Declining Industry: Consider two competing firms in a declining industry that cannot support both firms profitably. Each firm has three possible choices, as it must decide whether or not to exit the industry immediately, at the end of this quarter, or at the end of the next quarter. If a firm chooses to exit then its payoff is 0 from that point onward. Each quarter that both firms operate yields each a loss equal to -1, and each quarter that a firm operates alone yields it a payoff of 2. For example, if firm 1 plans to exit at the end of this quarter while firm 2 plans to exit at the end of the next quarter then the payoffs are (-1, 1) because both firms lose -1 in the first quarter and firm 2 gains 2 in the second. The payoff for each firm is the sum of its quarterly payoffs.
a. Write down this game in matrix form.
b. Are there any strictly dominated strategies? Are there any weakly dominated strategies?
c. Find the pure-strategy Nash equilibria.
d. Find the unique mixed-strategy Nash equilibrium. (Hint: you can use your answer to (b) to make things easier.)