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 -56, 14 -21, -56
Leave 28, 0 56, -7

Ben decides to Leave. Anna decides to Help with probability 2 / 7.

What is Ben expected payoff?

Ben's expected payoff is

$$ \begin{align*} E ( Leave ) &= -56 \times 2 / 7 -7 \times 5 / 7 \\ &= -16 -5 \\ &= -21 \end{align*} $$