Characterizing Path-Independent Fees: A Route to Zero Impermanent Loss in CPMMs

2026-04-30

AI Summary

This study addresses the path dependence inherent in traditional constant-product automated market makers (CPMMs) with fixed-fee structures, which induces impermanent loss and undermines protocol composability under automatic reinvestment. The authors provide the first complete characterization of a class of path-independent fee functions by designing fees that depend solely on the invariant \(k = xy\). They formulate the pool dynamics as a system of ordinary differential equations and derive closed-form swap formulas. While proving that no universal fee function can eliminate impermanent loss for all initial states, they construct a parameterized family of fee functions that achieve zero impermanent loss for specific initial conditions. The theoretical findings are validated through mathematical modeling, analytical solutions, and simulations, which also quantify the impact on arbitrage windows and slippage, offering a deployable and incentive-compatible framework for AMM fee mechanism design.

Abstract

Constant Product Market Makers use fees that are typically fixed proportions of trade size. When these fees are automatically reinvested into the pool, as in Uniswap~V2 and some designs of Uniswap V4, the final state after a trade can depend on how the trade is split into smaller transactions. This path dependence complicates the risk assessment for liquidity providers and affects composability guarantees. We characterize the functional class of fee structures that ensure path independence: the combined fee factor must depend only on the current pool invariant k=xy. For this class, we derive a system of ordinary differential equations governing pool dynamics and obtain a closed-form integral exchange formula. Within this class, we construct a parametric family of fee functions that achieve zero Impermanent Loss for a given initial pool state, and prove that no universal fee function can eliminate Impermanent Loss for all initial states simultaneously. We analyze implications for arbitrage windows and slippage, and validate our theory through controlled simulations. Our framework provides protocol designers with a principled approach to fee optimization that aligns liquidity provider and trader incentives while preserving composability.

Problem

• Path Independence
• Impermanent Loss
• Constant Product Market Makers
• Liquidity Provision
• Fee Structure

Innovation

• path independence
• impermanent loss
• constant product market maker
• fee structure
• liquidity provision