[Paper Review] Automated Market Makers for Decentralized Finance (DeFi)

Yongge Wang|arXiv (Cornell University)|Sep 3, 2020

Financial Markets and Investment StrategiesEconomics, Econometrics and Finance8 references19 citations

TL;DR

This paper proposes constant circle and ellipse-based cost functions for automated market makers in DeFi, offering computationally efficient, convex cost curves that are more robust against front-running attacks than existing constant product models. The approach enables stable price oracles with fixed price amplitude, reducing manipulation risk even with low initial liquidity.

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.

Motivation & Objective

• To address the limitations of existing automated market maker (AMM) models in DeFi, particularly their vulnerability to front-running and price manipulation.
• To develop a cost function that is computationally efficient, convex, and maintains stable price amplitude for reliable price oracles.
• To improve resilience in low-liquidity scenarios by fixing the price range and reducing attacker advantage.
• To provide a practical alternative to constant product and LMSR-based models that balance efficiency, security, and usability in DeFi applications.

Proposed method

• Proposes constant circle and ellipse-based cost functions as alternatives to traditional constant product, LMSR, and LS-LMSR models.
• Uses geometric constraints to define the cost function, ensuring convexity and computational efficiency via multiplication and square root operations.
• Designs the market maker so that token prices are bounded within a fixed amplitude, preventing extreme price swings.
• Introduces a mechanism for adjusting liquidity by adding tokens to the market to shift the price range without external reconfiguration.
• Analyzes the mapping between outstanding shares and actual token holdings to ensure price consistency after liquidity adjustments.
• Demonstrates that automatic state adjustment without external input is impossible due to irreversible share-to-coin mapping changes.

Experimental results

Research questions

• RQ1 Can a computationally efficient, convex cost function be designed for DeFi AMMs that outperforms constant product models?
• RQ2 How does the fixed price amplitude of circle/ellipse-based AMMs affect resistance to front-running and price manipulation?
• RQ3 What are the limitations of LS-LMSR models in practice, particularly regarding liquidity adjustment and computational complexity?
• RQ4 Can a market maker maintain price stability and security with minimal initial liquidity using the proposed cost function?
• RQ5 Is it feasible to automatically reconfigure an AMM's state without external liquidity input?

Key findings

• Constant circle/ellipse cost functions are computationally efficient, requiring only multiplication and square root operations, unlike LS-LMSR which requires costly logarithmic and exponential functions.
• The proposed cost functions have convex cost curves, aligning better with the economic principle of supply and demand compared to the concave curves of LS-LMSR.
• Circle/ellipse-based AMMs are more robust against front-running attacks because the price amplitude is fixed, limiting the potential profit from manipulation.
• Even with minimal initial liquidity, the fixed price range prevents attackers from significantly distorting prices, reducing risk for downstream DeFi applications using the AMM as a price oracle.
• It is impossible to automatically reconfigure an AMM’s state to a new price range without external liquidity input, as share-to-coin mappings are irreversible once set.
• The paper demonstrates that LS-LMSR markets may fail when price ratios approach their theoretical minimum, requiring active liquidity injection to remain functional.