Nash Equilibrium in mixed strategies

We don’t always have a pure strategy NE, but every game has at least one mixed strategy Nash equilibrium.

For a game with payoff matrices $A$ (to $i$) and $B$ (to $j$), a mixed strategy $(x^{\ast },y^{\ast })$ is a Nash equilibrium solution if:

$$\begin{array}{r}\forall x,x^{\ast }A{y}^{\ast T}\ge xA{y}^{\ast T}\\ \forall y,x^{\ast }A{y}^{\ast T}\ge x^{\ast }B{y}^T\end{array}$$

In other words:

• $x^{\ast }$ gives higher expected value to $i$ than any other strategy when $j$ plays $y^{\ast }$.
• $y^{\ast }$ gives higher expected value to $j$ than any other strategy when $i$ plays $x^{\ast }$.