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Competition between DEXs through Dynamic Fees

Leonardo Baggiani, Martin Herdegen, Leandro Sanchez-Betancourt

Abstract

We find an approximate Nash equilibrium in a game between decentralized exchanges (DEXs) that compete for order flow by setting dynamic trading fees. We characterize the equilibrium via a coupled system of partial differential equations and derive tractable approximate closed-form expressions for the equilibrium fees. Our analysis shows that the two-regime structure found in monopoly models persists under competition: pools alternate between raising fees to deter arbitrage and lowering fees to attract noise trading and increase volatility. Under competition, however, the switching boundary shifts from the oracle price to a weighted average of the oracle and competitors' exchange rates. Our numerical experiments show that, holding total liquidity fixed, an increase in the number of competing DEXs reduces execution slippage for strategic liquidity takers and lowers fee revenue per DEX. Finally, the effect on noise traders' slippage depends on market activity: they are worse off in low-activity markets but better off in high-activity ones.

Competition between DEXs through Dynamic Fees

Abstract

We find an approximate Nash equilibrium in a game between decentralized exchanges (DEXs) that compete for order flow by setting dynamic trading fees. We characterize the equilibrium via a coupled system of partial differential equations and derive tractable approximate closed-form expressions for the equilibrium fees. Our analysis shows that the two-regime structure found in monopoly models persists under competition: pools alternate between raising fees to deter arbitrage and lowering fees to attract noise trading and increase volatility. Under competition, however, the switching boundary shifts from the oracle price to a weighted average of the oracle and competitors' exchange rates. Our numerical experiments show that, holding total liquidity fixed, an increase in the number of competing DEXs reduces execution slippage for strategic liquidity takers and lowers fee revenue per DEX. Finally, the effect on noise traders' slippage depends on market activity: they are worse off in low-activity markets but better off in high-activity ones.
Paper Structure (7 sections, 2 theorems, 52 equations, 12 figures, 4 tables)

This paper contains 7 sections, 2 theorems, 52 equations, 12 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Two-player model
  3. Numerical results
  4. Simulations
  5. Conclusion
  6. Multiple-player model
  7. Simulations

Key Result

Theorem 2.2

For $l \in \{-N^{b}, \dots , N^{b} \}$ and $h \in \{-N^{a}, \dots , N^{a} \}$ define the matrices $\mathbf{A}^{b,l} : = ( \mathbf{A}^{l}_{i,j} )_{0 \leq i \leq j \leq 2N}$ and $\mathbf{A}^{a} : = ( \mathbf{A}^{a,h}_{i,l} )_{0 \leq i \leq l \leq 2N}$ by Denote with $\mathbf{1}$ the unit vectors of $\mathbb{R}^{2N^{a} + 1}$ and $\mathbb{R}^{2N^{b} + 1}$. Define the functions $w^{a}: [0,T] \times

Figures (12)

  • Figure 1: Optimal fees for selling $\mathfrak{p}^{*}_{a}(t,y_{t-})$ (solid line) and for buying $\mathfrak{m}^{*}_{a}(t,y_{t-})$ (dashed line) at time $t=0.5$ as a function of the quantity of asset $Y$ in the pool.
  • Figure 2: Linear approximation of the fees for selling $\mathfrak{p}^{*}_{a}(t,y_{t-})$ (solid line) and for buying $\mathfrak{m}^{*}_{a}(t,y_{t-})$ (dashed line) at time $t=0.5$ as a function of the quantity of asset $Y$ in the pool.
  • Figure 3: Plots of the optimal fees for buying and selling as a function of the agent’s inventory (x-axis) and the opponent’s inventory (y-axis).
  • Figure 4: Optimal fees for selling $\mathfrak{p}^{*}_{a}(t,y_{t-})$ (solid line) and for buying $\mathfrak{m}^{*}_{a}(t,y_{t-})$ (dashed line) at time $t=0.5$ as a function of the quantity of asset $Y$ in the pool for $k^{a,b} =0.1$.
  • Figure 5: Optimal fees for selling $\mathfrak{p}^{*}_{a}(t,y_{t-})$ (solid line) and for buying $\mathfrak{m}^{*}_{a}(t,y_{t-})$ (dashed line) at time $t=0.5$ as a function of the quantity of asset $Y$ in the pool for $k^{a,0} =0.1$.
  • ...and 7 more figures

Theorems & Definitions (6)

  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 4.1
  • Remark 4.2
  • Theorem A.1