Competition between DEXs through Dynamic Fees

This paper characterizes the approximate Nash equilibrium of competing decentralized exchanges setting dynamic fees, revealing that while the two-regime fee structure persists, competition shifts the fee-switching boundary to a weighted average of oracle and competitor rates, ultimately reducing slippage for strategic traders while having activity-dependent effects on noise traders.

The Gist

Imagine a bustling digital marketplace where people trade digital assets (like crypto tokens). In this world, there are no human bankers or stockbrokers. Instead, the trading happens automatically through "pools" of money managed by computer code. These are called Decentralized Exchanges (DEXs).

Think of these pools like lemonade stands.

• The Liquidity Providers (LPs) are the people who put their lemons and sugar into the stands so others can buy lemonade. They want to make a profit.
• The Liquidity Takers are the customers buying the lemonade.
• The Fee is the price the stand charges for every cup sold.

For a long time, these stands had a fixed price (a static fee). But recently, technology allowed them to change their prices dynamically based on what's happening outside. This paper asks a big question: What happens when multiple lemonade stands compete against each other by changing their prices in real-time?

Here is the story of the paper, broken down into simple concepts:

1. The Two Types of Customers

To understand the stands' strategy, we need to know who is buying:

• The "Noise" Customers: These are regular people just thirsty for lemonade. They don't care about the price too much; they just want a drink. The stands love these customers because they provide steady business.
• The "Arbitrageurs" (The Smart Shoppers): These are professional traders who watch the prices of all the stands. If Stand A sells lemonade for $1.00 and Stand B sells it for $1.05, the Smart Shopper buys from A and sells to B instantly to make a free profit. This is bad for the stands because it drains their inventory without them making enough fee money.

2. The Old Way (Monopoly)

Imagine there is only one lemonade stand in town.

• If the Smart Shoppers get too active, the stand raises its price (fee) to scare them away.
• If the stand is too expensive, it loses the regular thirsty customers.
• The stand has a "switching point": when the price gets too high, it lowers the fee to attract regulars; when the price gets too low, it raises the fee to stop the Smart Shoppers.

3. The New Reality (Competition)

Now, imagine two (or more) stands right next to each other. This is the core of the paper.

• The Game: Each stand tries to set the perfect fee to make the most money, but they have to guess what the other stand is doing.
• The "Weighted Average" Rule: In the old monopoly world, a stand only looked at the "official" market price (the Oracle) to decide when to change fees.
  • The Analogy: Now, the stand doesn't just look at the official price. It looks at the average of the official price AND the price of its rival stand.
  • If the rival stand lowers its price, the first stand must react faster. The "switching point" moves. It's no longer just about the outside world; it's about the neighborhood war.

4. The Surprising Results

The authors ran thousands of computer simulations to see how this competition plays out. Here is what they found:

• The "Smart Shoppers" Win: When there are more stands competing, the Smart Shoppers can route their orders to the cheapest stand. This means they get better deals (less "slippage," or extra cost).
• The "Regular Customers" Pay the Price (Sometimes):
  • In a quiet market (low activity), competition makes the stands charge higher fees to regular customers to survive. The regulars get worse deals.
  • In a busy market (high activity/volatility), the stands are so busy that they actually lower fees to grab volume. In this case, the regular customers get better deals!
• The Stands Lose Money: This is the sad part for the lemonade stand owners. Even though the total amount of lemonade sold might stay the same, the total profit for the stands goes down. Why? Because they are fighting over the same customers, driving prices down.
  • The Metaphor: It's like a price war at a gas station. If two stations are right next to each other, they keep lowering prices to steal customers. Eventually, both stations make less money than if one of them had just been alone.

5. The "Linear" Secret

The math in the paper is very complex (involving partial differential equations), but the authors found a simple pattern:

• The optimal fee isn't a crazy, unpredictable curve. It's almost a straight line.
• The Metaphor: You don't need a supercomputer to run the stand. You can just say, "If I have too much inventory, I lower the price. If I have too little, I raise it." This simple rule works almost as well as the complex math, which is great news for saving on computer costs (gas fees).

Summary: Who Wins and Who Loses?

• The Winners: The Smart Traders (Arbitrageurs) and, in busy markets, the Regular Customers. They get better prices because the stands are fighting for them.
• The Losers: The Lemonade Stand Owners (Liquidity Providers). They have to work harder for less profit because the competition eats into their margins.
• The Platform: The platform hosting the stands (like Uniswap) doesn't care much; it takes a cut of the fees, and the total volume stays roughly the same.

In a nutshell: This paper proves that when automated markets compete, they become more efficient for the traders but less profitable for the people providing the money. It's a classic economic trade-off: Competition helps the buyer, but it squeezes the seller.

Technical Summary

Here is a detailed technical summary of the paper "Competition between DEXs through Dynamic Fees" by Baggiani, Herdegen, and Sánchez-Betancourt.

1. Problem Statement

The paper addresses the strategic interaction between multiple Decentralized Exchanges (DEXs) operating as Automated Market Makers (AMMs). While existing literature largely focuses on optimal fee setting in a monopoly setting (a single pool) or assumes static fees, this work investigates the Nash equilibrium in a competitive environment where multiple pools (specifically a duopoly, extended to $M$ players) compete for order flow by setting dynamic trading fees.

The core challenge is that a pool's fee decision affects not only its own inventory and revenue but also the order flow distribution across competing pools. Liquidity takers (traders) route orders to the pool offering the most attractive execution price (lowest slippage + fees), creating a coupled game where pools must balance:

1. Deterring Arbitrageurs: Raising fees to prevent losses from "Loss-Versus-Rebalancing" (LVR) when prices deviate from the oracle.
2. Attracting Noise Traders: Lowering fees to capture volume from non-strategic traders, which increases volatility and fee revenue.

2. Methodology

Model Framework

• Setting: A finite-horizon game ($T > 0$) involving $M$ pools trading a riskless asset $X$ and a risky asset $Y$.
• Market Structure:
  • Oracle Price ($S_t$): The price of $Y$ on a centralized exchange (CEX), modeled as a Brownian motion ($S_t = S_0 + \sigma W_t$).
  • Pool Dynamics: Each pool is a Constant Function Market Maker (CFMM). The authors use a discrete grid for the inventory of asset $Y$ to facilitate numerical solutions.
  • Order Flow: Modeled via point processes with stochastic intensities. The arrival rate of buy/sell orders depends on:
    • The misalignment between the pool's effective quote (including fees) and the Oracle price.
    • The misalignment between the pool's quote and the competitors' quotes.
• Control Variables: Each pool $i$ chooses two predictable fee processes: $m^i_t$ (fee for buying $Y$) and $p^i_t$ (fee for selling $Y$). These fees are state-dependent, relying on the pool's own inventory and the inventories (and thus quotes) of all competitors.

Mathematical Formulation

• Objective: Each pool maximizes expected cumulative fee revenue over the horizon $[0, T]$.
• Hamilton-Jacobi-Bellman (HJB) Equations: The problem is formulated as a stochastic differential game. The value function $v_i(t, s, y_1, \dots, y_M)$ satisfies a coupled system of HJB equations.
  • Unlike the monopoly case (where fees depend only on own inventory and $S_t$), the competition case requires fees to depend on the joint inventory state of all players.
• Approximation Strategy: To derive tractable closed-form solutions, the authors employ two key approximations:
  1. Negligible Cross-Impact: They assume that a change in one agent's inventory has a negligible first-order impact on the value function of other agents (though it still affects their quotes and order flow).
  2. Linearization: They transform the non-linear HJB system into a linear system of Partial Differential Equations (PDEs) using an exponential ansatz ($w_i = e^{k_i v_i}$).
• Solution: The system is solved using matrix exponentials on the joint inventory grid, yielding explicit expressions for the optimal fees.

3. Key Contributions

1. Characterization of Competitive Equilibrium: The paper derives an approximate Nash equilibrium for dynamic fee setting in a multi-DEX environment. It proves that the equilibrium fees are linear functions of both the pool's own inventory and the competitors' inventories.
2. Shift in Switching Boundary: In monopoly models, the fee regime switches (from high fees to deter arbitrage to low fees to attract noise) based solely on the distance from the Oracle price. Under competition, this switching boundary shifts to a weighted average of the Oracle price and the competitors' exchange rates. This reflects that the trader's "outside option" is now a combination of the CEX and rival DEXs.
3. Welfare Analysis: The paper provides a rigorous analysis of how competition affects different market participants:
   • Strategic Liquidity Takers: Benefit significantly from competition due to tighter spreads and better routing options.
   • Liquidity Providers (LPs): Suffer from reduced revenue per pool as order flow is fragmented.
   • Noise Traders: Experience a non-monotonic effect; they are worse off in low-activity markets (higher slippage due to fragmented liquidity) but better off in high-activity markets (benefiting from increased competition and tighter spreads).

4. Key Results & Numerical Findings

• Fee Structure: The optimal fees exhibit a "two-regime" structure similar to the monopoly case but with a shifted threshold.
  • High Fee Regime: Activated when the pool is mispriced relative to the weighted average of the Oracle and competitors, deterring arbitrage.
  • Low Fee Regime: Activated when the pool is well-aligned, aiming to capture noise trading volume.
• Impact of Competition Intensity ($k$):
  • As the sensitivity to competitors' prices increases (higher $k_{i,j}$), the fee schedules converge toward the monopoly benchmark (fees become less responsive to competitors).
  • Fees scale inversely with the aggregate sensitivity parameter ($1/k$).
• Slippage and Execution:
  • Strategic Traders: In a duopoly or triopoly, the "best ask" is the minimum of all pools' asks, and the "best bid" is the maximum. Order statistics show that increasing the number of venues reduces the expected spread for strategic traders, lowering their execution costs.
  • Noise Traders: In low-volatility/low-volume regimes, fragmentation leads to higher average slippage for noise traders because liquidity is shallower in each individual pool. In high-volatility regimes, the increased competition drives down slippage.
• Revenue Distribution:
  • Total market fee revenue remains roughly constant between monopoly and competition (assuming fixed total liquidity).
  • However, revenue per pool declines as the number of competitors increases.
  • If fixed entry costs exist for creating a pool, high competition could drive individual LP revenues below the break-even point, potentially limiting the number of active pools.

5. Significance

• Theoretical Advancement: This work bridges the gap between single-agent optimal control in AMMs and multi-agent game theory. It extends the seminal Avellaneda-Stoikov market-making framework to the specific context of CFMMs with dynamic fees.
• Practical Implications for DeFi:
  • Uniswap v4 & Hooks: The findings are directly relevant to the new capabilities in Uniswap v4, where LPs can program dynamic fees. The paper suggests that LPs should not just react to the Oracle price but must monitor competitor pools to optimize their fee schedules.
  • Market Design: It highlights a trade-off: while competition improves execution for sophisticated traders (arbitrageurs and large institutions), it may fragment liquidity to the detriment of small, noise traders in calm markets.
  • Sustainability: The results suggest a limit to the number of viable competing pools for a single asset pair, as excessive fragmentation erodes LP profitability.

In summary, the paper demonstrates that while dynamic fees allow DEXs to adapt to market conditions, the introduction of competition fundamentally alters the equilibrium by making fees a function of the entire market's state, not just the local pool's state. This leads to more efficient execution for strategic traders but creates a "race to the bottom" in fee revenue for individual liquidity providers.