Game Theory with Applications to Finance and Marketing

whereq2 ∗ = arg max q 2 (3 − E[qq 1 ∗ ] − q 2 − 1),2E[q ∗ 1 ] = πq∗ 1 (1) + (1 − π)q∗ 1 (0).The (necessary and sufficient) first-order conditions of these three maximizationsgive three equations with three unknowns. Solving, we haveq ∗ 1 (1) = 5 − π6, q ∗ 1 (0) = 8 − π6, q2 ∗ = 1 + π .3Let Y i (c; π) be the equilibrium profit of firm i before subtracting theexpenditure on F , given that firm 1’s marginal cost is c and firm 2’sbeliefs are such that firm 1 spends F with probability 1 − π. Then, onecan show thatY 1 (1; π) = [ 5 − π6] 2 , Y 1 (0; π) = [ 8 − π ] 2 .6Suppose that there exist pure strategy NE’s. In a pure strategy NE,firm 1 either spends F with prob. 1 or does not spend F with prob. 1.Suppose first that there is a pure strategy NE in which firm 1 invests.Then this is expected correctly by firm 2, and hence π = 0. From thepreceding discussion, this requires, for firm 1 to follow its equilibriumstrategy,Y 1 (0; 0) − F ≥ Y 1 (1; 0),or F ≤ 3936 .Next, suppose instead that there is a pure strategy NE in which firm1 does not invest. The earlier discussion implies that π = 1 and inequilibrium it must be thatY 1 (0; 1) − F ≤ Y 1 (1; 1),or F ≥ 3336 .Finally, consider mixed strategy NE’s. For firm 1 to randomize betweento and not to invest with respectively prob. (1 − π) and π, 17Y 1 (0; π) − F = Y 1 (1; π),17 Note again that neither firm 1 nor firm 2 can randomize in the quantity-setting stage(why?), and hence a mixed strategy NE can at most involve firm 1 randomly makinginvestment decisions.33