Mixed Equilibrium in a Pure Location Game: The Case of n 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 firms is characterized as the solution of a well-behaved boundary value problem. The analysis suggests that, in contrast to the cases n=3 and n→∞, the equilibrium for a finite number of n�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. DOI: https://doi.org/10.1515/bejte-2014-0066 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-106959 Accepted Version Originally published at: Ewerhart, Christian (2015). Mixed equilibrium in a pure location game: the case of n�4 firms. The BE Journal in Theoretical Economics, 15(2):457-472. DOI: https://doi.org/10.1515/bejte-2014-0066 Mixed Equilibrium in a Pure Location Game: The Case of n 4 Firms* Christian Ewerhart** Abstract 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 firms is characterized as the solution of a well-behaved boundary value problem. The analysis suggests that, in contrast to the cases n = 3 and n ! ∞, the equilibrium for a finite number of n 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.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 firms is characterized as the solution of a well-behaved boundary value problem. The analysis suggests that, in contrast to the cases n = 3 and n ! ∞, the equilibrium for a finite number of n 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. JEL Codes C72; D43; D72; D82; L13 KeywordsHotelling game; mixed-strategy equilibrium; shootingmethod; strategic forecasting *) The problem studied below was kindly suggested to me by Marco Ottaviani. Useful comments on an earlier version were received by Burkhard Schipper and an anonymous referee. For useful discussions, I am grateful to Pavlo Blavatskyy and Jean Gabszewicz. **) University of Zurich, Department of Economics, Schönberggasse 1, CH-8001 Zürich, Switzerland. Phone +41-44-6343733. Email: [email protected].