G601: Problem Set 3: Mixed Strategies, January 24, 1999

A. Problem 3.1. from Games and Information .

B. Three companies provide tires to the Australian market. The total cost curve for a firm making Q tires is TC = 5 + 20Q, and the demand equation is P = 100-N, where N is the total number of tires on the market.

(a) According to the Cournot model, what will the total industry output be?

(b) If M firms produce, rather than three, what will be the individual firm's output, total output, price, and profit?

C. Consider a Cournot duopoly in which two firms have constant marginal costs, but different ones. Firm 1's marginal cost is 3. Firm 2's marginal cost is either 3.25 (high) or 2.75 (low), and Firm 1 assesses equal probabilities of these. The demand curve is P = 4-Q.

An equilibrium for this game consists of output Q1 for firm 1 and outputs Q_L and Q_H for the low-cost and high-cost type firm 2. Find the equilibrium. (Hint: Start by finding the reaction curve for firm 2. )

D. Conjecture: If neither player has a weakly dominant strategy in a game, then the game has at least one mixed strategy equilibrium.

Show that this conjecture is false by finding a counterexample (or by any other proof). Hint: the counterexample I found has simultaneous moves, with three actions for Row and two for Column, and a unique pure-strategy Nash equilibrium.

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