# Mixed Strategy Nash Equilibrium

In a Mixed Strategy Equilibrium, players chooses probabilities that makes the other players indifferent between one strategy or another.

## Example

A horde of zombies is attacking the village where Anna and Ben live. They could either Help or Leave.

Ben

Anna
Help Leave
Help 3, 2 0, 0
Leave 0, 0 2, 1

## Anna's mixed strategy

Anna chooses to Help with probability p that makes Ben indifferent between Helping and Leaving. \begin{align*} E(Help) &= E(Leave) \\ 2 p + 0 (1 - p) &= 0p + 1(1-p) \\ 2p &= 1 - p \\ 3p = 1 \\ p = \frac{1}{3} \end{align*}

## Ben's mixed strategy

Ben chooses to Help with probability q that makes Anna indifferent between Helping and Leaving. \begin{align*} E(Help) &= E(Leave) \\ 3 q + 0 (1 - q) &= 0q + 2(1-q) \\ 3q &= 2 - 2q \\ 5q = 2 \\ q = \frac{2}{5} \end{align*}

## Conclusion

In equilibrium:

• Anna Helps with probability \frac{1}{3} and Leaves with probability \frac{2}{3}.
• Ben Helps with probability \frac{2}{5} and Leaves with probability \frac{3}{5}.

### Question

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