Pareto efficiency as alternative to Nash equilibrium in competitive discrete location under delivered pricing

We address the facility location problem for firms that sell an homogeneous product and compete on delivered prices. If the firms set equilibrium prices at each market, the problem can be seen as a location game for which there exists a Nash equilibrium if demand is inelastic. However, this equilibrium may be Pareto-inefficient and the firms could decide to collude if they can guarantee themselves higher payoffs than those prescribed by the equilibrium. The aim of this paper is to study the joint profit maximization problem to obtain a collusive solution of the location game (Pareto-efficient) and determine if such a solution is also a Nash equilibrium. We develop Integer Linear Programming models to find Pareto-efficient solutions for multiple and single facility location. Nash equilibria are obtained by the best response procedure. An empirical investigation is performed to compare the joint profit of the firms and the profit of each one of the firms, got through the Pareto-efficient locations, with those got through the Nash equilibrium.