We study axiomatic foundations for different classes of constant-function automated market makers (CFMMs). We focus particularly on separability and on different invariance properties under scaling. Our main results are an axiomatic characterization of a natural generalization of constant product market makers (CPMMs), popular in decentralized finance, on the one hand, and a characterization of the Logarithmic Scoring Rule Market Makers (LMSR), popular in prediction markets, on the other hand. The first class is characterized by the combination of independence and scale invariance, whereas the second is characterized by the combination of independence and translation invariance. The two classes are therefore distinguished by a different invariance property that is motivated by different interpretations of the num\'eraire in the two applications. However, both are pinned down by the same separability property. Moreover, we characterize the CPMM as an extremal point within the class of scale invariant, independent, symmetric AMMs with non-concentrated liquidity provision. Our results add to a formal analysis of mechanisms that are currently used for decentralized exchanges and connect the most popular class of DeFi AMMs to the most popular class of prediction market AMMs.

翻译：本文研究不同类别常数函数自动做市商（CFMMs）的公理化基础，重点聚焦于可分离性及缩放下的不同不变性性质。主要成果包括：一方面，对去中心化金融中广泛使用的常数乘积自动做市商（CPMMs）的自然泛化形式进行了公理化刻画；另一方面，对预测市场中流行的对数评分规则市场做市商（LMSR）进行了公理化刻画。前者由独立性与尺度不变性共同刻画，后者则由独立性与平移不变性共同刻画。两类做市商的区别源于两种应用场景中对计价单位的不同诠释所导向的差异化不变性性质，但二者均受同一可分离性约束。此外，我们还将CPMM刻画为具有非集中流动性供给的尺度不变、独立、对称AMM类中的极值点。本研究为当前去中心化交易所运行机制的正式分析提供了理论支撑，并建立了DeFi领域最主流AMM与预测市场最主流AMM之间的理论关联。