The theory of the firm and industry equilibrium

Martin J. Osborne
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7.2 Exercises on strategic games

  1. Find the Nash equilibria of the following strategic game.

    L R
    T 2,2 0,0
    B 0,0 1,1
    Solution
    (T,L)
    Neither player can increase its payoff by choosing a different action, so this action profile is a Nash equilibrium.
    (T,R)
    Player 1 can increase her payoff from 0 to 1 by choosing the action B rather than the action T. Thus this action profile is not a Nash equilibrium.
    (B,L)
    Firm 1 can increase its payoff from 0 to 2 by choosing the action T rather than the action B. Thus this action profile is not a Nash equilibrium.
    (B,R)
    Neither firm can increase its payoff by choosing a different action, so this action profile is a Nash equilibrium.

    We conclude that the game has two Nash equilibria, (T,L) and (B,R).

  2. 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(t1, t2) = −t1(t1 − t2 − 2)
    and country 2's payoff is
    u2(t1, t2) = −t2(t2 − t1 − 8).
    Find the Nash equilibria of the strategic game that models this situation.

    Solution

    • Find the countries' best response functions. To find the best response function of country 1, solve
      maxt1[−t1(t1 − t2 − 2)].
      We find b1(t2) = (1/2)(t2 + 2). Similarly the best response function of country 2 is b2(t1) = (1/2)(t1 + 8).
    • Solve for the Nash equilibria. A Nash equilibrium is a pair (t*1,t*2) such that t*1 = b1(t*2) and t*2 = b2(t*1). These two equations have a unique solution, (t1, t2) = (4, 6).
    We conclude that the game has a unique Nash equilibrium, (t*1, t*2) = (4, 6).