# Chapter 6 - Nash equilibria in mixed strategies

## Recap

In the previous chapter

- The definition of Nash equilibria;
- Identifying Nash equilibria in pure strategies;
- Solving the duopoly game;

This brings us to a very important part of the course. We will now consider equilibria in mixed strategies.

## Recall of expected utility calculation

In the matching pennies game discussed previously:

Recalling Chapter 2 a strategy profile of \(\sigma_1=(.2,.8)\) and \(\sigma_2=(.6,.4)\) implies that player 1 plays heads with probability .2 and player 2 plays heads with probability .6.

We can extend the utility function which maps from the set of pure strategies to \(\mathbb{R}\) using *expected payoffs*. For a two player game we have:

## Obtaining equilibria

Let us investigate the best response functions for the matching pennies game.

If we assume that player 2 plays a mixed strategy \(\sigma_2=(y,1-y)\) we have:

and

shown:

- If \(y<1/2\) then \(r_2\) is a best response for player 1.
- If \(y>1/2\) then \(r_1\) is a best response for player 1.
- If \(y=1/2\) then player 1 is indifferent.

If we assume that player 1 plays a mixed strategy \(\sigma_1=(x,1-x)\) we have:

and

shown:

Thus we have:

- If \(x<1/2\) then \(c_1\) is a best response for player 2.
- If \(x>1/2\) then \(c_2\) is a best response for player 2.
- If \(x=1/2\) then player 2 is indifferent.

Let us draw both best responses on a single diagram, indicating the best responses in each quadrant. The arrows show the deviation indicated by the best responses.

If either player plays a mixed strategy other than \((1/2,1/2)\) then the other player has an incentive to modify their strategy. Thus the Nash equilibria is:

This notion of “indifference” is important and we will now prove an important theorem that will prove useful when calculating Nash Equilibria.

## Equality of payoffs theorem

### Definition of the support of a strategy

In an \(N\) player normal form game the **support** of a strategy \(\sigma\in\Delta S_i\) is defined as:

I.e. the support of a strategy is the set of pure strategies that are played with non zero probability.

For example, if the strategy set is \({A,B,C}\) and \(\sigma=(1/3,2/3,0)\) then \(\mathcal{S}(\sigma)={A,B}\).

### Theorem of equality of payoffs

In an \(N\) player normal form game if the strategy profile \((\sigma_i,s_{-i})\) is a Nash equilibria then:

### Proof

If \(|\mathcal{S}(\sigma_i)|=1\) then the proof is trivial.

We assume that \(|\mathcal{S}(\sigma_i)|>1\). Let us assume that the theorem is not true so that there exists \(\bar s\in\mathcal{S}(\sigma)\) such that

Without loss of generality let us assume that:

Thus we have:

Giving:

which implies that \((\sigma_i,s_{-i})\) is not a Nash equilibrium.

### Example

Let’s consider the matching pennies game yet again. To use the equality of payoffs theorem we identify the various supports we need to try out. As this is a \(2\times 2\) game we can take \(\sigma_1=(x,1-x)\) and \(\sigma_2=(y,1-y)\) and assume that \((\sigma_1,\sigma_2)\) is a Nash equilibrium.

from the theorem we have that \(u_1(\sigma_1,\sigma_2)=u_1(r_1,\sigma_2)=u_1(r_2,\sigma_2)\)

Thus we have found player 2’s Nash equilibrium strategy by finding the strategy that makes player 1 indifferent. Similarly for player 1:

Thus the Nash equilibria is:

To finish this chapter we state a famous result in game theory:

### Nash’s Theorem

Every normal form game with a finite number of pure strategies for each player, has at least one Nash equilibrium.