Lecture6a - CMSC 498T Game Theory 6a Repeated Games Dana Nau University of Maryland Updated Nau Game Theory 1 Repeated Games Used by game theorists

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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 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 Ø G is called the stage game Usually (but not always) a normal- form game Ø 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 Ø 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 3, 3 0, 5 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 : Ø 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 3, 3 0, 5 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 Ø Regardless of what move you make, the next state is always the same Another instance of the stage game Ø The only difference is how many points you’ve accumulated so far First calculate the SPE actions for round n
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This note was uploaded on 01/13/2012 for the course CMSC 498T taught by Professor Staff during the Fall '11 term at Maryland.

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Lecture6a - CMSC 498T Game Theory 6a Repeated Games Dana Nau University of Maryland Updated Nau Game Theory 1 Repeated Games Used by game theorists

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