# Nash equilibrium

A Nash equilibrium of a game is a set of (possibly mixed) strategies $\sigma=(\sigma_{1},\ldots,\sigma_{n})$ such that, if each player $i$ believes that that every other player $j$ will play $\sigma_{j}$, then $i$ should play $\sigma_{i}$. That is, when $u_{i}$ is the utility function for the $i$-th player:

 $\sigma_{i}\neq\sigma^{\prime}_{i}\rightarrow u_{i}(\sigma_{i},\sigma_{-1})>u_{% i}(\sigma^{\prime}_{i},\sigma_{-1})$
 $\forall i\leq n\text{ and }\forall\sigma^{\prime}_{i}\in\Sigma_{i}$

Translated, this says that if any player plays any strategy other than the one in the Nash equilibrium then that player would do worse than playing the Nash equilibrium.

Title Nash equilibrium NashEquilibrium 2013-03-22 12:51:35 2013-03-22 12:51:35 Henry (455) Henry (455) 8 Henry (455) Definition msc 91A99