Differential variational inequalities and shipper dynamic oligopolistic network competition

In this paper, we show that the differential Cournot–Nash game describing dynamic oligopolistic network competition may be articulated as a differential variational inequality involving both control and state variables. We exploit this formulation to establish necessary conditions, an existence theorem and a nonconvex functional mathematical programming formulation. We show by example that, despite its nonconvexity, this mathematical program may be solved by the multi-start global optimization scheme found in the off-the-shelf software package GAMS when used in conjunction with the commercial solver MINOS. We also present a detailed numerical example. For that example, freight trip tables, formed ex post from the shipment patterns that are solutions to the dynamic oligopolistic network competition model we present, exhibit substantial temporal fluctuations. These observations are significant for they suggest that shippers must be extremely astute and capable of dramatically and rapidly altering production and distribution schedules if they are to compete in the final goods market successfully. If distribution services are provided by separate freight carriers, those carriers must be able to survive in a feast-or-famine environment.