Definition of Nash Equilibrium

Definition: Nash equilibria are sets of strategies for players in a noncooperative game such that no single one of them would be better off switching strategies unless others did.

Formally: Using the normal form[/link] definitions, let utility functions as functions of payoffs for the n players u₁() ... u_n() and sets of possible actions A=A₁ x ... x A_n, be common knowledge to all the players. Also define a_-i as the vector of actions of the other players besides player i. Then a Nash equilibrium is an array of actions a^* in A such that u_i(a^*) >= u_i(a_-i^* | a_i) for all i and all a_i in A_i.

In a two-player game that can be expressed in a payoff matrix, one can generally find Nash equilibria if there are any by, first, crossing out strictly dominated strategies for each player. After crossing out any strategy, consider again all the strategies for the other player. When done crossing out strategies, consider which of the remaining cells fail to meet the criteria above, and cross them out too. At the end of the process, each player must be indifferent among his remaining choices, GIVEN the action of the others. (Econterms)

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