Question

Problem 5. (20 points) There are two competing firms. Each firm decides when to exit the market: (i) immediately, (ii) after 6 months, or (iii) after 1 year. If a firm decides to exit the market, it does not get any payoff from that point on (that is, its utility stays the same). After every 6 months firms gain or lose utilities as follow: if both firms are still in the market, they both lose -1 and if only one firm is in the market, that firm gains +2. a. Write down the game in matrix form. b. Find the pure-strategy Nash equilibria. c. Find the unique mixed strategy Nash equilibrium. For simplicity, assume that exiting after 6 months will not be a part of a Nash equilibrium.

Answer 1

a)the game in matrix form when two firms have two decisions to make 1)exit and 2)not exit

FIRM1,FIRM2	EXIT(q)	NOT EXIT(1-q)
EXIT(r)	0,0	0,2
NOT EXIT(1-r)	2,0	-1,-1

b)the nash equillibria will be (2,0) or (0,2) because both of these are best strategy by the firm.

when firm 2 is fixed at choosing to EXIT the best payoff for firm1 is 2(not exiting) when firm2 is fixed at choosing to not exit the best payoff for firm1 is 0(exit).Now when firm1 is fixed to exit the best payoff for firm2 is 2(not exit) and when firm1 is fixed to not exiting the best payoff for firm2 is 0(exit) so there are two equilibrium points.

c)to determine mixed strategy nash equilibrium,

STRATEGY1 : 0q+0(1-q) STRATEGY2 : 2q+(-1)(1-q)

0q+0(1-q) = 2q+(-1)(1-q)

by solving we get, q=1/3 (There is no pure strategy nash equilibrium)

similarly r=1/3

(r,q)=(1/3,1/3)