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
Consider the following payoff matrix:
| Ben | |||
|---|---|---|---|
Anna |
Help | Leave | |
| Help | 7, -4 | -2, -8 | |
| Leave | -5, -8 | 3, 0 | |
What is the Mixed Strategy Nash Equilibrium.
Anna Helps with probability `2 / 3`. Ben Helps with probability `8 / 17`.
Anna's mixed strategy
Anna Helps with probability `p` so that Ben is indifferent between Helping and Leaving:
$$ \begin{align*} -4 \times p -8 \times (1 - p) &= -8 \times p + 0 \times (1 - p) \\ 4 p -8 &= -8 p + 0 \\ 12 p &= 8 \\ p &= \frac{8}{12} \end{align*} $$
Ben's mixed strategy
Ben Helps with probability `q` so that Anna is indifferent between Helping and Leaving:
$$ \begin{align*} 7 \times q -5 \times (1 - q) &= -2 \times q + 3 \times (1 - q) \\ 12 q -5 &= -5 q + 3 \\ 17 q &= 8 \\ q &= \frac{8}{17} \end{align*} $$Conclusion
In equilibrium:
- Anna Helps with probability `2 / 3` and Leaves with probability `1 / 3`.
- Ben Helps with probability `8 / 17` and Leaves with probability `9 / 17`.