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 10, -15 -45, 40
Leave 35, 5 -20, 50

Anna decides to Help. Ben decides to Help with probability 4 / 5.

What is Anna expected payoff?

Anna's expected payoff is

$$ \begin{align*} E ( Help ) &= 10 \times 4 / 5 -45 \times 1 / 5 \\ &= 8 -9 \\ &= -1 \end{align*} $$