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Nash Equlibrium

Nash Equlibrium - 2 Mixed-Strategy Nash Equilibrium...

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Lecture 4: Nash Equilibrium Econ 171 Spring 2010 Overview Nash equilibrium Mixed-strategy Nash equilibrium 1
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Nash Equilibrium Begin with three 2 × 2 games that are not dominance solvable: X Y A 4 , 0 0 , 2 C 4 , 2 4 , 0 O B O 2 , 1 0 , 0 B 0 , 0 1 , 2 Battle of the Sexes H T H 1 , - 1 - 1 , 1 T - 1 , 1 1 , - 1 Matching Pennies Figure 1: Three games that are not dominance solvable Definition 1 A strategy profile s S is a Nash equilibrium if and only if s i BR i ( s - i ) for each player i . That is, u i ( s i , s - i ) u i ( s 0 i , s - i ) for each s 0 i S i and each player i . C D C 2 , 2 0 , 3 D 3 , 0 1 , 1 Prisoners’ Dilemma (again) N B N 0 , 0 3 , 1 B 1 , 3 2 , 2 Chicken A B A 1 , 1 0 , 0 B 0 , 0 1 , 1 Coordination Figure 2: Three (more) of the classic normal-form games A B A 2 , 2 0 , 0 B 0 , 0 1 , 1 Pareto Coordination O B O 4 , 2 2 , 3 B 6 , - 1 0 , 0 Pigs Figure 3: The last two classic games
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Unformatted text preview: 2 Mixed-Strategy Nash Equilibrium Denition 2 Consider a strategy prole = ( 1 , 2 ,..., n ) , where i S i for each player i . Prole is a mixed-strategy Nash equilibrium if and only if u i ( i ,-i ) u i ( i ,-i ) for each s i S i and each player i . That is, i is a best response to -i for every player i . A B A 7 , 1 3 , 2 B 5 , 1 4 , C 4 , 3 6 , A B A , 7 , 4 B 3 , 9 , C 2 , 2 , 4...
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