Methods for computing Nash equilibria of a location-quantity game

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