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 (a₁, a₂) 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

• 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(a₁, a₂) = 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

  max_a1a₁(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

u₁(t₁, t₂) = t₁(t₁  t₂  2)

u₂(t₁, t₂) = t₂(t₂  t₁  8).

[Solution]