Hotelling competition with discrete consumers

Populating the Hotelling line with finitely many consumers produces a model of spatial competition that is challenging to analyse as it does not possess a Nash equilibrium in pure strategies. This paper characterises mixed-strategy Nash equilibrium (MSNE) for the model. The MSNE is unique, symmetric, and its support converges to the equilibrium of the standard Hotelling model—with a continuum of consumers—from above. The results suggest that the standard model is a reasonable approximation of the MSNE for a market with a large number of discrete buyers. By contrast, the intensity of competition is significantly lower when the market is home to a small numbers of buyers. In this case the MSNE expected prices and profits are well above the equilibrium prices and profits of the standard model.