Mixed strategy

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In his famous paper, John Forbes Nash proved that there is an equilibrium for every finite game. One can divide Nash equilibria into two types. Pure strategy Nash equilibria are Nash equilibria where all players are playing pure strategies. Mixed strategy Nash equilibria are equilibria where at least one player is playing a mixed strategy A game can have a pure-strategy or a mixed-strategy Nash equilibrium.

A pure strategy is an unconditional, defined choice that a person makes in a situation or game. A mixed strategy is an assignment of probability to all choices in the strategy set.

Definition

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Nash equilibrium in mixed strategy

A mixed strategy Nash equilibrium involves at least one player playing a randomized strategy and no player being able to increase his or her expected payoff by playing an alternate strategy. A Nash equilibrium in which no player randomizes is called a pure strategy Nash equilibrium.

Example: Matching pennies

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