Two firms, A and B, are in a market that is declining in size. The game starts in period 0, and the firms can compete in periods 0, 1, 2, 3, · · · (i.e. indefinitely) if they so choose. Duopoly profits (which are earned if both firms are in the market) in period t for firm A are equal to 101 ? 10t and they are 10.5 ? t for firm B. Monopoly profits (which are earned if a firm is the only one left in the market) are 500 ? 25t for firm A and 51 ? 2t for firm B.
Suppose that at the start of each period, each firm must decide to “stay in” or to “exit” if it is still active (the two firms do so simultaneously if both are still active). Once a firm exits, it is out of the market forever and earns zero in that period and each period thereafter. Firms’ payoffs are equal to the sum of their profits.
(a) How many information sets are there for each firm in period t? (Your answer will depend on t.)
(b) For periods t ? 26, what action must each firm play in any subgame perfect Nash equilibrium? Give reasons.
(c) For periods t ? 10, what action must each firm play in any subgame perfect Nash equilibrium? Give reasons.
(d) Find a sub game perfect Nash equilibrium for this game.