[Paper Review] Mixing Constant Sum and Constant Product Market Makers

Alexander Port, Neelesh Tiruviluamala|arXiv (Cornell University)|Mar 23, 2022

Evolutionary Algorithms and Applications4 citations

TL;DR

This paper introduces a novel homotopy-based framework for blending constant sum market makers (CSMM) and constant product market makers (CPMM), enabling smooth, geometrically meaningful transitions between their behaviors. By parameterizing the blend via a stability parameter t and using non-uniform homotopy, the method allows precise control over price stability, impermanent loss, and slippage, with key results showing that lower stability increases portfolio value and improves sensitivity to user behavior.

ABSTRACT

Two popular forms of automated market makers are constant sum and constant product (CSMM and CPMM respectively). Each has its advantages and disadvantages: CSMMs have stable exchange rates but are vulnerable to arbitrage and can sometimes fail to provide liquidity, while a CPMM can have large impermanent loss due to exchange rate changes but are always able to provide liquidity to participants. A significant amount of work has been done in order to get the best of both constant sum and constant product characteristics. Perhaps most the relevant to this paper is Stableswap, which has an "amplification coefficient" parameter controlling the balance between the two types of behavior [Ego19]. Alternative approaches, such as in [AEC21], involve constructing AMMs using portfolio value functions. However, there is still much work to be done on these fronts. This paper presents multiple novel methods for mixing market makers and demonstrates new tools for designing markets with specific features.

Motivation & Objective

• To design a new class of automated market makers (AMMs) that combine the price stability of constant sum market makers (CSMM) with the liquidity resilience of constant product market makers (CPMM).
• To overcome limitations in existing hybrid models like Stableswap, which rely on fixed parametrization and lack geometric consistency in blending.
• To develop a mathematically rigorous homotopy framework where the blending parameter t has a clear geometric interpretation as the proportional movement from CSMM to CPMM.
• To enable non-uniform blending through t(s) parametrization, allowing dynamic adjustment of stability and bias for tailored market behavior.
• To analyze the impact of stability and bias on portfolio value functions and AMM performance, particularly under price shocks.

Proposed method

• Proposes a homotopy construction H(s,t) = (1-t)·f(s) + t·g(s) to smoothly interpolate between CSMM and CPMM curves, where f(s) and g(s) represent the initial and final states.
• Defines normalized CSMM and CPMM surfaces A₀(X) = (∑aⱼxⱼ)/∑aⱼxⱼⁱ = 1 and A₁(X) = ∏(xⱼ/xⱼⁱ)ᵃʲ = 1, ensuring both pass through the initial point Xⁱ.
• Introduces a non-uniform homotopy via t(s), a quadratic function of s, to allow variable blending rates across the curve, enabling custom stability and bias profiles.
• Uses the portfolio value function V(P) = P·X to evaluate AMM performance, with U(r₁,…,rₙ₋₁) = V(r₁,…,rₙ₋₁,1) enabling 2D visualization of value under different price vectors.
• Applies λ(s,t) parametrization to ensure numerical stability and prevent drift during computation, preserving exactness in curve evaluation.
• Demonstrates that lower stability (higher t) leads to higher slippage but also higher portfolio value, as the minimizing X point moves closer to the initial Xⁱ.

Experimental results

Research questions

• RQ1 How can a continuous, geometrically meaningful blend between constant sum and constant product market makers be constructed to avoid the limitations of arithmetic or geometric mixing?
• RQ2 What is the impact of non-uniform blending (via t(s)) on price stability, impermanent loss, and portfolio value in AMMs?
• RQ3 How does the stability parameter t relate to real-world AMM behavior, particularly in terms of slippage and sensitivity to user trades?
• RQ4 Why does a decrease in stability lead to an increase in portfolio value, and how does this affect long-term market maker performance?
• RQ5 Can the homotopy framework be generalized to higher-dimensional AMMs and used to design AMMs with dynamic, adaptive behavior under market shocks?

Key findings

• The non-uniform homotopy construction allows for precise, customizable blending between CSMM and CPMM behaviors, with t(s) enabling dynamic adjustment of stability and bias along the curve.
• Lower stability (higher t) results in higher slippage but also increases the portfolio value function, as the optimal X point under a given price vector moves closer to the initial position Xⁱ.
• The geometric homotopy framework ensures that t always represents the exact fractional movement from CSMM to CPMM, providing a clear and consistent interpretation of the blending parameter.
• The portfolio value function increases with decreasing stability, indicating that higher slippage AMMs can preserve more value under certain conditions.
• The λ(s,t) parametrization prevents numerical drift and enables exact computation, making the method suitable for production deployment in decentralized exchanges.
• Visual analysis confirms that lower stability curves have higher value because the minimizing X point (where gradient aligns with price vector) is closer to Xⁱ, increasing P·X.