3.1 Nash Equilibrium

This is a chapter from Jim Ratliff's Graduate-Level Game-Theory Course. See outline for the entire course. I no longer maintain, update, or correct these notes. However, I would appreciate hearing from people who download these notes and find them useful. Also I *may* eventually post problem sets and their solutions. Let me know if you'd like to be notified of such changes. (Please email me.)

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When all players are rational and their rationality is common knowledge, the resulting outcome will be rationalizable. If a player does not correctly forecast the play of an opponent, her beliefs to which she played a best response were incorrect. Her actual action may not be a best response to the actual play of her opponents. Therefore in a rationalizable outcome one or more players can have *ex post* regret about their choices.

When rational players correctly forecast the strategies of their opponents they are not merely playing best responses to their *beliefs* about their opponents' play; they are playing best responses to the *actual* play of their opponents. When all players correctly forecast their opponents' strategies, and play best responses to these forecasts, the resulting strategy profile is a *Nash equilibrium*.

Nash equilibrium has been justified as representing a *self-enforcing agreement*. I present an example of Aumann's inteneded to show that a Nash equilibrium need not be self-enforcing. Nash equilibria can also be vulnerable to multiplayer deviations. We discuss a difficulty of justifying Nash equilibrium as the outcome of a dynamic process: if it is known that the game is repeated, then the repeated game is itself a new, more-complicated game.

A game need not have a pure-strategy Nash equilibrium. However, every strategic-form game does have a (possibly degenerate) mixed-strategy Nash equilibrium. We learn (thanks to John Nash) that a Nash equilibrium exists if and only if there exists a fixed point of a particular best-response correspondence. Kakutani's fixed-point theorem guarantees the existence of this fixed point when the correspondence satisfies certain conditions. We verify that the best-response correspondence satisfies the Kakutani conditions and hence prove the existence of a Nash equilibrium.

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