# Expected Payoff

The expected payoff is the average between the payoff when the other randomizes their strategy.

## 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 -18, -36 -90, 63
Leave -27, 63 -18, -36

Ben decides to Help. Anna decides to Help with probability 8 / 9.

What is Ben expected payoff?

Ben's expected payoff is

\begin{align*} E ( Help ) &= -36 \times 8 / 9 + 63 \times 1 / 9 \\ &= -32 + 7 \\ &= -25 \end{align*}