A method to determine oligopolistic market equilibria based on a convex optimization formulation

An approach to determine oligopolistic market equilibria using a sequence of mathematical programming problems have recently been presented by Murphy, Sherali and Soyster. Harker has presented an alternative approach based on a variational inequality representation of the problem. In Harker’s paper the author applies a diagonalization/relaxation algorithm to the variational inequality formulation of the oligopolistic market equilibrium problem. This paper uses another method to solve an oligopolistic equilibrium problem in its variational inequality formulation. The method is based on the convex optimization formulation for variational inequalities used by Nguyen and Dupuis for the solution of assymmetric traffic assignment problems. As is the case for the general traffic equilibrium problem the oligopolistic market equilibrium problem can with the use of this method be solved by postoptimizing a linear subproblem in each iteration. However due to the simplicity of the constraint set an alternative search method which does not require any linear programming problem to be solved can be used. The method is simple and only elementary calculations are used.