Hotelling Games with Three, Four, and More Players*

Abstract

This paper extends the interval Hotelling model with quadratic transport costs to the n-player case. For a large set of locations including potential equilibrium configurations, we show for n > 2 that firms neither maximize differentiation-as in the duopoly model-nor minimize differentiation-as in the multi-firm game with linear transport cost. Subgame perfect equilibria for games with up to nine players are characterized by a U-shaped price structure and interior corner firm locations. Results are driven by an asymmetry between firms. Interior firms are weaker competitors than their rivals at the corners. Increasing the number of firms shifts even more power to the corner firms. As a result, there is too much differentiation from the social perspective if n 3, while adding firms leads to a level of differentiation in equilibrium below the social optimum.

Multiproduct Firms in Hotelling’s Spatial Competition

“…Such a U‐shape is also obtained in the case of five single‐outlet firms by Economides (1993) and Brenner (2005).…”

“…Suppose next there are three firms i = a , b , c , each of which establishes one outlet. Then, Brenner (2005) shows that there exists a unique SPNE locations given by . Observe that neither maximum differentiation nor minimum differentiation holds in the triopolistic market.…”

Sequential location in a discrete directional market with three or more players

“…Both Economides (1989) and Kats (1995) resemble such equal distance locations by allowing the location choices of firms in a circular market. On the other hand, Brenner (2005) discusses the equilibrium locations in a linear market with more than three players and achieves the result of unequal distance locations. directional market.…”

The equivalence of convex and concave transport cost in a circular spatial model with and without zoning

“…Unexpectedly, the convex quadratic transport cost function cannot deliver price equilibrium for any location of firms. Thus, the equilibrium properties associated with this function are not as robust as many researchers suggest, D 'Aspremont et al (1979), Anderson (1986Anderson ( ), (1988, Gabszewicz et al (1986), Lambertini (1994), Böckem (1994), Tabuchi et al (1995), Junichiro et al (2004) and Brenner (2005). Notably, the concave quadratic transport cost function yields perfect price-location equilibrium.…”