Nash Equilibrium solution

A pair of strategies $(i^{\ast },j^{\ast })$ is a Nash equilibrium solution to the game $(A,B)$ if:

$$\begin{array}{rl}\forall i,a_{i\ast ,j\ast }& \ge a_{i,j\ast }\\ \forall j,b_{i\ast ,j\ast }& \ge a_{i\ast ,j}\end{array}$$

That is, $(i^{\ast },j^{\ast })$ is a Nash equilibrium if:

• if $j$ plays $j^{\ast }$ then $i^{\ast }$ gives the best outcome for $i$
• if $i$ plays $i^{\ast }$ then $j^{\ast }$ gives the best outcome for $i$

Unfortunately:

1. not every interaction scenario has a pure strategy NE
2. some interaction scenarios have more than one NE