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

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.

• The paper introduces a functional characterization of fees that ensures swap outcomes remain consistent despite trade fragmentation.
• It derives closed-form expressions for CPMM pool dynamics and shows how state-aware fee functions can eliminate impermanent loss near fixed pool states.
• The work demonstrates that universal zero-IL fees are impossible, highlighting the need for adaptive, state-dependent fee designs in decentralized exchanges.

Characterizing Path-Independent Fees and Eliminating Impermanent Loss in Constant Product Market Makers

Introduction

This paper provides a rigorous structural and quantitative analysis of fee design in Constant Product Market Makers (CPMMs), focusing primarily on the risk of Impermanent Loss (IL) faced by liquidity providers and establishing the conditions for path independence in fee structures. The authors introduce a functional characterization of fees that ensures the outcome of any swap is invariant to trade fragmentation, and derive a parametric family of fee functions capable of eliminating IL for a fixed initial pool state. They analyze practical implications, confirm theoretical claims with numerical experiments, and clarify fundamental design trade-offs in the context of decentralized exchanges (DEX) and Automated Market Makers (AMM).

Path-Independence in Fee Structures

The primary theoretical contribution is the characterization of the class of path-independent fee functions. CPMMs with reinvested fees exhibit path dependence—the final pool state after a trade can vary depending on how the trade is split—when fees are not appropriately designed. Path independence is shown to require that the effective combined fee factor depends uniquely on the current pool invariant $k=xy$. Specifically, a fee structure is path-independent if and only if there exists a function $\phi(k)$ such that the combined fee factor satisfies $1 - \gamma_1(x,y) \gamma_2(x,y) = \phi(xy)$, with $\gamma_1$ and $\gamma_2$ denoting input and output fee fractions.

This result reduces the space of desirable fee functions to single-variable functions of the invariant, justifying that only state-dependent designs built around the invariant can preserve composability and deterministic risk for liquidity providers. The path-independence property is validated experimentally, with machine-precision outcomes across trade fragmentation for appropriately designed fee functions.

Integral Exchange Formula and Pool Dynamics

The paper derives a closed-form integral exchange formula for CPMMs with path-independent fees. The pool dynamics under such fees are governed by a system of ODEs, enabling rigorous analysis and allowing the outcome of any swap to be computed directly via an integral of the fee function, rather than iterative simulation. The exchange outcome is given by $G(k_f) - G(k_0) = \ln(1+\Delta x)$, where $G(k)$ is defined as an integral involving $\phi(k)$, and $k_0$, $k_f$ denote the initial and final invariants, respectively. This analytic tractability simplifies composability analysis and arbitrage/slippage modeling in on-chain or protocol-deployed systems.

Zero Impermanent Loss Fee Construction

A significant claim in the paper is the explicit construction of a parametric family of state-aware fee functions capable of achieving zero IL for any reference pool state. For a chosen initial invariant $\phi(k)$0, a fee function $\phi(k)$1 can be constructed such that the IL is exactly zero across all trade sizes originating from $\phi(k)$2. The fee function starts at zero for infinitesimal trades at the reference state but increases rapidly as the invariant diverges from $\phi(k)$3, offsetting IL with accumulated fee revenue.

This adaptive behavior is demonstrated analytically and numerically. Notably, for practical trade sizes, the zero-IL fee quickly exceeds standard fixed fees, and is less than fixed fees only for infinitesimal trades near the reference state. The parametric structure highlights that exact IL elimination is highly localized and cannot be achieved with a constant fee across all pool states.

Impossibility of Universal Zero-IL Fees

The authors provide a formal proof that no universal fee function exists that can eliminate IL across all possible initial pool states and trade sizes simultaneously. The required fee rate must vary with the initial state, implying that only state-aware or dynamically updated fee functions can guarantee IL mitigation. This fundamental limitation necessitates periodic recomputation, targeted zero-IL for fixed liquidity ranges, or acceptance of residual IL in practical systems.

Experimental Validation and Practical Implications

Controlled numerical experiments confirm the theoretical predictions:

• Path-independent fees yield invariant outcomes across fragmented trades with errors at machine precision.
• Standard protocols (e.g., Uniswap V2) with discretized fee compounding exhibit negligible errors for typical parameters but do not achieve strict path independence.
• Zero-IL fee functions produce exact IL elimination only for trades near the reference state, trading off increased effective fees and slippage for larger trades.

For protocol design, path-independent fees preserve composability and enable deterministic pricing and risk modeling. Zero-IL constructions are feasible but require state awareness and may entail operational complexity in managing liquidity changes. The framework provides designers with a principled approach to fee optimization, aligning incentives while maintaining predictable reserve evolution.

Limitations and Future Directions

The analysis is confined to two-asset CPMMs and does not generalize directly to concentrated liquidity protocols (e.g., Uniswap V3) or multi-asset pools. Extending this framework to more complex market maker designs, adaptive fee mechanisms, and decentralized governance of fee states represents a natural direction for future research. Additionally, the computational and gas cost of deploying state-aware fee mechanisms demands further engineering investigation.

Conclusion

This paper rigorously characterizes path-independent fee structures in CPMMs, providing closed-form expressions for pool dynamics and constructing state-aware fees that eliminate Impermanent Loss for fixed reference states. The impossibility result for universal zero-IL fees clarifies structural limitations and guides protocol design trade-offs. The work offers a robust foundation for sustainable liquidity provision and composable fee optimization in decentralized exchanges, setting the stage for advanced adaptive mechanisms in future DeFi infrastructure.