## 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 bi.
• Find the pair (a1a2) 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: a1 = b1(a2) and a2 = b2(a1).

### 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 a1 of advertising and firm 2 chooses the amount a2 of advertising then the payoff (profit) of firm 1 is
a1(c + a2  a1)
and the payoff (profit) of firm 2 is
u2(a1a2) = a2(c + a1  a2),
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 a2 of firm 2, fix a2 and solve
maxa1a1(c + a2  a1).
The derivative is c + a2  2a1, so the maximizer is a1 = (c + a2)/2. Thus firm 1's best response function is given by
b1(a2) = (c + a2)/2.
Similarly, firm 2's best response function is given by
b2(a1) = (c + a1)/2.
• A Nash equilibrium is a pair (a1*,a2*) such that a1* = b1(a2*) and a2* = b2(a1*). Thus a Nash equilibrium is a solution of the equations
a1* = (c + a2*)/2
a2* = (c + a1*)/2.
Substituting the second equation in the first equation, we get (a1*,a2*) = (c,c).

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

### Exercise

Each of two countries chooses a tariff rate to impose on imports. If country 1 chooses the rate t1 and country 2 chooses the rate t2 then country 1's payoff is
u1(t1t2) = t1(t1  t2  2)
and country 2's payoff is
u2(t1t2) = t2(t2  t1  8).
Find the Nash equilibria of the strategic game that models this situation.

[Solution]