A Dynamic Equilibrium Model for Automated Market Makers

This paper develops a dynamic equilibrium model for Automated Market Makers that reveals how the strategic interaction between informed arbitrageurs, noise traders, and liquidity providers—shaped by intrinsic buy-sell asymmetry and execution costs—leads to a non-monotonic, hump-shaped relationship between volatility and optimal liquidity provision.

The Gist

Imagine a bustling, automated digital marketplace called a Decentralized Exchange (DEX). Unlike a traditional stock exchange with a human manager matching buyers and sellers, this marketplace runs on a robot program called an Automated Market Maker (AMM).

Here is the cast of characters:

• The Liquidity Providers (LPs): These are the "shop owners." They deposit their own money (crypto assets) into the robot's vault to make sure there's always stock available for people to buy. They earn a small fee on every trade, hoping the fees will outweigh the risks.
• The Arbitrageurs: These are the "smart shoppers." They constantly scan the robot's prices and the prices on big, centralized exchanges (like Binance). If the robot is selling something cheaper than the big exchange, they buy it instantly to make a quick profit.
• The Noise Traders: These are regular people buying crypto for fun, to pay for something, or just because they feel like it. They aren't trying to make a profit; they just need to trade.

The Big Mystery

For a long time, economists were confused. They knew that the "smart shoppers" (arbitrageurs) were constantly taking money out of the "shop owners'" (LPs) pockets because they were trading on information the shop owners didn't have. This is called adverse selection.

The big question was: Why would anyone ever want to be a shop owner (LP) if the smart shoppers keep stealing their profits?

The Paper's Discovery: The "Buy-Sell" Trap

The authors of this paper built a mathematical model to solve this mystery. They found a hidden trap in the robot's design.

The Analogy of the Curved Slide:
Imagine the robot's pricing rule is like a curved slide.

• Buying is like sliding down a steep, slippery slope. It's hard to get a good deal because the robot's price shoots up quickly as you buy more.
• Selling is like sliding down a gentle, flat path. It's easier to get a fair price.

The paper proves that because of this shape, buying is always more expensive and risky for the shop owner than selling. Even if the market price isn't moving, the robot's geometry makes it harder to buy than to sell.

The "Empty Shop" Problem

In their first, simple model, the authors assumed only smart shoppers (arbitrageurs) were trading.

• The Result: The shop owners lost money on every trade. The fees they earned weren't enough to cover the losses from the smart shoppers.
• The Conclusion: In a world with only smart shoppers, nobody would open a shop. The vault would be empty, and the market would collapse.

But wait! Real markets don't collapse. Vaults are full. So, what's missing?

The Real-World Twist: Chaos and Competition

The authors looked at real data from Ethereum and found the missing pieces. They realized the market isn't just smart shoppers; it's a chaotic mix of three things:

1. Real Noise: Regular people trading for non-profit reasons. They pay fees that do help the shop owners.
2. The Gas Fee Race: On the blockchain, if you want your trade to go through fast, you have to pay a "tip" (gas fee) to the miners.
3. The "Overrun" Effect: This is the most creative part. When a price difference appears, hundreds of smart shoppers race to grab it.
   • The Winner: The first person to click "buy" gets the profit.
   • The Losers (Overrun Arbitrageurs): The next 99 people try to buy, but the price has already changed by the time their transaction goes through. They end up buying at a bad price and losing money.

The Analogy of the Flash Sale:
Imagine a store has a flash sale on a TV.

• The Winner: One person grabs the TV at the sale price.
• The Losers: 50 other people rush to the counter, but the TV is gone. They end up paying full price (or even more) because they were too slow.
• The Shop Owner: Even though the 50 losers lost money, the store collected a massive amount of "rush fees" from all of them trying to get in.

The "Hump-Shaped" Solution

The paper's final, brilliant insight is how much liquidity (money) shop owners should provide based on how crazy the market is (volatility).

• Low Volatility (Calm Market): Not many smart shoppers are racing. The fees are low. Shop owners don't see much point in providing huge amounts of money.
• Medium Volatility (Just Right): This is the sweet spot. The market is active enough that smart shoppers are racing, creating a flood of "overrun" trades. The shop owners collect massive fees from all these failed races. This is when shop owners provide the most liquidity.
• High Volatility (Chaos): The market is so crazy that prices jump wildly. The "overrun" losses become so huge that the fees can't cover them. Also, the "tips" (gas fees) to get trades through become so expensive that people stop trading. Shop owners pull their money out to avoid disaster.

The Bottom Line

The paper explains that Liquidity Providers don't just sit there and hope. They are playing a complex game against:

1. Smart sharks (arbitrageurs) who try to steal their money.
2. Clumsy sharks (overrun arbitrageurs) who try to steal money but fail, accidentally paying fees to the shop owner.
3. Regular people who just want to trade.

The market works because the "clumsy sharks" and "regular people" generate enough fees to pay for the losses caused by the "smart sharks." And the amount of money shop owners are willing to put in follows a hump shape: they put in the most money when the market is moderately chaotic, but pull back when it's either too boring or too terrifying.

This model finally explains why these digital markets stay open and full of money, even when the math says they should be empty!

Technical Summary

Here is a detailed technical summary of the paper "A Dynamic Equilibrium Model for Automated Market Makers" by Zang, Wang, and Zhang.

1. Problem Statement

The paper addresses a fundamental economic puzzle in Decentralized Finance (DeFi): Why do Liquidity Providers (LPs) continue to supply capital to Automated Market Makers (AMMs) despite facing severe adverse selection from informed arbitrageurs?

• The Paradox: In Constant Function Market Makers (CFMMs) like Uniswap V2, LPs passively absorb order flow. When arbitrageurs exploit price discrepancies between the AMM and external centralized exchanges (CEX), they extract value from LPs (impermanent loss).
• The Gap: Existing theoretical models often assume a stable flow of "noise traders" (uninformed traders) to compensate LPs for these losses. However, it is unclear if such noise exists in sufficient quantities on-chain, or if the equilibrium can be sustained solely by informed agents.
• The Goal: To develop a dynamic equilibrium framework that formalizes the strategic interaction between arbitrageurs and LPs, incorporating realistic market frictions (slippage, fees, gas costs) and agent heterogeneity to explain observed liquidity provision.

2. Methodology

The authors employ a hybrid approach combining continuous-time stochastic control theory with empirical analysis of on-chain data.

A. Theoretical Framework

1. Baseline Model (Pure Informed Trading):

   • Setting: A constant-product AMM ($xy=k$) with proportional fees ($\gamma$).
   • Agents: LPs (maximizing CRRA utility) and Informed Arbitrageurs (maximizing instantaneous profit).
   • Dynamics: CEX prices follow Geometric Brownian Motion (GBM). Arbitrageurs trade when the AMM price deviates from the CEX price by more than the fee band.
   • LP Wealth: Modeled as a jump-diffusion process where "jumps" represent arbitrage corrections that cause impermanent loss.

2. Extended Model (Heterogeneous Agents & Frictions):

   • Stochastic Volatility: CEX price volatility follows a mean-reverting process (CIR/Heston type).
   • Endogenous Gas Fees: Gas fees are modeled as a convex, increasing function of volatility ($g(v_t)$), reflecting network congestion during high-volatility arbitrage races.
   • Agent Heterogeneity:
     • Real Noise Traders: Trade for idiosyncratic reasons (payment, privacy) independent of mispricing.
     • Overrun Arbitrageurs: Profit-seeking agents who fail to win the "priority race" (e.g., due to gas bidding wars) and execute trades on stale prices, resulting in ex-post losses.

B. Empirical Analysis

• Data: Transaction-level data from major Uniswap and Binance-AMM pools (e.g., ETH/USDT, BNB/USDT) combined with high-frequency CEX price data.
• Classification: Trades are classified as "profitable" or "unprofitable" based on ex-post comparison against fee-adjusted CEX prices.
• Metrics: Analysis of buy-sell asymmetry, volume distribution, and the correlation between CEX volatility, gas fees, and unprofitable trade volume.

3. Key Contributions

Contribution 1: Intrinsic Buy-Sell Asymmetry

The paper derives and validates a structural asymmetry in CFMM price impact.

• Theory: Due to the geometric curvature of the constant-product invariant ($xy=k$), buying an asset (depleting the scarce reserve) induces higher slippage than selling the same notional amount.
• Result: Even without directional price trends, the cost of buying is systematically higher than selling. This implies that profitable arbitrage opportunities are harder to find on the buy-side than the sell-side in stable markets.

Contribution 2: The "Degenerate Equilibrium" in Baseline Models

In a baseline model with only informed arbitrageurs:

• Finding: Providing liquidity is strictly dominated by holding the asset directly.
• Mechanism: The negative jumps in LP wealth caused by arbitrage corrections (adverse selection) are never fully offset by fee revenue.
• Implication: An AMM populated solely by rational, informed arbitrageurs would collapse into a degenerate equilibrium with minimal liquidity provision. This contradicts real-world observations, suggesting the baseline model is missing critical components.

Contribution 3: The "Overrun" Mechanism and Hump-Shaped Liquidity

The extended model introduces arbitrage races and overrun order flow to explain the persistence of liquidity.

• Mechanism: During high volatility, multiple arbitrageurs compete. The winner captures the spread; the "losers" (overrun arbitrageurs) trade against an already corrected pool, incurring losses. These losses effectively act as a subsidy to LPs (similar to noise trading fees).
• Gas Fee Dynamics: High volatility increases gas fees, which acts as a friction.
• Result: Optimal liquidity provision ($\theta^*$) exhibits a hump-shaped relationship with volatility:
  • Low Volatility: Low fees and low arbitrage activity; LPs provide moderate liquidity.
  • Medium Volatility: Increased frequency of "overrun" trades generates significant fee revenue that outweighs impermanent loss. LPs increase liquidity.
  • High Volatility: Gas costs become prohibitive, and price jumps become too large/severe. The risk of massive impermanent loss dominates, causing LPs to withdraw liquidity.

4. Key Results

1. Empirical Validation of Asymmetry: On-chain data confirms that in drift-neutral regimes, sell-side trades are significantly more likely to be profitable than buy-side trades, validating the theoretical price-impact asymmetry.
2. Heterogeneity of Traders:
   • Profitable Trades: Represent ~10% of transaction count but ~55% of volume (large, institutional/arbitrage).
   • Unprofitable Trades: Represent ~90% of count but smaller volume. Crucially, the volume of unprofitable trades co-moves positively with CEX volatility.
3. Endogenous Gas Fees: The study finds that CEX price volatility Granger-causes gas prices, supporting the model's assumption that gas fees are endogenous to market stress rather than purely exogenous.
4. Hump-Shaped Liquidity: The extended model successfully predicts that LPs will supply the most liquidity at intermediate volatility levels. This is empirically observed in ETH/USDC pools, where liquidity depth peaks at moderate volatility and drops at extreme volatility.

5. Significance

• Theoretical Advancement: The paper moves beyond static equilibrium models to a dynamic, micro-founded framework that explains why liquidity persists. It resolves the "adverse selection paradox" by showing that "failed" arbitrage (overrun trades) acts as a necessary source of fee revenue for LPs.
• Policy and Design Implications:
  • Gas Fees: Highlights that gas fees are not just a cost but a structural mechanism that regulates arbitrage intensity and protects LPs from excessive adverse selection during extreme volatility.
  • AMM Design: Suggests that future AMM designs (e.g., concentrated liquidity) must account for the non-monotonic relationship between volatility and liquidity provision.
• Market Understanding: Provides a rigorous explanation for the "noise" observed in DeFi. Much of what appears to be random noise trading is actually competitive arbitrage failure, a structural feature of the AMM mechanism under high volatility.

In summary, the paper establishes that AMM liquidity is sustained not by a mythical flow of uninformed noise traders, but by a complex equilibrium where competitive arbitrage failures (overruns) and volatility-dependent gas frictions generate the necessary fee revenue to compensate LPs for the risks of informed trading.