3 + 3 + 3 + ... /n = 3 V (T/T) = 3 + 3 + 3 + ... /n = 3 V (D/C) = 5 + 5 + 5 + ... /n = 5 V (D/T) = 5 + 1 + 1 + ... /n = 1 + ε V (C/D) = 0 + 0 + 0 + ... /n = 0 V (C/T) = 3 + 3 + 3 + /n = 3 V (T/D) = 0 + 1 + 1 + ... /n = 1 - ε V (T/C) = 3 + 3 + 3 + ... /n = 3

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Repeated Games Now draw the matrix form of this game: 1x T 3,3 0,5 3,3 C 3,3 0,5 3,3 D 5 ,0 1 ,1 5 ,0 C D T
Repeated Games T 3,3 1- ε, 1+ ε 3 ,3 C 3,3 0,5 3 ,3 D 5 ,0 1 ,1 1+ ε ,1- ε C D T If the game is repeated, ALWAYS DEFECT is no longer dominant. Time Average Payoffs

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Repeated Games T 3,3 1- ε, 1+ ε 3 ,3 C 3,3 0,5 3 ,3 D 5 ,0 1 ,1 1+ ε ,1- ε C D T … and TRIGGER achieves “a NE with itself.”
Repeated Games Time Average Payoffs T(emptation) > R(eward) > P(unishment)> S(ucker) T R,R P - ε, P + ε R ,R C R,R S,T R ,R D T ,S P ,P P + ε , P - ε C D T

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Discounting The discount parameter , δ , is the weight of the next payoff relative to the current payoff. In a indefinitely repeated game, δ can also be interpreted as the likelihood of the game continuing for another round (so that the expected number of moves per game is 1/(1- δ )). The
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## This note was uploaded on 03/21/2010 for the course ECON 1050 taught by Professor Neugeboren during the Summer '09 term at Harvard.

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ECS1050.06.post - Unit III: The Evolution of Cooperation...

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