Nash Equilibria in Network Facility Location Under Delivered Prices

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.