A Competitive Network Design Problem with Pricing

Abstract

This paper presents a simple model of competition between transport firms that captures the interaction of system design, price setting, and consumer choice. Transport competition is modeled as a noncooperative game where firms first select network designs, then prices for transportation between any two nodes. The goal is to find a Nash equilibrium in prices and system designs for all competing firms. Competition is studied under two alternate assumptions about consumer choice: customers can bundle separately purchased legs and customers cannot. If bundling cannot occur, it is shown that unique Nash equilibrium prices exist and that each firm's profit can be written as the difference between two minimum cost flow problems. Sufficient conditions for the existence of equilibrium network designs are also developed. If bundling can occur, it is shown that a price equilibrium may not exist, and if it does, the price equilibrium may not be unique. Lack of existence or uniqueness implies that firm profit is not a well defined function of network designs. This shows that the network design problem with bundling is difficult. With bundling, some results are possible in the case of duopoly competition: an equilibrium in prices always exists but equilibrium prices may not be unique. However, when each firm chooses a network design to maximize the lower bound of its profit, the equilibrium network designs chosen are the same as those chosen when bundling is ignored.