## 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

This question is available to anyone who signs up.