# gameL3 - Mixed Strategy Nash Equilibrium Let G = hN, (Ai),...

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Mixed Strategy Nash Equilibrium Let G = h N, ( A i ) , ( u i ) i be a strategic game. Prefer- ences must be speci f ed over lotteries on A ,wh i chw e assume are represented by the expectation of u i ( a ) . Let ( A i ) be the set of probability distributions over A i . α i ( A i ) is called a mixed strategy and a i A i is called a pure strategy . It is assumed that randomizations according to ( α 1 ,...,α n ) are performed independently. Thus, ( α 1 ,...,α n ) induces a lottery on A ,where a A has probability Y j N α j ( a j ) where α j ( a j ) is the probability of action a j under the mixed strategy

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De f nition 32.1: The mixed extension of the strategic game h N, ( A i ) , ( u i ) i is the strategic game h N, ( ( A i )) , ( U i ) i , where U i : × j N ( A j ) R assigns each α =( α 1 ,...,α n ) the expected value under u i of the lottery induced by α . For f nite games, we have U i ( α )= X a A Y j N α j ( a j ) u i ( a ) . Also, letting e ( a i ) denote the degenerate mixed strategy assigning probability one to a i ,wecanwr ite U i ( α )= X a i A i α i ( a i ) U i ( α i ,e ( a i )) . (1) De f nition 32.3: A mixed strategy Nash equilibrium of a strategic game is a Nash equilibrium of its mixed exten-
Claim: If α is a mixed strategy Nash equilibrium of G such that α i = e ( a i ) for all i N ,then a is a (pure strategy) Nash equililibrium of G , and conversely. Since there is no element of ( A i ) that yields higher expected utility than e ( a i ) ,thenno e ( a 0 i ) can yield higher utility. Thus, a is a (pure strategy) Nash equililibrium. From (1), we can write U i ( α i ,a i )= X a i A i α i ( a i ) u i ( a i ,a i ) . Because u i ( a i ,a i ) u i ( a i ,a i ) for all a i A i ,we have U i ( α i ,a i ) U i ( α i ,a i ) for all α i ( A i ) .

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Proposition 33.1: Every f nite strategic game has a mixed strategy Nash equilibrium. The strategy set is now convex, and payo
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## This note was uploaded on 07/17/2008 for the course ECON 817 taught by Professor Peck during the Fall '07 term at Ohio State.

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gameL3 - Mixed Strategy Nash Equilibrium Let G = hN, (Ai),...

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