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lecture6a - CMSC 498T Game Theory 6a Repeated Games Dana...

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Nau: Game Theory 1 Updated 10/17/11 CMSC 498T, Game Theory 6a. Repeated Games Dana Nau University of Maryland
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2 Updated 10/17/11 Repeated Stag Hunt Repeated Games Used by game theorists, economists, social and behavioral scientists as highly simplified models of various real-world situations Roshambo Iterated Chicken Game Repeated Matching Pennies Iterated Prisoner’s Dilemma Repeated Ultimatum Game Iterated Battle of the Sexes
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Nau: Game Theory 3 Updated 10/17/11 Finitely Repeated Games In repeated games, some game G is played multiple times by the same set of agents Øඏ G is called the stage game Usually (but not always) a normal- form game Øඏ Each occurrence of G is called an iteration , round , or stage Usually each agent knows what all the agents did in the previous iterations, but not what they’re doing in the current iteration Øඏ Thus, imperfect information with perfect recall Usually each agent’s payoff function is additive Agent 1 : Agent 2 : C C D C Round 1: Round 2: Prisoner’s Dilemma: 3+0 = 3 3+5 = 5 Total payoff: Iterated Prisoner’s Dilemma, 2 iterations: C D C 3, 3 0, 5 D 5, 0 1, 1
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Nau: Game Theory 4 Updated 10/17/11 Iterated Prisoner’s Dilemma with 2 iterations: Strategies The repeated game has a much bigger strategy space than the stage game One kind of strategy is a stationary strategy : Øඏ Use the same strategy in every stage game More generally, an agent’s play at each stage may depend on what happened in previous iterations C D C D C D C D C D c d c d c d c d c d c d c d c d c d c d (6, 6) (8, 3) (3, 8) (4, 4) (3, 8) (5, 5) (0,10) (1, 6) (8, 3) (10,0) (5, 5) (6, 1) (4, 4) (6, 1) (1, 6) (2, 2) 2 2 2 2 2 2 2 2 1 1 1 1 2 2 1 C D C 3, 3 0, 5 D 5, 0 1, 1
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Nau: Game Theory 5 Updated 10/17/11 Examples Some well-known IPD strategies: AllC : always cooperate AllD : always defect Grim : cooperate until the other agent defects, then defect forever Tit-for-Tat (TFT) : on 1 st move, cooperate. On n th move, repeat the other agent’s ( n –1) th move Tit-for-Two-Tats (TFTT) : like TFT, but only only retaliates if the other agent defects twice in a row Tester : defect on round 1. If the other agent retaliates, play TFT. Otherwise, alternately cooperate and defect Pavlov : on 1st round, cooperate. Thereafter, win => use same action on next round; lose => switch to the other action ( “win” means 3 or 5 points, “lose” means 0 or 1 point) C C TFT Tester C C C C C C D C D C C C D D TFT or Grim AllD C D D D D D D D D D D D AllC, Grim, TFT, or Pavlov AllC, Grim, TFT, or Pavlov C C C C C C C C C C C C TFTT Tester C C C D C D C C D C C C C D Pavlov AllD C D C D C D D D D D D D
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Nau: Game Theory 6 Updated 10/17/11 Backward Induction If the number of iterations is finite and all players know what it is, we can use backward induction to find a subgame-perfect equilibrium This time it’s simpler than game-tree search Øඏ Regardless of what move you make, the next state is always the same Another instance of the stage game Øඏ
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