## 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 |

## Example 1:

Ben decides to flip a roll a dice. If he rolls a 1 or a 2, he would Help (probability `p=\frac{1}{3}`). Otherwise, he would Leave (probability `1 - p=\frac{2}{3}`).

Anna decides to help. Her expected payoff is

$$ E(Help) = 1 \times p + 0 \left( 1 - p \right)= 1 \times \frac{1}{3} + 0 \times \frac{2}{3} = \frac{1}{3} $$## Example 2:

Anna decides to Help with probability `p=\frac{3}{4}`. If Ben decides to Leave, his expected payoff is

$$ E(Leave) = 0 \times \frac{3}{4} + 2 \times \frac{1}{4} = 0.5 $$### Question

Consider the following payoff matrix:

Ben | |||
---|---|---|---|

Anna |
Help | Leave | |

Help | -24, -80 | 8, -8 | |

Leave | 32, -40 | -24, 64 |

Ben decides to Help. Anna decides to Help with probability 7 / 8.

What is Ben expected payoff?

Ben's expected payoff is

$$ \begin{align*} E ( Help ) &= -80 \times 7 / 8 -40 \times 1 / 8 \\ &= -70 -5 \\ &= -75 \end{align*} $$