A New Framework for Modelling Liquidity Pools as Mean Field Games

In this work, we present an innovative 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. Through our approach, we achieve results that support the existence of solutions to the Mean Field Game and, additionally, the existence of approximate Nash equilibria for the proposed problem. These results not only offer a new perspective for representing liquidity pools in AMMs but also open promising opportunities for future research in this emerging field.

💡 Research Summary

This paper introduces a novel application of the probabilistic weak formulation of mean‑field games (MFG) to the modeling of liquidity pools in constant‑product automated market makers (AMMs), a cornerstone of decentralized finance (DeFi). The authors start by motivating the need for a game‑theoretic framework that captures the strategic interaction of a large number of traders who trade against a shared pool rather than against each other, as is the case in traditional order‑book markets.

In the model, there are N traders each holding two assets, ETH and USDC. The ETH inventory of trader i follows the stochastic differential equation

 dXᵢₜ = αᵢₜ dt + σᵢ dWᵢₜ,

where αᵢₜ is the trader’s control (the trading rate) and σᵢ is a volatility parameter. The USDC inventory evolves according to

 dYᵢₜ = –(αᵢₜ Pₜ + cₚ(αᵢₜ)) dt,

with Pₜ the price of ETH in USDC units derived from the pool’s reserves. For a constant‑product AMM the reserves satisfy Xₜ Yₜ = k, so the price can be expressed as Pₜ = k / Xₜ². The pool’s ETH reserve Xₜ is the initial reserve X₀ minus the average cumulative trading of all agents, i.e.

 Xₜ = X₀ – ∫₀ᵗ (1/N)∑ⱼαⱼₛ ds.

Applying Itô’s lemma to the total portfolio value Vᵢₜ = Yᵢₜ + Xᵢₜ Pₜ yields a dynamics that contains a term proportional to the average trading rate of the whole population, thereby coupling each individual’s decision to the collective state of the pool.

Traders are assumed risk‑neutral and aim to maximize the expected terminal wealth minus running inventory costs h(t,Xᵢₜ) and a terminal penalty l(Xᵢ_T):

 Jᵢ(α) = E