Algorithms for finding global and local equilibrium points of Nash-Cournot equilibrium models involving concave cost

TL;DR

This paper introduces algorithms to find global and local equilibrium points in Nash-Cournot models with separable concave costs, using convex envelope approximations and adaptive bisection for efficient solutions.

Contribution

The paper presents novel algorithms that efficiently compute both global and local equilibria in models with concave costs, extending beyond linear and convex cost frameworks.

Findings

Algorithms effectively find global equilibria for moderate-sized models.

Local equilibrium algorithms handle larger models efficiently.

Convex envelope approximation simplifies complex concave cost functions.

Abstract

We consider Nash-Cournot oligopolistic equilibrium models involving separable concave cost functions. In contrast to the models with linear and convex cost functions, in these models a local equilibrium point may not be a global one. We propose algorithms for finding global and local equilibrium points for the models having separable concave cost functions. The proposed algorithms use the convex envelope of a separable concave cost function over boxes to approximate a concave cost model with an affine cost one. The latter is equivalent to a strongly convex quadratic program that can be solved efficiently. To obtain better approximate solutions the algorithms use an adaptive rectangular bisection which is performed only in the space of concave variables Computational results on a lot number of randomly generated data show that the proposed algorithm for global equilibrium point are efficient for the models with moderate number of concave cost functions while the algorithm for local equilibrium point can solve efficiently the models with much larger size.