# nash equilibrium

from: http://en.wikipedia.org/wiki/Nash_equilibrium#Formal_definition

Let *(S, f)* be a game with *n* players, where *S _{i}* is the strategy set for player

*i*,

*S=S*is the set of strategy profiles and

_{1}X S_{2}... X S_{n}*f=(f*is the payoff function for

_{1}(x), ..., f_{n}(x))*x*

*S*. Let *x _{i}* be a strategy profile of player

*i*and

*x*be a strategy profile of all players except for player

_{-i}*i*. When each player

*i*

{1, ..., n} chooses strategy *x _{i}* resulting in strategy profile

*x = (x*then player

_{1}, ..., x_{n})*i*obtains payoff

*f*. Note that the payoff depends on the strategy profile chosen, i.e., on the strategy chosen by player

_{i}(x)*i*as well as the strategies chosen by all the other players. A strategy profile

*x*

^{*} *S* is a Nash equilibrium (NE) if no unilateral deviation in strategy by any single player is profitable for that player, that is

A game can have either a pure-strategy or a mixed Nash Equilibrium, (in the latter a pure strategy is chosen stochastically with a fixedfrequency). Nash proved that if we allow mixed strategies, then every game with a finite number of players in which each player can choose from finitely many pure strategies has at least one Nash equilibrium.

When the inequality above holds strictly (with > instead of

) for all players and all feasible alternative strategies, then the equilibrium is classified as a **strict Nash equilibrium**. If instead, for some player, there is exact equality between

and some other strategy in the set *S*, then the equilibrium is classified as a **weak Nash equilibrium**.

[edit]