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 | -1, 5 | -4, 3 | |
| Leave | -8, -5 | 4, 5 | |
What is the Mixed Strategy Nash Equilibrium.
Anna Helps with probability `5 / 6`. Ben Helps with probability `4 / 5`.
Anna's mixed strategy
Anna Helps with probability `p` so that Ben is indifferent between Helping and Leaving:
$$ \begin{align*} 5 \times p -5 \times (1 - p) &= 3 \times p + 5 \times (1 - p) \\ 10 p -5 &= -2 p + 5 \\ 12 p &= 10 \\ p &= \frac{10}{12} \end{align*} $$
Ben's mixed strategy
Ben Helps with probability `q` so that Anna is indifferent between Helping and Leaving:
$$ \begin{align*} -1 \times q -8 \times (1 - q) &= -4 \times q + 4 \times (1 - q) \\ 7 q -8 &= -8 q + 4 \\ 15 q &= 12 \\ q &= \frac{12}{15} \end{align*} $$Conclusion
In equilibrium:
- Anna Helps with probability `5 / 6` and Leaves with probability `1 / 6`.
- Ben Helps with probability `4 / 5` and Leaves with probability `1 / 5`.