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 1, -10 -9, -9
Leave -10, 4 6, 3

What is the Mixed Strategy Nash Equilibrium.

Anna Helps with probability `1 / 2`. Ben Helps with probability `8 / 13`.

Anna's mixed strategy

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

$$ \begin{align*} -10 \times p + 4 \times (1 - p) &= -9 \times p + 3 \times (1 - p) \\ -14 p + 4 &= -12 p + 3 \\ -2 p &= -1 \\ p &= \frac{-1}{-2} \end{align*} $$

Ben's mixed strategy

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

$$ \begin{align*} 1 \times q -10 \times (1 - q) &= -9 \times q + 6 \times (1 - q) \\ 11 q -10 &= -15 q + 6 \\ 26 q &= 16 \\ q &= \frac{16}{26} \end{align*} $$

Conclusion

In equilibrium:

  • Anna Helps with probability `1 / 2` and Leaves with probability `1 / 2`.
  • Ben Helps with probability `8 / 13` and Leaves with probability `5 / 13`.