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:

$${\sigma}_{i}\ne {\sigma}_{i}^{\prime}\to {u}_{i}({\sigma}_{i},{\sigma}_{1})>{u}_{i}({\sigma}_{i}^{\prime},{\sigma}_{1})$$ 


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