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.

中文翻译：

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去中心化金融 (DeFi) 的自动化做市商

本文比较了自动做市商的数学模型，包括对数市场评分规则 (LMSR)、流动性敏感 LMSR (LS-LMSR)、常数乘积/均值/总和等。结果表明，尽管 LMSR 可能不是去中心化金融 (DeFi) 应用程序的好模型，但与基于恒定乘积/均值的自动化做市商相比，LS-LMSR 具有几个优势。然而，LS-LMSR 需要复杂的计算（即对数和幂）并且成本函数曲线是凹的。在某些 DeFi 应用中，优选具有凸曲线的计算高效成本函数，以符合供求原则。本文为自动化做市商提出并分析了基于恒定圆/椭圆的成本函数。所提出的成本函数在计算上是高效的（只需要乘法和平方根计算）并且比广泛部署的恒定乘积成本函数有几个优点。例如，提议的做市商对领跑者（滑点）攻击更具鲁棒性。