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lecture3 notes - 6.254 Game Theory with Engineering...

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6.254 : Game Theory with Engineering Applications Lecture 3: Strategic Form Games - Solution Concepts Asu Ozdaglar MIT February 9, 2010 1

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Game Theory: Lecture 3 Introduction Outline Review Examples of Pure Strategy Nash Equilibria Mixed Strategies and Mixed Strategy Nash Equilibria Characterizing Mixed Strategy Nash Equilibria Rationalizability Reading: Fudenberg and Tirole, Chapters 1 and 2. 2
Game Theory: Lecture 3 Nash Equilibrium Pure Strategy Nash Equilibrium Definition (Nash equilibrium) A (pure strategy) Nash Equilibrium of a strategic game �I , ( S i ) i ∈I , ( u i ) i ∈I is a strategy profile s S such that for all i ∈ I u i ( s i , s i ) u i ( s i , s i ) for all s i S i . Why is this a “reasonable” notion? No player can profitably deviate given the strategies of the other players. Thus in Nash equilibrium, “best response correspondences intersect”. Put differently, the conjectures of the players are consistent : each player i chooses s i expecting all other players to choose s i , and each player’s conjecture is verified in a Nash equilibrium. 3

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Game Theory: Lecture 3 Examples Example: Second Price Auction Second Price Auction (with Complete Information) The second price auction game is specified as follows: An object to be assigned to a player in { 1, .., n } . Each player has her own valuation of the object. Player i ’s valuation of the object is denoted v i . We further assume that v 1 > v 2 > ... > 0. Note that for now, we assume that everybody knows all the valuations v 1 , . . . , v n , i.e., this is a complete information game. We will analyze the incomplete information version of this game in later lectures. The assignment process is described as follows: The players simultaneously submit bids, b 1 , .., b n . The object is given to the player with the highest bid (or to a random player among the ones bidding the highest value). The winner pays the second highest bid. The utility function for each of the players is as follows: the winner receives her valuation of the object minus the price she pays, i.e., v i b j ; everyone else receives 0. 4
Game Theory: Lecture 3 Examples Second Price Auction (continued) Proposition In the second price auction, truthful bidding, i.e., b i = v i for all i, is a Nash equilibrium. Proof: We want to show that the strategy profile ( b 1 , .., b n ) = ( v 1 , .., v n ) is a Nash Equilibrium— a truthful equilibrium . First note that if indeed everyone plays according to that strategy, then player 1 receives the object and pays a price v 2 . This means that her payoff will be v 1 v 2 > 0, and all other payoffs will be 0. Now, player 1 has no incentive to deviate, since her utility can only decrease. Likewise, for all other players v i = v 1 , it is the case that in order for v i to change her payoff from 0 she needs to bid more than v 1 , in which case her payoff will be v i v 1 < 0.

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