Mixed Strategy Nash Equilibrium

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

Anna's mixed strategy

Anna chooses to Help with probability `p` that makes Ben indifferent between Helping and Leaving. $$ \begin{align*} E(Help) &= E(Leave) \\ 2 p + 0 (1 - p) &= 0p + 1(1-p) \\ 2p &= 1 - p \\ 3p = 1 \\ p = \frac{1}{3} \end{align*} $$

Ben's mixed strategy

Ben chooses to Help with probability `q` that makes Anna indifferent between Helping and Leaving. $$ \begin{align*} E(Help) &= E(Leave) \\ 3 q + 0 (1 - q) &= 0q + 2(1-q) \\ 3q &= 2 - 2q \\ 5q = 2 \\ q = \frac{2}{5} \end{align*} $$

Conclusion

In equilibrium:

• Anna Helps with probability `\frac{1}{3}` and Leaves with probability `\frac{2}{3}`.
• Ben Helps with probability `\frac{2}{5}` and Leaves with probability `\frac{3}{5}`.

Question