Computers & Operations Research 35 (2008) 3311 – 3330 www.elsevier.com/locate/cor Methods for computing Nash equilibria of a location–quantity game M. Elena Sáiz ∗ , Eligius M.T. Hendrix Wageningen Universiteit. Hollandseweg 1, 6706 KN Wageningen, The Netherlands Available online 4 March 2007 Abstract A two-stage model is described where firms take decisions on where to locate their facility and on how much to supply to which market. In such models in literature, typically the market price reacts linearly on supply. Often two competing suppliers are assumed or several that are homogeneous, i.e., their cost structure is assumed to be identical. The focus of this paper is on developing methods to compute equilibria of the model where more than two suppliers are competing that each have their own cost structure, i.e., they are heterogeneous. Analytical results are presented with respect to optimality conditions for the Nash equilibria in the two stages. Based on these analytical results, an enumeration algorithm and a local search algorithm are developed to find equilibria. Numerical cases are used to illustrate the results and the viability of the algorithms. The methods find an improvement of a result reported in literature.  2007 Elsevier Ltd. All rights reserved. Keywords: Iterative algorithms; Networks; Discrete location; Spatial models; Competition; Oligopoly; n-person games n> 2 1. Introduction Many studies in literature describe a so-called non-cooperative game where competing firms decide on production locations and supply quantities to markets. To make a game theoretic analysis tractable, often a limited number of suppliers are considered, or alternatively homogeneous firms and markets are assumed. We focus on situations where companies can be as well similar as not similar. In supply chains, farm cooperatives, etc., many decisions appear in which preferences cannot be assumed to be homogeneous. Also symmetric behaviour, finite strategy set or a two or few actors setting are strong assumptions in literature. Decisions are influenced by differences on prices or cost (“player” depending) between actors and between the location of the facilities. Our focus is on constructing solution methods for games in which players are: asymmetric, heterogeneous and facing multiple decisions in several stages. Different competitive location models are available in the literature, see for instance the survey papers [1–3] and the references therein. They vary in the ingredients which form the model. For instance, the location space may be the plane, a network or a discrete set. In [4] the idea of a Cournot oligopoly equilibrium was introduced, where two firms compete on the same market. Due to price reaction of the market on the total quantity offered, a price equilibrium appears. Hotelling [5] added the idea of having a freedom in choice of location, where the possible location area is a simple line in between the markets. A generally applicable concept is that of a Nash equilibrium [6] which is defined by the situation where none of the firms (players) is better off by changing its current (equilibrium) strategy. Because choice of location is usually prior ∗ Corresponding author. Tel.: +31 317485644; fax: +31 317485646. E-mail addresses: Elena.Saiz@wur.nl (M.E. Sáiz), Eligius.Hendrix@wur.nl (E.M.T. Hendrix). 0305-0548/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2007.02.022