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MixedNash-1

# MixedNash-1 - Nash Equilibria in Mixed Strategies Here is a...

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Nash Equilibria in Mixed Strategies Here is a general method for finding Nash equilibria in mixed strategies. The key is to think of a game as a list of functions which specify the payoffs for each of the players as functions of the action taken by the given player and the actions taken by the other player(s). A best-reply for a player then is a simply a choice that maximizes the payoff of that player given the actions taken by the other players. A best-reply function is just that function which describes the best-reply for the player as a function of the actions taken by the other players. Nash equilibria are then combinations of strategies which are mutually best replies to one another. This "maximization" approach works for mixed-strategies as well as pure-strategies - all that's required is to think of what's being chosen in the right way. Let's consider the game of chicken: Dean James Swerve Straight q-Mix Swerve 0 0 1 -1 1-q q-1 Straight -1 1 -2 -2 q-2 3q-2 p-Mix p-1 1-p 3p-2 p-2 -2+3p+q-2pq -2+p+3q-2pq The "q-Mix" column gives the payoffs if Dean plays "swerve" with probability q and "straight" with probability 1-q. Similarly, the "p-Mix" row gives the payoffs if James plays "swerve" with probability p and "straight" with probability 1-p. The payoffs

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MixedNash-1 - Nash Equilibria in Mixed Strategies Here is a...

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