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Automated Market Makers for Decentralized Finance (DeFi)

Yongge Wang

TL;DR

The paper evaluates LMSR, LS-LMSR, and constant product/mean/sum AMMs for DeFi, identifying LMSR limitations in blockchain contexts and noting LS-LMSR’s advantages but its concavity and gas-inefficiency. It introduces constant ellipse AMMs as a convex, computation-friendly alternative with bounded price amplitude, and demonstrates a CoinSwap prototype. Through Solidity deployment and gas-cost benchmarking, the work shows notable improvements in gas efficiency and slippage control relative to Uniswap V2/V3, while preserving path independence and liquidity sensitivity. The findings suggest constant ellipse AMMs offer a practical balance of pricing robustness, front-running resistance, and on-chain performance for DeFi applications, alongside explicit tradeoffs such as non-translation-invariant pricing and fixed price-amplitude characteristics.

Abstract

This paper compares mathematical models for automated market makers including logarithmic market scoring rule (LMSR), liquidity sensitive LMSR (LS-LMSR), constant product/mean/sum, and others. It is shown that though LMSR may not be a good model for Decentralized Finance (DeFi) applications, LS-LMSR has several advantages over constant product/mean based automated market makers. However, LS-LMSR requires complicated computation (i.e., logarithm and exponentiation) and the cost function curve is concave. In certain DeFi applications, it is preferred to have computationally efficient cost functions with convex curves to conform with the principle of supply and demand. This paper proposes and analyzes constant circle/ellipse based cost functions for automated market makers. The proposed cost functions are computationally efficient (only requires multiplication and square root calculation) and have several advantages over widely deployed constant product cost functions. For example, the proposed market makers are more robust against front-runner (slippage) attacks.

Automated Market Makers for Decentralized Finance (DeFi)

TL;DR

The paper evaluates LMSR, LS-LMSR, and constant product/mean/sum AMMs for DeFi, identifying LMSR limitations in blockchain contexts and noting LS-LMSR’s advantages but its concavity and gas-inefficiency. It introduces constant ellipse AMMs as a convex, computation-friendly alternative with bounded price amplitude, and demonstrates a CoinSwap prototype. Through Solidity deployment and gas-cost benchmarking, the work shows notable improvements in gas efficiency and slippage control relative to Uniswap V2/V3, while preserving path independence and liquidity sensitivity. The findings suggest constant ellipse AMMs offer a practical balance of pricing robustness, front-running resistance, and on-chain performance for DeFi applications, alongside explicit tradeoffs such as non-translation-invariant pricing and fixed price-amplitude characteristics.

Abstract

This paper compares mathematical models for automated market makers including logarithmic market scoring rule (LMSR), liquidity sensitive LMSR (LS-LMSR), constant product/mean/sum, and others. It is shown that though LMSR may not be a good model for Decentralized Finance (DeFi) applications, LS-LMSR has several advantages over constant product/mean based automated market makers. However, LS-LMSR requires complicated computation (i.e., logarithm and exponentiation) and the cost function curve is concave. In certain DeFi applications, it is preferred to have computationally efficient cost functions with convex curves to conform with the principle of supply and demand. This paper proposes and analyzes constant circle/ellipse based cost functions for automated market makers. The proposed cost functions are computationally efficient (only requires multiplication and square root calculation) and have several advantages over widely deployed constant product cost functions. For example, the proposed market makers are more robust against front-runner (slippage) attacks.

Paper Structure

This paper contains 15 sections, 12 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Existing models for prediction market makers
  3. Prediction market and market makers
  4. Logarithmic market scoring rules (LMSR)
  5. Constant product/sum/mean AMMs
  6. Constant ellipse AMMs
  7. Supply-and-demand, liquid sensitivity, and price fluctuation
  8. LS-LMSR
  9. Constant product and constant mean
  10. Constant ellipse
  11. Front running attacks based on slippage
  12. Price amplitude
  13. Implementation and performance
  14. Gas cost and comparison
  15. Conclusion

Figures (9)

  • Figure 1: LMSR market maker cost function curves for $C(x,y,z)=100$ and $C(x,y)=100$
  • Figure 2: LS-LMSR market maker cost function curves for $C(x,y,z)=100$ and $C(x,y)=100$
  • Figure 3: Constant product cost function curves for $xyz=100$ and $xy=100$
  • Figure 4: Constant mean cost function curves for $xy^2z^3=100$ and $x^2y^3=100$
  • Figure 5: Constant sum market maker cost function curves for $x+y+z=100$ and $x+y=100$
  • ...and 4 more figures

Theorems & Definitions (2)

  • Definition 2.1
  • Example 1