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Partially Active Automated Market Makers

Sunghun Ko

TL;DR

This paper addresses adverse selection in constant function market makers (CFMMs) by introducing partially active automated market makers (PA-AMMs) that partition reserves into an active and a passive portion, enabling trading only against the active liquidity with a block-level rebalancing rule governed by the activeness parameter $\lambda$. By deriving the top-gap dynamics, stationary distribution, and LVR under a two-asset geometric mean market maker (G3M) invariant, the authors formulate a discounted stochastic-control problem to optimally choose $\lambda$ to balance price-tracking error and loss-versus-rebalancing (LVR). They solve for the optimal policy, yielding a state-dependent form that collapses to a simple closed-form $\lambda^*(\gamma)$ in the small-$\Delta t$ limit, where $\gamma$ encodes the relative weight of LVR against tracking error. The results show a fundamental trade-off: smaller $\lambda$ reduces LVR and can increase LP wealth, but increases the price gap and tracking error; larger $\lambda$ lowers tracking error but raises LVR. Overall, PA-AMMs offer a principled mechanism to reduce arbitrage losses while delivering guidance on when to trade off rebalancing speed against price accuracy, with implications for liquidity provisioning and potential applications in prediction markets.

Abstract

We introduce a new class of automated market maker (AMM), the \emph{partially active automated market maker} (PA-AMM). PA-AMM divides its reserves into two parts, the active and the passive parts, and uses only the active part for trading. At the top of every block, such a division is done again to keep the active reserves always being \(λ\)-portion of total reserves, where \(λ\in (0, 1]\) is an activeness parameter. We show that this simple mechanism reduces adverse selection costs, measured by loss-versus-rebalancing (LVR), and thereby improves the wealth of liquidity providers (LPs) relative to plain constant-function market makers (CFMMs). As a trade-off, the asset weights within a PA-AMM pool may deviate from their target weights implied by its invariant curve. Motivated by the optimal index-tracking problem literature, we also propose and solve an optimization problem that balances such deviation and the reduction of LVR.

Partially Active Automated Market Makers

TL;DR

This paper addresses adverse selection in constant function market makers (CFMMs) by introducing partially active automated market makers (PA-AMMs) that partition reserves into an active and a passive portion, enabling trading only against the active liquidity with a block-level rebalancing rule governed by the activeness parameter . By deriving the top-gap dynamics, stationary distribution, and LVR under a two-asset geometric mean market maker (G3M) invariant, the authors formulate a discounted stochastic-control problem to optimally choose to balance price-tracking error and loss-versus-rebalancing (LVR). They solve for the optimal policy, yielding a state-dependent form that collapses to a simple closed-form in the small- limit, where encodes the relative weight of LVR against tracking error. The results show a fundamental trade-off: smaller reduces LVR and can increase LP wealth, but increases the price gap and tracking error; larger lowers tracking error but raises LVR. Overall, PA-AMMs offer a principled mechanism to reduce arbitrage losses while delivering guidance on when to trade off rebalancing speed against price accuracy, with implications for liquidity provisioning and potential applications in prediction markets.

Abstract

We introduce a new class of automated market maker (AMM), the \emph{partially active automated market maker} (PA-AMM). PA-AMM divides its reserves into two parts, the active and the passive parts, and uses only the active part for trading. At the top of every block, such a division is done again to keep the active reserves always being -portion of total reserves, where is an activeness parameter. We show that this simple mechanism reduces adverse selection costs, measured by loss-versus-rebalancing (LVR), and thereby improves the wealth of liquidity providers (LPs) relative to plain constant-function market makers (CFMMs). As a trade-off, the asset weights within a PA-AMM pool may deviate from their target weights implied by its invariant curve. Motivated by the optimal index-tracking problem literature, we also propose and solve an optimization problem that balances such deviation and the reduction of LVR.
Paper Structure (19 sections, 7 theorems, 67 equations, 2 figures, 1 algorithm)

This paper contains 19 sections, 7 theorems, 67 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and Related Works
  3. Our Contributions
  4. Roadmap
  5. Preliminaries
  6. Constant Function Market Makers
  7. Setup
  8. Loss-versus-rebalancing
  9. PA-AMM and Its Properties
  10. The Partial Usage of Reserves and Periodic Rebalancing
  11. The Dynamics
  12. Finding Optimal Lambda
  13. The Problem
  14. Solution
  15. Conclusion
  16. ...and 4 more sections

Key Result

Proposition 1

Loss-versus-rebalancing takes the form where we define, for $P \geq 0$, the instantaneous LVR by In particular, $\mathrm{LVR}$ is a non-negative, non-decreasing, and predictable process.

Figures (2)

  • Figure 1: Left. The efficient frontier of instantaneous LVR and the variance of price gap. Middle. The cumulative LVR over time for $\lambda \in [0.25, 0.5, 0.75, 1]$. Right. The price gap over time for $\lambda \in [0.25, 0.5, 0.75, 1]$. For the middle and right panels, we used the historical ETH price from May 2025 to October 2025. All of the pools were initialized with $1000$ ETH and an equivalent amount of USDT.
  • Figure 2: (Left) Leading-order loss $\lambda\mapsto \mathbb{E}[((1-\lambda)^2+\gamma\lambda)g^2]$ for $\gamma=4$ under the stationary AR(1) approximation. (Right) The small-$\Delta t$ asymptotic optimizer $\lambda^\ast(\gamma)$ for $\gamma\in[0,10]$.

Theorems & Definitions (16)

  • Example 1
  • Proposition 1: Theorem 1 of milionis2022automated
  • Example 2
  • Proposition 2
  • proof
  • Remark 1
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Lemma 1
  • ...and 6 more