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Lecture13-new

# Lecture13-new - 6.254 Game Theory with Engineering...

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6.254 : Game Theory with Engineering Applications Lecture 13: Extensive Form Games Asu Ozdaglar MIT March 18, 2010 1

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Game Theory: Lecture 13 Introduction Outline Extensive Form Games with Perfect Information One-stage Deviation Principle Applications Ultimatum Game Rubinstein-Stahl Bargaining Model Reading: Fudenberg and Tirole, Sections 4.1-4.4. 2
Game Theory: Lecture 13 Extensive Form Games Introduction We have studied extensive form games which model sequential decision making. Equilibrium notion for extensive form games: Subgame Perfect (Nash) Equilibrium . It requires each player’s strategy to be “optimal” not only at the start of the game, but also after every history. For finite horizon games, found by backward induction . Backward induction refers to starting from the “last” subgames of a finite game, finding the best response strategy profiles or the Nash equilibria in the subgames, then assigning these strategies profiles and the associated payoffs to be subgames, and moving successively towards the beginning of the game. For finite/infinite horizon games, characterization in terms of one-stage deviation principle. 3

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Game Theory: Lecture 13 One-stage Deviation Principle One-stage Deviation Principle Focus on multi-stage games with observed actions (or perfect information games). One-stage deviation principle is essentially the principle of optimality of dynamic programming. We first state it for finite horizon games. Theorem (One-stage deviation principle) For finite horizon multi-stage games with observed actions , s * is a subgame perfect equilibrium if and only if for all i, t and h t , we have u i ( s * i , s * - i | h t ) u i ( s i , s * - i | h t ) for all s i satisfying s i ( h t ) 6 = s * i ( h t ) , s i | h t ( h t + k ) = s * i | h t ( h t + k ) , for all k > 0, and all h t + k G ( h t ) . Informally, s is a subgame perfect equilibrium (SPE) if and only if no player i can gain by deviating from s in a single stage and conforming to s thereafter. 4
Game Theory: Lecture 13 One-stage Deviation Principle One-stage Deviation Principle for Infinite Horizon Games The proof of one-stage deviation principle for finite horizon games relies on the idea that if a strategy satisfies the one stage deviation principle then that strategy cannot be improved upon by a finite number of deviations. This leaves open the possibility that a player may gain by an infinite sequence of deviations, which we exclude using the following condition. Definition Consider an extensive form game with an infinite horizon, denoted by G . Let h denote an -horizon history, i.e., h = ( a 0 , a 1 , a 2 ... ) , is an infinite sequence of actions. Let h t = ( a 0 , ... a t - 1 ) be the restriction to first t periods. The game G is continuous at infinity if for all players i, the payoff function u i satisfies sup h , ˜ h s.t. h t = ˜ h t | u i ( h ) - u i ( ˜ h ) | → 0 as t .

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