Mixed Equilibrium in a Pure Location Game: The Case of n≥4$$n \ge 4$$ Firms

The Hotelling game of pure location allows interpretations in spatial competition, political theory, and strategic forecasting. In this paper, the doubly symmetric mixed-strategy equilibrium for n≥4$n \ge 4$ firms is characterized as the solution of a well-behaved boundary value problem. The analysis suggests that, in contrast to the cases n=3$n = 3$ and n→∞$n \to \infty $, the equilibrium for a finite number of n≥4$n \ge 4$ firms tends to overrepresent locations at the periphery of its support interval. Moreover, in the class of examples considered, an increase in the number of firms universally leads to a wider range of location choices and to a more dispersed distribution of individual locations.