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 | -21, 49 | -35, -70 | |
Leave | 28, -49 | 56, -14 |
Anna decides to Help. Ben decides to Help with probability 6 / 7.
What is Anna expected payoff?
Anna's expected payoff is
$$ \begin{align*} E ( Help ) &= -21 \times 6 / 7 -35 \times 1 / 7 \\ &= -18 -5 \\ &= -23 \end{align*} $$