Nash equilibrium

A Nash equilibrium of a game is a set of (possibly mixed) strategies $\sigma =({\sigma }_1,\mathrm{\dots },{\sigma }_n)$ 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:

$${\sigma }_i\ne {\sigma }_i^{\prime }\to u_i({\sigma }_i,{\sigma }_{-1})>u_i({\sigma }_i^{\prime },{\sigma }_{-1})$$

$$\forall i\le n\text{ and }\forall {\sigma }_i^{\prime }\in {\mathrm{\Sigma }}_i$$

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.