On the Existence of Unique Equilibria in Location Models

In this paper, we study a two-stage location-then-price game where consumers are distributed piecewise uniformly, each piece being referred to as an interval.Although the firms face a coordination problem, it is obvious that, for any given locations and prices, there is a unique indifferent consumer.So only the exact interval in which the indifferent consumer is located may be uncertain for the firms.Therefore, we encompass the firms with beliefs about the interval in which the indifferent consumer is located.Given their beliefs, the firms' expected profit functions are quasi-concave.We consider the situation where firms first choose beliefs and then maximize the corresponding expected profit in two stages as a psychological game.We show that there exists a unique psychological equilibrium for this game, which consists of a subgame perfect Nash equilibrium for the two-stage game given certain beliefs, and of beliefs such that the equilibrium outcome is consistent with these beliefs.This equilibrium outcome is found easily by applying a coordination argument.