1. Spatial competition

Consider the Hoteling model of spatial competition. Suppose a continuum of consumers are uniformly distributed on the interval of real numbers [0, 1]. There is a homogeneous good whose price is fixed at 1. Each firm i competes by choosing their location zi [0, 1]. Each consumer buys one unit from the closest firm. If there is a tie in which firms are closest to a consumer, the consumer's demand is split equally. Assume each firm seeks to maximize its profit, and that there are no fixed or marginal costs for the firm.

Recall that in the case with two firms, numbered 1 and 2, the unique Nash equilibrium was for the two firms to locate at the median of the distribution, which for the uniform distribution was at z1 = z2 = 0.5. Suppose now there are three firms, numbered 1, 2, and 3.

(a) Explain why z1 = z2 = z3 = 0.5 is not a Nash equilibrium.

(b)Prove that there is no Nash equilibrium (in pure strategies).

(c) Suppose that firm 3 is an incumbent firm, whose location has already been fixed at z1 = 0.2. Find all pure strategy Nash equilibria for the location game between firms 1 and 2.

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