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 10, -1 -1, -2
Leave -3, -6 3, 3

What is the Mixed Strategy Nash Equilibrium.

Anna Helps with probability `9 / 10`. Ben Helps with probability `6 / 17`.

Anna's mixed strategy

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

$$ \begin{align*} -1 \times p -6 \times (1 - p) &= -2 \times p + 3 \times (1 - p) \\ 5 p -6 &= -5 p + 3 \\ 10 p &= 9 \\ p &= \frac{9}{10} \end{align*} $$

Ben's mixed strategy

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

$$ \begin{align*} 10 \times q -3 \times (1 - q) &= -1 \times q + 3 \times (1 - q) \\ 13 q -3 &= -4 q + 3 \\ 17 q &= 6 \\ q &= \frac{6}{17} \end{align*} $$

Conclusion

In equilibrium:

  • Anna Helps with probability `9 / 10` and Leaves with probability `1 / 10`.
  • Ben Helps with probability `6 / 17` and Leaves with probability `11 / 17`.