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, 3 | -10, 10 | |
Leave | 0, 9 | 3, -10 |
What is the Mixed Strategy Nash Equilibrium.
Anna Helps with probability `19 / 26`. Ben Helps with probability `1 / 2`.
Anna's mixed strategy
Anna Helps with probability `p` so that Ben is indifferent between Helping and Leaving:
$$ \begin{align*} 3 \times p + 9 \times (1 - p) &= 10 \times p -10 \times (1 - p) \\ -6 p + 9 &= 20 p -10 \\ -26 p &= -19 \\ p &= \frac{-19}{-26} \end{align*} $$
Ben's mixed strategy
Ben Helps with probability `q` so that Anna is indifferent between Helping and Leaving:
$$ \begin{align*} -7 \times q + 0 \times (1 - q) &= -10 \times q + 3 \times (1 - q) \\ -7 q + 0 &= -13 q + 3 \\ 6 q &= 3 \\ q &= \frac{3}{6} \end{align*} $$Conclusion
In equilibrium:
- Anna Helps with probability `19 / 26` and Leaves with probability `7 / 26`.
- Ben Helps with probability `1 / 2` and Leaves with probability `1 / 2`.