A New Framework for Modelling Liquidity Pools as Mean Field Games

Abstract

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.

Figures (6)

• Figure 1: Monthly trading volume on DEXs.
• Figure 2: Mean field equilibrium of the baseline scenario ($\lambda=2$, $\eta=0.3$, $\rho=5$, $A=[-1.5, 1.5]$). Top panel: aggregate equilibrium control $q^*(t)$, strictly negative for all $t$, indicating sustained ETH selling. Middle panel: price trajectory $P(t)$ induced by $q^*$, strictly decreasing from $P_0 \approx 998$ to $P_T \approx 892$. Bottom panel: remaining liquidity $L(t) = X_0 - \int_0^t q^*(s)\,ds$, which remains bounded below ($L(t) \geq 10.01$), confirming no pool depletion.
• Figure 3: Mean field equilibrium with incentive to accumulate ETH ($x^* = 15 > x_0 = 10$, $\lambda=2$, $\eta=0.3$, $\rho=5$). Compare with Figure \ref{['fig:qstar_precio_baseline']}. Top panel:$q^*(t) > 0$ for all $t$, indicating sustained buying. Middle panel:$P(t)$ strictly increasing, from $P_0 \approx 1001$ to $P_T \approx 1059$. Bottom panel: remaining liquidity $L(t)$ decreasing, reflecting the extraction of ETH from the pool by traders.
• Figure 4: Local perturbation sensitivity at the computed fixed point control $q^\star$. We report the empirical slope $\gamma(\varepsilon)$ obtained by re-solving the best-response problem under a small perturbation of the mean-field input around $q^\star$. Smaller values indicate a more locally contractive response map and hence a more stable fixed point in practice.
• Figure 5: Unilateral deviation check at $q^\star$ (frozen-policy $\varepsilon$-Nash test). For each agent $i$, we keep all other agents on their baseline trajectories and re-optimize only agent $i$; the histogram shows $\Delta J_i := J_i(\text{deviate})-J_i(\text{baseline})$. Values concentrated near $0$ indicate no profitable unilateral deviations at the tested resolution.

Theorems & Definitions (19)

• Remark
• Remark
• Remark
• Remark 2.1
• Definition 2.2
• Corollary 3.1
• proof
• Corollary 3.2
• proof
• Corollary 3.3