Mixed Strategy Nash Equilibrium

In a Mixed Strategy Equilibrium, players chooses probabilities that makes the other players indifferent between a strategy or another.

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 -6, -1 0, 2
Leave -5, 2 -7, -1

What is the Mixed Strategy Nash Equilibrium.

Anna Helps with probability `1 / 2`. Ben Helps with probability `1 / 4`.

Anna's mixed strategy

Anna Helps with probability `p` so that Ben is indifferent between Helping and Leaving:

$$ \begin{align*} -1 \times p + 2 \times (1 - p) &= 2 \times p -1 \times (1 - p) \\ -3 p + 2 &= 3 p -1 \\ -6 p &= -3 \\ p &= \frac{-3}{-6} \end{align*} $$

Ben's mixed strategy

Ben Helps with probability `q` so that Anna is indifferent between Helping and Leaving:

$$ \begin{align*} -6 \times q -5 \times (1 - q) &= 0 \times q -7 \times (1 - q) \\ -1 q -5 &= 7 q -7 \\ -8 q &= -2 \\ q &= \frac{-2}{-8} \end{align*} $$

Conclusion

In equilibrium:

  • Anna Helps with probability `1 / 2` and Leaves with probability `1 / 2`.
  • Ben Helps with probability `1 / 4` and Leaves with probability `3 / 4`.