Production, Manufacturing and Logistics On competitive sequential location in a network with a decreasing demand intensity Daniel Granot a , Frieda Granot a , Tal Raviv b, * a Sauder School of Business, University of British Columbia Vancouver, BC, Canada b Industrial Engineering Department, Tel Aviv University, Tel Aviv 69978, Israel article info Article history: Received 10 November 2007 Accepted 21 December 2009 Available online 4 January 2010 Keywords: Location Game theory abstract We introduce and analyze a Hotelling like game wherein players can locate in a city, at a fixed cost, according to an exogenously given order. Demand intensity is assumed to be strictly decreasing in dis- tance and players locate in the city as long as it is profitable for them to do so. For a linear city (i) we explicitly determine the number of players who will locate in equilibrium, (ii) we fully characterize and compute the unique family of equilibrium locations, and (iii) we show that players’ equilibrium expected profits decline in their position in the order. Our results are then extended to a city represented by an undirected weighted graph whose edge lengths are not too small and co-location on nodes of the graph is not permitted. Further, we compare the equilibrium outcomes with the optimal policy of a monopolist who faces an identical problem and who needs to decide upon the number of stores to open and their locations in the city so as to maximize total profit. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction We introduce and analyze a Hotelling like game wherein play- ers can locate in a city, at a fixed cost, according to an exogenously given order. Demand intensity is assumed to be strictly decreasing in distance and players locate in the city as long as it is profitable for them to do so. Competitive location analysis in economics, and perhaps in gen- eral, has started with the seminal work of Hotelling (1929), who has introduced and analyzed a classical location and pricing game which is referred to as the Hotelling’s Game. Hotelling’s model con- sists of a linear city, represented by the [0, 1] segment, wherein consumers are spread uniformly. Two competing firms set their locations and sale prices. Each consumer in the city would pur- chase one unit of the identical product sold by the two competing firms at a cost which consists of the sale price and the transporta- tion cost. The transportation cost is proportional to the distance between the consumer and the firm, and each consumer patronizes the firm that offers her the lowest total cost. For fixed locations of the firms, Hotelling has characterized the unique Nash Equilibrium in prices. He has then claimed that if, for any given location of the firms, prices are determined by the corre- sponding Nash equilibrium, then the two firms will end up locating as close as possible to each other at about the midpoint of the city. However D’Aspermont et al. (1979) have shown that this part of Hotelling analysis is wrong and that there is no Nash equilibrium in pure strategies for this price-location game. For more related lit- erature see, e.g. Economides (1986), Anderson (1988) and Brenner (2005). Eaton and Lipsey (1975) introduced several important extensions of Hotelling game, including relaxing the assumption of even spread of customers over the market and the two dimen- sional topology. Another paradigm for competitive locations, wherein players make their decisions sequentially rather than simultaneously as in Hotelling’s model, was introduced by Teitz (1968) and Prescott and Visscher (1977). Teitz (1968) has considered a location game where a leader initially locates p stores in a linear city and a fol- lower subsequently locates r 6 p stores, and has shown that the pattern of the locations of the leader coincide with the social opti- mum solution. An important contribution in the same stream of re- search is due to Hakimi (1983), who has considered a location problem in a city represented by a network. Specifically, he has for- mally introduced the ðrjpÞ-medianoid and ðrjpÞ-centroid problems in a network, wherein consumers are assumed to be located only at the nodes. In the ðrjpÞ-medianoid, the follower is seeking best loca- tions for r facilities in a network on which p facilities of a compet- itor are already located. In the ðrjpÞ-centroid problem, the leader is seeking locations for p facilities that will minimize the profit of a follower who subsequently will locate r facilities in the network. Some important computational results regarding the medianoid and centroid problems in various spaces (i.e., plane, tree or net- work) are due, e.g., to Drezner (1981, 1982), Megiddo et al. (1983), Hakimi (1983), and Hansen and Labbé (1988). Berman and Krass (2002) have considered the problem of locating m new facilities by a firm which is already present in the market and com- petes with other firms. The new facilities will not only compete with previously located facilities by other firms, but may also 0377-2217/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.12.021 * Corresponding author. Tel.: +972 3 6406977; fax: +972 3 6407669. E-mail address: talraviv@eng.tau.ac.il (T. Raviv). European Journal of Operational Research 205 (2010) 301–312 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor