Hotelling Competition With Discrete Consumers: Comparing Equilibrium Concepts in Spatial Competition

Abstract

Combining the Hotelling line with the realistic assumption that markets contain finitely many consumers, produces a model of spatial competition that does not possess a Nash equilibrium in pure strategies. This paper characterises both the ex-ante stable mixed strategy Nash equilibrium and the ex-post stable undercut-proof equilibrium for the model, and compares them with the equilibrium of the standard Hotelling model with a continuum of consumers. The mixed strategy equilibrium is unique and converges to the equilibrium of the standard model from above, indicating that the use of a continuum is a reasonable approximation of the mixed-strategy equilibrium for a market with a large number of buyers. By contrast, the undercut-proof equilibrium converges to half this level as marginal competition is more intense when the location of the point of indifference relative to the marginal consumers can be observed ex-post. Incorporating location choice into the model shows that the greater intensity of competition in the undercut-proof equilibrium, leads firms to locate closer to the centre of the line. These results provide a novel explanation for the tendency of product characteristics to converge in mature markets.