# Chapter 4 - Best responses

## Recap

In the previous chapter we discussed:

- Predicting rational behaviour using dominated strategies;
- The CKR;

We did discover certain games that did not have any dominated strategies.

## Best response functions

### Definition of a best response

In an \(N\) player normal form game. A strategy \(s^*\) for player \(i\) is a best response to some strategy profile \(s_{-i}\) if and only if \(u_i(s^*,s_{-i})\geq u_{i}(s,s_{-i})\) for all \(s\in S_i\).

We can now start to predict rational outcomes in pure strategies by identifying all best responses to a strategy.

We will underline the best responses for each strategy giving (\(r_i\) is underlined if it is a best response to \(s_j\) and vice versa):

We see that \((r_3,c_3)\) represented a pair of best responses. What can we say about the long term behaviour of this game?

## Best responses against mixed strategies

We can identify best responses against mixed strategies. Let us take a look at the matching pennies game:

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

and

We see that:

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

Let us repeat this exercise for the battle of the sexes game.

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

and

We see that:

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

## Connection between best responses and dominance

### Definition of the undominated strategy set

In an \(N\) player normal form game, let us define the undominated strategy set \(UD_i\):

If we consider the following game:

We have:

### Definition of the best responses strategy set

In an \(N\) player normal form game, let us define the best responses strategy set \(B_i\):

In other words \(B_i\) is the set of functions that are best responses to some strategy profile in \(S_{-i}\).

Let us try to identify \(B_2\) for the above game. Let us assume that player 1 plays \(\sigma_1=(x,1-x)\). This gives:

Plotted:

We see that \(c_3\) is never a best response for player 2:

We will now attempt to identify \(B_1\) for the above game. Let us assume that player two plays \(\sigma_2=(x,y,1-x-y)\). This gives:

If we can find values of \(y\) that give valid \(\sigma_2=(x,y,1-x-y)\) and that make the above difference both positive and negative then:

\(y=1\) gives \(u_1(r_1,\sigma_2)-u_2(r_2,\sigma_2)=1>0\) (thus \(r_1\) is best response to \(\sigma_2=(0,1,0)\)). Similarly, \(y=0\) gives \(u_1(r_1,\sigma_2)-u_2(r_2,\sigma_2)=-2<0\) (thus \(r_2\) is best response to \(\sigma_2=(x,0,1-x)\) for any \(0\leq x \leq 1\)) as required.

We have seen in our example that \(B_i=UD_i\). This leads us to two Theorems (the proofs are omitted).

### Theorem of equality in 2 player games

In a 2 player normal form game \(B_i=UD_i\) for all \(i\in{1,2}\).

This is however not always the case:

### Theorem of inclusion in \(N\) player games

In an \(N\) player normal form game \(B_i\subseteq UD_i\) for all \(1 \leq i\leq n\).