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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 definitions, let utility functions as functions of payoffs for the n players u1() ... un() and sets of possible actions A=A1 x ... x An, 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 ui(a*) >= ui(a-i* | ai) for all i and all ai in Ai.

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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