1. Market Entry Game (50 points). For this problem, you will need to read the paper by Camerer and Lovallo (1999) posted on NYU Courses (but we consider a slightly simplified version in our theoretical analysis). The setup is as follows: 1. N players privately and simultaneously choose whether to enter a market with some known capacity, c, and profit, v. 2. The (N- E) players who stay out earn a payment of 0. 3. The E players who enter are randomly assigned an integer-valued ranking, r, ranging from 1 to E. 4. If the number of entrants, E, does not exceed the industry capacity, c, all entrants earn an equal share of the profit, v/E. 5. If the number of entrants exceeds capacity, then the c highest-rated entrants (ranked r c) earn u/E. The remaining (E-c) players (ranked r> c) each suffer a loss of K. Formally, we can denote the strategy space of player i {1,2,..., N) by S, = {0,1), where 0 denotes non-entry and I denotes entry. Throughout this problem, we assume that players maximize their expected monetary payoffs. (a) [7 pts] Explain why player i's expected payoff can be written as: 8 = 0 8 = 1, Ec -K () 8=1,E>c (b) [7 pts] Show that there are no pure-strategy Nash equilibrium in which the number of entrants is strictly smaller than c. [Hint: use the payoff functions from (a) to show that in such a scenario, there is always a profitable deviation for nonentrants.] (e) [7 pts] Consider next some pure strategy profiles under which the number of entrants is at least as large as the market capacity EZ c. In order for this to be a Nash equilibrium, all players must be best-responding to the other players' strategies. Show that in order for players who are entering to be best-responding, the following inequality must hold: Ev/K+c and that players whose strategy is to stay are only best responding if: E v/K+c-1