Economic Theory Bulletin

, Volume 4, Issue 2, pp 125–136 | Cite as

Competing over a finite number of locations

Research Article

Abstract

We consider a Hotelling location game where retailers can choose one of a finite number of locations. Consumers have strict preferences over the possible available store locations and retailers aim to attract the maximum number of consumers. We prove that a pure strategy equilibrium exists if the number of retailers is large enough. Moreover, as the number of retailers grows large, in equilibrium the distribution of retailers over the locations converges to the distribution of consumers’ preferences.

Keywords

Hotelling games Pure equilibria Large games Political competition 

JEL Classification

C72 D72 R30 R39 

References

  1. Balder, E.J.: An equilibrium closure result for discontinuous games. Econ. Theory 48(1), 47–65 (2011). doi:10.1007/s00199-010-0574-6 CrossRefGoogle Scholar
  2. Barelli, P., Meneghel, I.: A note on the equilibrium existence problem in discontinuous games. Econometrica 81(2), 813–824 (2013). doi:10.3982/ECTA9125 CrossRefGoogle Scholar
  3. Carmona, G.: Understanding some recent existence results for discontinuous games. Econ. Theory 48(1), 31–45 (2011). doi:10.1007/s00199-010-0532-3 CrossRefGoogle Scholar
  4. de Castro, L.I.: Equilibrium existence and approximation of regular discontinuous games. Econ. Theory 48(1), 67–85 (2011). doi:10.1007/s00199-010-0580-8 CrossRefGoogle Scholar
  5. Debreu, G.: A social equilibrium existence theorem. Proc. Natl. Acad. Sci. USA 38, 886–893 (1952)CrossRefGoogle Scholar
  6. Downs, A.: An Economic Theory of Democracy. Harper and Row, New York (1957)Google Scholar
  7. Eaton, B.C., Lipsey, R.G.: The principle of minimum differentiation reconsidered: some new developments in the theory of spatial competition. Rev. Econ. Stud. 42(1), 27–49 (1975)CrossRefGoogle Scholar
  8. Feldmann, R., Mavronicolas, M., Monien, B.: Nash equilibria for Voronoi games on transitive graphs. In: Leonardi, S. (ed.) Internet and Network Economics, Lecture Notes in Computer Science, vol. 5929, pp. 280–291. Springer, Berlin, Heidelberg (2009). doi:10.1007/978-3-642-10841-9_26
  9. Fournier, G., Scarsini, M.: Hotelling games on networks: efficiency of equilibria (2014). doi:10.2139/ssrn.2423345
  10. Glicksberg, I.L.: A further generalization of the Kakutani fixed theorem, with application to Nash equilibrium points. Proc. Am. Math. Soc. 3, 170–174 (1952)Google Scholar
  11. He, W., Yannelis, N.C.: Discontinuous games with asymmetric information: an extension of Reny’s existence theorem. Mimeo (2014a)Google Scholar
  12. He, W., Yannelis, N.C.: Equilibria with discontinuous preferences. Mimeo (2014b)Google Scholar
  13. He, W., Yannelis, N.C.: Existence of walrasian equilibria with discontinuous, non-ordered, interdependent and price-dependent preferences. Mimeo (2014c)Google Scholar
  14. Heijnen, P., Soetevent, A.R.: Price competition on graphs. Technical Report TI 2014–131/VII, Tinbergen Institute (2014)Google Scholar
  15. Hotelling, H.: Stability in competition. Econ. J. 39(153), 41–57 (1929)CrossRefGoogle Scholar
  16. Jackson, M.O., Swinkels, J.M.: Existence of equilibrium in single and double private value auctions. Econometrica 73(1), 93–139 (2005). doi:10.1111/j.1468-0262.2005.00566.x CrossRefGoogle Scholar
  17. Kalai, E.: Large robust games. Econometrica 72(6), 1631–1665 (2004). doi:10.1111/j.1468-0262.2004.00549.x CrossRefGoogle Scholar
  18. Laster, D., Bennet, P., Geoum, I.: Rational bias in macroeconomic forecasts. Q. J. Econ. 45(2), 145–186 (1999). doi:10.1007/s00355-010-0495-0 Google Scholar
  19. Mavronicolas, M., Monien, B., Papadopoulou, V.G., Schoppmann, F.: Voronoi games on cycle graphs. In: Mathematical Foundations of Computer Science 2008, Lecture Notes in Computer Science, vol. 5162, pp. 503–514. Springer, Berlin (2008). doi:10.1007/978-3-540-85238-4_41
  20. McLennan, A., Monteiro, P.K., Tourky, R.: Games with discontinuous payoffs: a strengthening of Reny’s existence theorem. Econometrica 79(5), 1643–1664 (2011). doi:10.3982/ECTA8949 CrossRefGoogle Scholar
  21. Milgrom, P., Roberts, J.: Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica 58(6), 1255–1277 (1990). doi:10.2307/2938316 CrossRefGoogle Scholar
  22. Monderer, D., Shapley, L.S.: Potential games. Games Econ. Behav. 14(1), 124–143 (1996). doi:10.1006/game.1996.0044 CrossRefGoogle Scholar
  23. Osborne, M.J., Pitchik, C.: The nature of equilibrium in a location model. Int. Econ. Rev. 27(1), 223–237 (1986). doi:10.2307/2526617 CrossRefGoogle Scholar
  24. Ottaviani, M., Sorensen, P.N.: The strategy of professional forecasting. J. Fin. Econ. 81(2), 441–466 (2006). doi:10.1007/s00355-010-0495-0 CrossRefGoogle Scholar
  25. Pálvölgyi, D.: Hotelling on graphs. Mimeo (2011)Google Scholar
  26. Reny, P.J.: On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica 67(5), 1029–1056 (1999). doi:10.1111/1468-0262.00069 CrossRefGoogle Scholar
  27. Reny, P.J.: Strategic approximations of discontinuous games. Econ. Theory 48(1), 17–29 (2011). doi:10.1007/s00199-010-0518-1 CrossRefGoogle Scholar
  28. Rosenthal, R.W.: A class of games possessing pure-strategy Nash equilibria. Int. J. Game Theory 2, 65–67 (1973)CrossRefGoogle Scholar
  29. Salop, S.C.: Monopolistic competition with outside goods. Bell J. Econ. 10(1), 141–156 (1979)CrossRefGoogle Scholar
  30. Schmeidler, D.: Equilibrium points of nonatomic games. J. Stat. Phys. 7(4), 295–300 (1973). doi:10.1007/BF01014905 CrossRefGoogle Scholar
  31. Topkis, D.: Equilibrium points in nonzero-sum \(n\)-person submodular games. SIAM J. Control Optim. 17(6), 773–787 (1979). doi:10.1137/0317054 CrossRefGoogle Scholar

Copyright information

© Society for the Advancement of Economic Theory 2015

Authors and Affiliations

  1. 1.THEMA (UMR CNRS 8184)Université de Cergy-Pontoise, UFR d’Economie et GestionCergy-Pontoise CedexFrance
  2. 2.Dipartimento di Economia e FinanzaLUISSRomeItaly

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