Question

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 or 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. Write down this game in matrix form. b. Are there any strictly dominated strategies? Are there any weakly dominated strategies? 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.) a.

Answer 1

Answer- Suppose there are two fimm that is firm 1 and firm 2, each firm has three possible choice as it must decide whether or not to (1) exit at the industry immediately (E). (ii) exit at the end of the quarter (T). (iii) exit at the end of the next quarter (N). a) Matrix forcy FIRM-2 T wt Z firm-IT E 0.0 2.0 4.0 0.2 -1,-1 1,-1 N. 0,4 -1,1 -2.2

C) The pure strategy nash equilibrium is (N.E) and (EN) are both. Because of T never best response to any one of the pure strategies. Thus, the pure strategy nash equilibrium we consider only N and E. Therefore, it is easy to say that (N.E) and (E.N) are both pure strategy nash equilibrium

d) If suppose fimml plays E and N with probability 12 each. This has a payoff 2.42,-1 against the pure strategies E.T. N, but the pure strategy T has payoffs 2, -1, -1 against the pure strategiesE, T, N respectively. So, T is weakly dominated by the mixed strategy (1/2.0.1/2). Therefore there are mixed strategy NE in which neither player puts any probability on strategy T. Now let p be probability that Firm 1 exits immediately (E), and q is probability stays until next term (N). Since T is weakly dominated, we have q= 1- p. Now the expected utility to Fimm 2 of choosing E is zero and the expected utility for choosing N is 4p - 2q. The expected utilities of E and N to Fim 2 should match in Nash equilibrium if it places positive probability on both strategies. Thus, we have 0=4p - 2q = 4p-2 (1-p) = 6p - 2 Which means p= 1/3 and q= 2/3. Thus, the mixed strategy Nash equilibrium for Firm-1 (1/3.0.2/3) and for fimm-2 also plays mixed strategy (1/3, 0,2/3).