## Examples and exercises on finding Nash equilibria of two-player games using best response functions

### Procedure

- Find each player's best response function by finding the action that maximizes its payoff for any given action of the other player. Denote the best response function of player
*i* by *b*_{i}.
- Find the pair (
*a*_{1}, *a*_{2}) of actions with the property that player 1's action is a best response to player 2's action, and player 2's action is a best response to player 1's action: *a*_{1} =
*b*_{1}(*a*_{2}) and *a*_{2} = *b*_{2}(*a*_{1}).

### Example

Consider the strategic game in which
- the players are two firms
- each player can choose its amount of advertising (any nonnegative number)
- if firm 1 chooses the amount
*a*_{1} of advertising and firm 2 chooses the amount *a*_{2} of advertising then the payoff (profit) of firm 1 is
*a*_{1}(*c* + *a*_{2} *a*_{1})

and the payoff (profit) of firm 2 is
*u*_{2}(*a*_{1}, *a*_{2}) = *a*_{2}(*c* + *a*_{1} *a*_{2}),

where *c* is a positive constant.

What are the Nash equilibria?

- Find the firms' best response functions. To find the best response of firm 1 to any action
*a*_{2} of firm 2, fix *a*_{2} and solve
max_{a}_{1}*a*_{1}(*c* + *a*_{2} *a*_{1}).

The derivative is *c* + *a*_{2} 2*a*_{1}, so the maximizer is *a*_{1} = (*c* + *a*_{2})/2. Thus firm 1's best response function is given by
*b*_{1}(*a*_{2}) = (*c* + *a*_{2})/2.

Similarly, firm 2's best response function is given by
*b*_{2}(*a*_{1}) = (*c* + *a*_{1})/2.

- A Nash equilibrium is a pair (
*a*_{1}*,*a*_{2}*) such that *a*_{1}* = *b*_{1}(*a*_{2}*) and *a*_{2}* =
*b*_{2}(*a*_{1}*). Thus a Nash equilibrium is a solution of the equations
*a*_{1}* = (*c* + *a*_{2}*)/2

*a*_{2}* = (*c* + *a*_{1}*)/2.

Substituting the second equation in the first equation, we get (*a*_{1}*,*a*_{2}*) = (*c*,*c*).

We conclude that the game has a unique Nash equilibrium, in which each firm's amount of advertising is *c*.

Each of two countries chooses a tariff rate to impose on imports. If country 1 chooses the rate *t*_{1} and country 2 chooses the rate *t*_{2} then country 1's payoff is
*u*_{1}(*t*_{1}, *t*_{2}) = *t*_{1}(*t*_{1} *t*_{2} 2)

and country 2's payoff is
*u*_{2}(*t*_{1}, *t*_{2}) = *t*_{2}(*t*_{2} *t*_{1} 8).

Find the Nash equilibria of the strategic game that models this situation.
[Solution]

Copyright © 1997 by Martin J. Osborne