On Solutions to Cournot-Nash Equilibria Equations on the Sphere

TL;DR

This paper analyzes equations related to Cournot-Nash equilibria on the sphere, focusing on their mathematical properties and conditions for smooth solutions, using optimal transport and Monge-Ampère equations.

Contribution

It extends the analysis of Cournot-Nash equilibrium equations by connecting them to Monge-Ampère type PDEs with nonlocal terms and providing conditions for solution smoothness.

Findings

Solutions are smooth under certain conditions.

The equations relate to optimal transport problems with nonlocal terms.

Standard PDE techniques can be adapted to this context.

Abstract

We discuss equations associated to Cournot-Nash Equilibria as put forward recently by Blanchet and Carlier. These equations are related to an optimal transport problem in which the source measure is known, but the target measure is part of the problem. The resulting equation is a Monge-Amp\`ere type with possible nonlocal terms. If the cost function is of a particular form, the equation is vulnerable to standard optimal transportation PDE techniques, with some modifications to deal with the new terms. We give some conditions on the problem from which we can conclude that solutions are smooth.