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 | -9, 0 | 5, 7 | |
| Leave | 9, 1 | 7, -5 | |
What is the Mixed Strategy Nash Equilibrium.
Anna Helps with probability `6 / 13`. Ben Helps with probability `1 / 8`.
Anna's mixed strategy
Anna Helps with probability `p` so that Ben is indifferent between Helping and Leaving:
$$ \begin{align*} 0 \times p + 1 \times (1 - p) &= 7 \times p -5 \times (1 - p) \\ -1 p + 1 &= 12 p -5 \\ -13 p &= -6 \\ p &= \frac{-6}{-13} \end{align*} $$
Ben's mixed strategy
Ben Helps with probability `q` so that Anna is indifferent between Helping and Leaving:
$$ \begin{align*} -9 \times q + 9 \times (1 - q) &= 5 \times q + 7 \times (1 - q) \\ -18 q + 9 &= -2 q + 7 \\ -16 q &= -2 \\ q &= \frac{-2}{-16} \end{align*} $$Conclusion
In equilibrium:
- Anna Helps with probability `6 / 13` and Leaves with probability `7 / 13`.
- Ben Helps with probability `1 / 8` and Leaves with probability `7 / 8`.