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 4, -2 0, -6
Leave 9, -1 10, 0

What is the Mixed Strategy Nash Equilibrium.

Anna Helps with probability `1 / 5`. Ben Helps with probability `1 / 5`.

Anna's mixed strategy

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

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

Ben's mixed strategy

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

$$ \begin{align*} 4 \times q + 9 \times (1 - q) &= 0 \times q + 10 \times (1 - q) \\ -5 q + 9 &= -10 q + 10 \\ 5 q &= 1 \\ q &= \frac{1}{5} \end{align*} $$

Conclusion

In equilibrium:

  • Anna Helps with probability `1 / 5` and Leaves with probability `4 / 5`.
  • Ben Helps with probability `1 / 5` and Leaves with probability `4 / 5`.