A New Framework for Modelling Liquidity Pools as Mean Field Games

TL;DR

This paper applies mean field game theory to model liquidity pools in decentralized finance's AMMs, providing theoretical existence results and numerical validation of equilibrium properties.

Contribution

It introduces a novel MFG framework for AMMs, extending traditional price impact models to nonlinear reserve-based pricing, with proven existence of solutions and equilibria.

Findings

Established existence of MFG solutions for AMMs.

Validated equilibrium stability and convergence through numerical simulations.

Demonstrated sensitivity of the model to parameters and incentives.

Abstract

In this work, we present an application of the probabilistic weak formulation of mean field games (MFG) for modeling liquidity pools in a constant product automated market maker (AMM) protocol in the context of decentralized finance. Our work extends one of the most conventional applications of MFG, which is the price impact model in an order book, by incorporating an AMM instead of a traditional order book. The key structural difference is that in the AMM setting, the price is determined by the pool's reserves through a nonlinear mechanism, replacing the linear price-impact function used in classical models. Through our approach, we establish the existence of solutions to the Mean Field Game and, additionally, the existence of approximate Nash equilibria for the finite-player game. We complement the theoretical results with a comprehensive numerical study that validates the equilibrium structure: stability under perturbations, the $\varepsilon$-Nash property via unilateral deviations, finite-player convergence at propagation-of-chaos rates, and sensitivity to cost parameters and incentive targets. These results offer a new game-theoretic perspective for representing strategic behavior in AMM-based liquidity pools and open promising opportunities for future research in this emerging field.