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Unformatted text preview: Lecture V: Mixed Strategies Markus M. M¨obius February 24, 2004 • Gibbons, sections 1.31.3.A • Osborne, chapter 4 1 The Advantage of Mixed Strategies Consider the following RockPaperScissors game: Note that RPS is a zero sum game. S P R R P S 0,01,1 1,1 1,1 0,01,11,1 1,1 0,0 This game has no purestrategy Nash equilibrium. Whatever pure strategy player 1 chooses, player 2 can beat him. A natural solution for player 1 might be to randomize amongst his strategies. Another example of a game without purestrategy NE is matching pen nies. As in RPS the opponent can exploit his knowledge of the other player’s action. 11,1 1,1 1,11,1 H T H T Fearing this what might the opponent do? One solution is to randomize and play a mixed strategy. Each player could flip a coin and play H with probability 1 2 and T with probability 1 2 . Note that each player cannot be taken advantage of. Definition 1 Let G be a game with strategy spaces S 1 , S 2 ,.., S I . A mixed strategy σ i for player i is a probability distribution on S i i.e. for S i finite a mixed strategy is a function σ i : S i → < + such that ∑ s i ∈ S i σ i ( s i ) = 1 . Several notations are commonly used for describing mixed strategies. 1. Function (measure): σ 1 ( H ) = 1 2 and σ 1 ( T ) = 1 2 2. Vector: If the pure strategies are s i 1 ,.. s iN i write ( σ i ( s i 1 ) ,..,σ i ( s iN i )) e.g. ( 1 2 , 1 2 ) . 3. 1 2 H + 1 2 T Class Experiment 1 Three groups of two people. Play RPS with each other 30 times. Calculate frequency with which each strategy is being played. • Players are indifferent between strategies if opponent mixes equally between all three strategies. • In games such as matching pennies, poker bluffing, football run/pass etc you want to make the opponent guess and you worry about being found out. 2 2 Mixed Strategy Nash Equilibrium Write Σ i (also Δ( S i )) for the set of probability distributions on S i . Write Σ for Σ 1 × .. × Σ I . A mixed strategy profile σ ∈ Σ is an Ituple ( σ 1 ,..,σ I ) with σ i ∈ Σ i . We write u i ( σ i ,σ i ) for player i’s expected payoff when he uses mixed strategy σ i and all other players play as in σ i . u i ( σ i ,σ i ) = X s i ,s i u i ( s i ,s i ) σ i ( s i ) σ i ( s i ) (1) Remark 1 For the definition of a mixed strategy payoff we have to assume that the utility function fulfills the VNM axioms. Mixed strategies induce lotteries over the outcomes (strategy profiles) and the expected utility of a lottery allows a consistent ranking only if the preference relation satisfies...
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This note was uploaded on 05/19/2010 for the course 412 002 taught by Professor Dingli during the Spring '10 term at École Normale Supérieure.
 Spring '10
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