# 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.