# Optimal Mixed Strategies?

I'm trying to understand how I would find the optimal mixed strategies in zero sum games. For example... given the following zero sum game in standard strategic form...

\begin{array}{r|r|} +8 & -2 \\ -4 & +20\\ \end{array}

How would I find the optimal mixed strategy for the given player?

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Optimal in what sense? –  Chris Eagle Dec 12 '11 at 13:18

Suppose hero chooses first strategy (payoffs +8/-2) with probability $p$. We want to have the same expectation independent of which strategy the other player chooses. So

$$8p -4(1-p) = -2p + 20(1-p) \Rightarrow p = \frac{12}{17}.$$

gives the optimal mixed strategy.

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There is an optimal mixed strategy for each of the row and the column players. A similar calculation will determine the optimal mixed strategy for the other player. Using p above you can find out whether the game advantages Row or Column or is fair. Assuming the payoffs above are from Row's point of view the expected value from the point of view of the other player will be the negative of the value of the game from that of Row. –  Joseph Malkevitch Dec 12 '11 at 20:46