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 | 10, 5 | -6, 3 | |
Leave | 0, 6 | 7, 10 |
What is the Mixed Strategy Nash Equilibrium.
Anna Helps with probability `2 / 3`. Ben Helps with probability `7 / 23`.
Anna's mixed strategy
Anna Helps with probability `p` so that Ben is indifferent between Helping and Leaving:
$$ \begin{align*} 5 \times p + 6 \times (1 - p) &= 3 \times p + 10 \times (1 - p) \\ -1 p + 6 &= -7 p + 10 \\ 6 p &= 4 \\ p &= \frac{4}{6} \end{align*} $$
Ben's mixed strategy
Ben Helps with probability `q` so that Anna is indifferent between Helping and Leaving:
$$ \begin{align*} 10 \times q + 0 \times (1 - q) &= -6 \times q + 7 \times (1 - q) \\ 10 q + 0 &= -13 q + 7 \\ 23 q &= 7 \\ q &= \frac{7}{23} \end{align*} $$Conclusion
In equilibrium:
- Anna Helps with probability `2 / 3` and Leaves with probability `1 / 3`.
- Ben Helps with probability `7 / 23` and Leaves with probability `16 / 23`.