["Mathematical Finance, EarlyView. ", "\nABSTRACT\nWe study axiomatic foundations for different classes of automated market makers (AMMs). Our main results are a characterization of a generalization of constant product market makers (CPMMs), popular in decentralized finance, and a characterization of the Logarithmic Market Scoring Rule AMM, popular in prediction markets. 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 numéraire in the two applications, but pinned down by the same separability property. Moreover, we characterize the CPMM as trader‐optimal among scale invariant, independent, symmetric and sufficiently funded AMMs. Our results add to a formal analysis of mechanisms used in practice and connect the most popular class of decentralized finance (DeFi) AMMs to the most popular class of prediction market AMMs."]