Expected Payoff

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


A horde of zombies is attacking the village where Anna and Ben live. They could either Help or Leave.


Help Leave
Help 3, 2 0, 0
Leave 0, 0 2, 1

Example 1:

Ben decides to 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 $$ E(Leave) = 0 \times \frac{3}{4} + 2 \times \frac{1}{4} = 0.5 $$


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