# Equilibrium in Hotelling's Model of Spatial Competition

https://doi.org/0012-9682(198707)55:4<911:EIHMOS>2.0.CO;2-X
p. 911-922

Carolyn Pitchik, Martin J. Osborne

We study Hotelling's two-stage model of spatial competition, in which two firms first simultaneously choose locations in the unit interval, then simultaneously choose prices. Under Hotelling's assumptions (uniform distribution of consumers, travel cost proportional to distance, inelastic demand of one unit by each consumer) the price-setting subgames possess equilibria in pure strategies for only a limited set of location pairs. Because of this problem (pointed out independently by Vickrey (1964) and d'Aspremont et al. (1979)), Hotelling's claim that there is an equilibrium of the two-stage game in which the firms locate close to each other is incorrect. A result of Dasgupta and Maskin (1986) guarantees that each price-setting subgame has an equilibrium in mixed strategies. We first study these mixed strategy equilibria. We are unable to provide a complete characterization of them, although we show that for a subset of location pairs all equilibria are of a certain type. We reduce the problem of finding an equilibrium of this type to that of solving three or fewer highly nonlinear equations. At each of a large number of location pairs we have computed approximate solutions to the system of equations. Next, we use our analytical results and computations to study the equilibrium location choices of the firms. There is a unique (up to symmetry) subgame perfect equilibrium in which the location choices of the firms are pure; in it, the firms locate 0.27 from the ends of the market. At this equilibrium, the support of the subgame equilibrium price strategy is the union of two short intervals. Most of the probability weight is in the upper interval, so that this strategy is reminiscent of occasional "sales" by the firms. We also find a subgame perfect equilibrium in which each firm uses a mixed strategy in locations. In fact, in the class of strategy pairs in which the firms use the same mixed strategy over locations, and this strategy is symmetric about .5, there is a single equilibrium. In this equilibrium most of the probability weight of the common strategy is between 0.2 and 0.4, and between 0.6 and 0.8. There is a wide range of pure Nash (as opposed to subgame perfect) equilibrium location pairs: the subgame strategies in which each firm threatens to charge a price of zero in response to a deviation support all but those location pairs in which the firms are very close.