Abstract
This paper addresses a location-price problem on a transportation network. We suppose that the competing firms select their facility locations, and then they compete on delivered prices with the aim of profit maximization. The firms sell an homogeneous product and the customers buy from the one that offers the lowest price. Under some general conditions, for any locations of the facilities, the existence of a unique price equilibrium is shown. Then the location price problem is reduced to a location game if the competing firms set the equilibrium prices. The aim of this paper is to study this location game for any number of competing firms which locate multiple facilities. For essential products, it is proved that the global minimizers of the social cost are location Nash equilibria. In particular, there exists at least one global minimizer of social cost at the nodes of the network if marginal delivered costs are concave. In this case, an Integer Linear Programming (ILP) formulation is proposed to minimize the social cost. For non essential products, the minimizers of social cost may not be location Nash equilibria. Then a best response algorithm is proposed to find a location Nash equilibrium. If marginal delivered costs are concave, an ILP formulation is given for profit maximization of one firm, assuming that the locations of its competitors are fixed. Finally, the selection of a location Nash equilibrium when there are multiple location Nash equilibria is discussed.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abellanas, M., Lillo, I., López, M.D., Rodrigo, J.: Electoral strategies in a dynamical democratic system. Geometric models. Eur. J. Oper. Res. 175 (2), 870–878 (2006)
Avella, P., Sassano, A., Vasilev, I.: Computational study of large-scale p-Median problems. Math. Program. Ser. A 109, 89–114 (2007)
Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, New York (2006)
De Palma, A., Ginsburgh, V., Thisse, J.F.: On existence of location equilibria in the 3-firm Hotelling problem. J. Ind. Econ. 36, 245–252 (1987)
Díaz-Bañez, J.M., Heredia, M., Pelegrín, B., Pérez-Lantero, P., Ventura, I.: Finding all pure strategy Nash Equilibria in a planar location game. Eur. J. Oper. Res. 214, 91–98 (2011)
Dorta-González, P., Santos-Peñate, D.R., Suárez-Vega, R.: Spatial competition in networks under delivered pricing. Pap. Reg. Sci. 84, 271–280 (2005)
D\(\ddot{\mathrm{u}}\) rr, C., Thang, N.K.: Nash equilibria in Voronoi games on graphs. Lect. Notes Comput. Sci 4698, 17–28 (2007)
Eichberger, J.: Game Theory for Economics. Academic, New York (1993)
Eiselt, H.A.: Hotelling’s duopoly on a tree. Ann. Oper. Res. 40, 195–207 (1992)
Eiselt, H.A., Laporte, G.: Locational equilibrium of two facilities on a tree. Recherche Operationnelle 25, 5–18 (1991)
Eiselt, H.A., Laporte, G.: The existence of equilibria in the 3-facility Hotelling model in a tree. Transp. Sci. 27, 39–43 (1993)
Fernández, J., Salhi, S., Toth, B.G.: Location equilibria for a continuous competitive facility location problem under delivered pricing. Comput. Oper. Res. 4, 185–195 (2014)
Gabszewicz, J.J., Thisse, J.F.: Location. In: Aumann, R., Hart, S. (eds.) Handbook of Game Theory with Economic Applications, pp. 281–304. Elsevier Science, New York (1992)
García, M.D., Fernández, P., Pelegrín, B.: On price competition in location-price models with spatially separated markets. TOP 12, 351–374 (2004)
García, M.D., Pelegrín, B., Fernández, P.: Location strategy for a firm under competitive delivered prices. Ann. Reg. Sci. 47, 1–23 (2011)
Gupta, P.: Competitive spatial price discrimination with strictly convex production costs. Reg. Sci. Urban Econ. 24, 265–272 (1994)
Hakimi, S.L.: Locations with spatial interactions: competitive locations and games. In: Francis, R.L., Mirchandani, P.B. (eds.) Discrete Location Theory. Wiley, New York (1990)
Hamilton, J.H., Thisse, J.F., Weskamp, A.: Spatial discrimination, Bertrand vs. Cournot in a model of location choice. Reg. Sci. Urban Econ. 19, 87–102 (1989)
Hoover, E.M.: Spatial price discrimination. Rev. Econ. Stud. 4, 182–191 (1936)
Labbe, M., Hakimi, S.L.: Market and location equilibrium for two competitors. Oper. Res. 39, 749–756 (1991)
Lederer, P.J., Hurter, A.P.: Competition of chains, discriminatory pricing and location. Econometrica 54, 623–640 (1986)
Lederer, P.J., Thisse, J.F.: Competitive location on networks under delivered pricing. Oper. Res. Lett. 9, 147–153 (1990)
Marianov, V., Serra, D.: Median problems in networks. In: Eiselt, H.A., Marianov, V. (eds.) Foundations of Location Analysis, pp. 39–59. Springer, Berlin (2011)
Osborne, M.J., Pitchik, C.: Equilibrium in Hotelling’s model of spatial competition. Econometrica 55, 911–922 (1987)
Pelegrín, B., Dorta, P., Fernández, P.: Finding location equilibria for competing firms under delivered pricing. J. Oper. Res. Soc. 62, 729–741 (2011)
Plastria, F., Vanhaverbeke, L.: Maximal covering location problem with price decision for revenue maximization in a competitive environment. OR Spectr. 31, 555–571 (2009)
Sarkar, J., Gupta, B., Pal, D.: Location equilibrium for Cournot oligopoly in spatially separated markets. J. Reg. Sci. 2, 195–212 (1997)
Serra, D., ReVelle, C.S.: Competitive location and pricing on networks. Geogr. Anal. 31, 109–129 (1999)
Acknowledgements
This research has been supported by the Ministry of Economy and Competitiveness of Spain under the research project MTM2015-70260-P, and the Fundación Séneca (The Agency of Science and Technology of the Region of Murcia) under the research project 19241/PI/14.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Pelegrı́n, B., Fernández, P., Garcı́a, M.D. (2017). Nash Equilibria in Network Facility Location Under Delivered Prices. In: Mallozzi, L., D'Amato, E., Pardalos, P. (eds) Spatial Interaction Models . Springer Optimization and Its Applications, vol 118. Springer, Cham. https://doi.org/10.1007/978-3-319-52654-6_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-52654-6_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-52653-9
Online ISBN: 978-3-319-52654-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)