A note on Cournot-Nash equilibria and Optimal Transport between unequal dimensions

TL;DR

This paper explores Cournot-Nash equilibria in games with a continuum of players using optimal transport theory, especially focusing on transport between unequal dimensions, supported by numerical simulations.

Contribution

It introduces a framework connecting Cournot-Nash equilibria with optimal transport between unequal dimensions, providing new insights and computational approaches.

Findings

Establishes a link between equilibria and optimal transport in unequal dimensions

Develops numerical methods for the proposed models

Provides simulations demonstrating the theoretical results

Abstract

This note is devoted to study a class of games with a continuum of players for which Cournot-Nash equilibria can be obtained by minimisation of some cost related to Optimal Transport. In particular we focus on the case of an Optimal Transport term between unequal dimension. We also present several numerical simulations.