[2021] Economics 482 Game Theory and Economics - Midterm Exam - Q3 Hotelling Game Nash Equilibrium
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3. Consider the following version of a Hotelling game of firm location. There are three firms who will choose locations on a line between 0 and 1. Assumptions about payoffs are the same as we have made previously: consumers are uniformly distributed on the line; any consumer buys at the location chosen by a firm that is closest to that consumer; firms who choose the same location split consumers who buy at that location equally; and a firm’s payoff is the fraction of consumers to whom it sells.
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In this version of the problem, firm 3 can only choose from a restricted set of locations, while firms 1 and 2 can choose any location between 0 and 1, including 0 or 1. Specifically, firm 3 can only choose locations x3 ≥ 3/4.
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1. (a) Find a Nash equilibrium of the game in pure strategies (there is one). A full-credit answer must establish that each firm is best-responding in the action profile that you claim is a Nash equilibrium. (12 points)
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2. (b) In non-strategic problems of choice, if two people have the same preferences, but one of them has larger set of choices than the other, the one with more choices must be better off. Explain how the equilibrium of this game shows that this principle is not generally true in games. (8 points)
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