Lecture4-new

# Lecture4-new - 6.254 Game Theory with Engineering...

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6.254 : Game Theory with Engineering Applications Lecture 4: Strategic Form Games - Solution Concepts Asu Ozdaglar MIT February 11, 2010 1

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Game Theory: Lecture 4 Introduction Outline Review Correlated Equilibrium Existence of a Mixed Strategy Equilibrium in Finite Games Reading: Fudenberg and Tirole, Chapters 1 and 2. 2
Game Theory: Lecture 4 Review Rationalizability A different solution concept in which a player’s belief about the other players’ actions is not assumed to be correct (as in a Nash equilibrium), but rather, simply constrained by rationality. (1) Players maximize with respect to some ( uncorrelated ) beliefs about opponent’s behavior (i.e., they are rational). (2) Beliefs have to be consistent with other players being rational, and being aware of each other’s rationality, and so on (but they need not be correct). Leads to an infinite regress: “I am playing strategy σ 1 because I think player 2 is using σ 2 , which is a reasonable belief because I would play it if I were player 2 and I thought player 1 was using σ 0 1 , which is a reasonable thing to expect for player 2 because σ 0 1 is a best response to σ 0 2 , . . . . 3

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Game Theory: Lecture 4 Review Never-Best Response and Strictly Dominated Strategies Definition A pure strategy s i is strictly dominated if there exists a mixed strategy σ i Σ i such that u i ( σ i , s - i ) > u i ( s i , s - i ) for all s - i S - i . Definition A pure strategy s i is a never-best response if for all beliefs σ - i there exists σ i Σ i such that u i ( σ i , σ - i ) > u i ( s i , σ - i ) . A strictly dominated strategy is a never-best response. Does the converse hold? Last time, we studied a 3-player example that illustrates a never-best response strategy which is not strictly dominated. 4
Game Theory: Lecture 4 Review Rationalizable Strategies Iteratively eliminating never-best response strategies yields rationalizable strategies. Start with ˜ S 0 i = S i . For each player i ∈ I and for each n 1, ˜ S n i = { s i ˜ S n - 1 i | ∃ σ - i j 6 = i ˜ Σ n - 1 j such that u i ( s i , σ - i ) u i ( s 0 i , σ - i ) for all s 0 i ˜ S n - 1 i } . Independently mix over ˜ S n i to get ˜ Σ n i . Let R i = n = 1 ˜ S n i . We refer to the set R i as the set of rationalizable strategies of player i . 5

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Game Theory: Lecture 4 Review Rationalizable Strategies Since the set of strictly dominated strategies is a strict subset of the set of never-best response strategies, set of rationalizable strategies represents a further refinement of the strategies that survive iterated strict dominance. Let NE i denote the set of pure strategies of player i used with positive probability in any mixed Nash equilibrium. Then, we have NE i R i D i , where R i is the set of rationalizable strategies of player i , and D i is the set of strategies of player i that survive iterated strict dominance.
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