Dynamic Cournot-Nash Equilibrium: The Non-Potential Case

TL;DR

This paper investigates the existence, uniqueness, and convergence of dynamic Cournot-Nash equilibria in non-potential games with evolving types, extending previous potential game results to more general settings and illustrating with an optimal liquidation example.

Contribution

It extends the analysis of dynamic Cournot-Nash equilibria beyond potential games, providing new existence, uniqueness, and convergence results for non-potential cases.

Findings

Existence and uniqueness of equilibria under certain conditions.

Convergence of fixed-point iteration in quadratic cases.

Application to a toy model of optimal liquidation with price impact.

Abstract

We consider a large population dynamic game in discrete time where players are characterized by time-evolving types. It is a natural assumption that the players' actions cannot anticipate future values of their types. Such games go under the name of dynamic Cournot-Nash equilibria, and were first studied by Acciaio et al., as a time/information dependent version of the games devised by Blanchet and Carlier for the static situation, under an extra assumption that the game is of potential type. The latter means that the game can be reduced to the resolution of an auxiliary variational problem. In the present work we study dynamic Cournot-Nash equilibria in their natural generality, namely going beyond the potential case. As a first result, we derive existence and uniqueness of equilibria under suitable assumptions. Second, we study the convergence of the natural fixed-point iterations scheme in the quadratic case. Finally we illustrate the previously mentioned results in a toy model of optimal liquidation with price impact, which is a game of non-potential kind.