Axioms for Constant Function Market Makers

TL;DR

This paper provides axiomatic characterizations of constant-function market makers, linking decentralized exchange mechanisms with prediction market models through invariance and separability properties.

Contribution

It introduces axiomatic foundations for CPMMs and LMSRs, highlighting their key invariance properties and connecting DeFi and prediction market mechanisms.

Findings

Characterization of CPMMs via independence and scale invariance.

Characterization of LMSRs via independence and translation invariance.

CPMMs are extremal within a class of symmetric AMMs.

Abstract

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.