Market and locational equilibrium for two competitors

The authors consider a two-stage location and allocation game involving two competiting firms. The firms first select the location of their facility on a network. Then the firms optimally select the quantities each wishes to supply to the markets, which are located at the vertices of the network. The criterion for optimality for each firm is maximizing its profit, which is the total revenue minus the production and transportation costs. Under reasonable assumptions regarding the revenue, the production cost and the transportation cost functions, the authors show that there is a Nash equilibrium for the quantities offered at the markets by each firm. Furthermore, if the quantities supplied (at the equilibrium) by each firm at each market are positive, then there is also a Nash locational equilibrium, i.e., no firm finds it advantageous to change its location.