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A Dynamic Equilibrium Model for Automated Market Makers

Chengqi Zang, Zhenghui Wang, Weitong Zhang

Abstract

Automated Market Makers (AMMs) are a central component of decentralized exchanges, yet their equilibrium foundations and microeconomic mechanisms remain incompletely understood. This paper develops a dynamic equilibrium framework for Constant Function Market Makers (CFMMs) that formalizes the strategic interaction between arbitrageurs and liquidity providers (LPs) over time. We make three main contributions. First, we derive and empirically validate an intrinsic buy-sell asymmetry in CFMM price impact. Even in the absence of directional price movements, the geometric structure of constant product AMMs implies systematically different execution costs for buying and selling, a prediction that we confirm using on-chain transaction data. Second, we characterize the optimization problems of arbitrageurs and LPs in closed form, incorporating slippage and fees. In a baseline environment with only informed arbitrageurs, we show that providing liquidity is strictly dominated for LPs: arbitrage-driven price corrections generate negative jump returns that cannot be offset by fees, yielding a degenerate equilibrium with minimal liquidity provision. Third, motivated by empirical evidence, we extend the model to include agent heterogeneity, endogenous gas fees, and time varying volatility. In this extended environment, noise trading, arbitrage races, and execution costs jointly determine LP returns, giving rise to an interior equilibrium in which optimal liquidity provision is non-monotonic in volatility and exhibits a hump-shaped relationship. Overall, this paper builds a dynamic equilibrium model calibrated on extensive data that characterize the complex interaction between informed arbitrageurs, noise traders, and liquidity providers.

A Dynamic Equilibrium Model for Automated Market Makers

Abstract

Automated Market Makers (AMMs) are a central component of decentralized exchanges, yet their equilibrium foundations and microeconomic mechanisms remain incompletely understood. This paper develops a dynamic equilibrium framework for Constant Function Market Makers (CFMMs) that formalizes the strategic interaction between arbitrageurs and liquidity providers (LPs) over time. We make three main contributions. First, we derive and empirically validate an intrinsic buy-sell asymmetry in CFMM price impact. Even in the absence of directional price movements, the geometric structure of constant product AMMs implies systematically different execution costs for buying and selling, a prediction that we confirm using on-chain transaction data. Second, we characterize the optimization problems of arbitrageurs and LPs in closed form, incorporating slippage and fees. In a baseline environment with only informed arbitrageurs, we show that providing liquidity is strictly dominated for LPs: arbitrage-driven price corrections generate negative jump returns that cannot be offset by fees, yielding a degenerate equilibrium with minimal liquidity provision. Third, motivated by empirical evidence, we extend the model to include agent heterogeneity, endogenous gas fees, and time varying volatility. In this extended environment, noise trading, arbitrage races, and execution costs jointly determine LP returns, giving rise to an interior equilibrium in which optimal liquidity provision is non-monotonic in volatility and exhibits a hump-shaped relationship. Overall, this paper builds a dynamic equilibrium model calibrated on extensive data that characterize the complex interaction between informed arbitrageurs, noise traders, and liquidity providers.
Paper Structure (58 sections, 14 theorems, 64 equations, 7 figures, 2 tables)

This paper contains 58 sections, 14 theorems, 64 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Institutional Background and Data
  4. Centralized Exchanges and AMM-DEX.
  5. Centralized Exchange (CEX).
  6. Automated Market Maker (AMM)
  7. Data
  8. On-chain AMM data.
  9. Centralized Exchange prices.
  10. Baseline Model
  11. Market and Asset Price Dynamics
  12. LP Fee Structure with Numeraire.
  13. Arbitrageur's Optimization Problem
  14. Arbitrageur's Problem.
  15. LP's Dynamic Optimization Problem
  16. ...and 43 more sections

Key Result

proposition 1

Assume that the reserve pool is always enough for any transaction and neither asset can be traded until the reserve is zero ($R^A_t > 0 \bigcap R^B_t > 0$). Suppose that the arbitrageurs only aims to maximize the instantaneous return, then the optimal swapping of asset $A$ and $B$ under case 1 and c where $\Delta A_{\max }=\frac{R^B_{t-} e^{\gamma}}{P_t} - R^A_{t-}$ and $\Delta A_{\text{aff}}=-\fr

Figures (7)

  • Figure 1: Buy-Sell Asymmetry in Profitability Rate of ETH/USDT pool by Volatility
  • Figure 2: Buy-Sell Asymmetry in Profitable Transactions in BNB/USDT Pool
  • Figure 3: Volume Distribution of Profitable (Green) v.s. Unprofitable (Red) Transactions ETH/USDT pool
  • Figure 4: Volume Distribution of Profitable (Green) v.s. Unprofitable (Red) Transactions BNB/USDT pool
  • Figure 5: Volume Distribution of Profitable (Green) v.s. Unprofitable (Red) Transactions POL/USDT pool
  • ...and 2 more figures

Theorems & Definitions (14)

  • proposition 1
  • proposition 2
  • proposition 3: LP Wealth Dynamics
  • lemma 1: Per-event proportional wealth return under fee-extracted CFMM correction
  • theorem 1: Liquidity Provision under Purely Informed Trading
  • lemma 2: Real Noise Trader Fee Generation
  • proposition 4: Race-game Quantity Choice, Entry, and Overrun Volume scaling
  • lemma 3: Per-event proportional wealth return under overrun arbitrageur overrun
  • proposition 5: Extended LP wealth dynamics
  • theorem 2: CRRA HJB reduction and correct first/second derivatives when $U$ depends on $\theta$ via $K(\theta$)
  • ...and 4 more