Question

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 If the number of entrants, E, does not exceed the industry capacity,e, entrants 5. If the number of entrants exceeds capacity, then the e highest-rated entrants (ranked from 1 to E earn an equal share of the profit, v/E r earn v/E. The remaining (E players (ranked r > c) each suffer a loss of Formally, we can denote the strategy space ofplayer i e 1,2,N by S-E (0,1, where 0 denotes non-entry and 1 denotes entry. Throughout this problem, we assume that players maximize their expected monetary payoffs (a) 7 ptsl Explain why player i's expected payoff can be written as -0 (b) [7 ptsl 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. (c)7 pts Consider next some pure strategy profiles under which the number of entrants is at least as large as the market capacity Ec. 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: and that players whose strategy is to stay are only best responding if.

(d) [4 pts] Combining the two inequalities derived in (c) we have:

v/K + c ? 1 ? E ? v/K + c

Use Equation(1) to show that if N = 12, c = 4, v = 15 and K = 5, the number entrants in Nash equilibrium is either 6 or 7.

Answer 1

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