Computable models of static and dynamic spatial oligopoly

Oligopolies are a fundamental economic market structure in which the number of competing firms is sufficiently small so that the profit of each firm is dependent upon the interaction of the strategies of all firms. There are alternative behavioral assumptions one may employ in forming a model of spatial oligopoly. In this chapter, we study the classical oligopoly problem based on Cournot’s theory. The Cournot-Nash solution of oligopoly models assumes that firms choose their strategy simultaneously and each firm maximizes their utility function while assuming their competitor’s strategy is fixed. We begin this chapter with the basic definition of Nash equilibrium and the formulation of static spatial and network oligopoly models as variational inequality (VI) which can be solved by several numerical methods that exist in the literature. We then move on to dynamic oligopoly network models and show that the differential Nash game describing dynamic oligopolistic network competition may be articulated as a differential variational inequality (DVI) involving both control and state variables. Finite-dimensional time discretization is employed to approximate the model as a mathematical program which 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 small-scale numerical example for a dynamic oligopolistic network.