A theory of Automated Market Makers in DeFi

TL;DR

This paper develops a formal theoretical framework for Automated Market Makers in DeFi, analyzing their properties and providing a general solution to arbitrage, enhancing understanding of their economic mechanisms.

Contribution

It introduces an abstract operational model of AMMs, proving fundamental properties and offering a universal arbitrage solution, advancing the theoretical foundation of DeFi AMMs.

Findings

Formal properties of AMMs characterized

Sufficient conditions for economic properties identified

A general arbitrage solution devised

Abstract

Automated market makers (AMMs) are one of the most prominent decentralized finance (DeFi) applications. AMMs allow users to trade different types of crypto-tokens, without the need to find a counter-party. There are several implementations and models for AMMs, featuring a variety of sophisticated economic mechanisms. We present a theory of AMMs. The core of our theory is an abstract operational model of the interactions between users and AMMs, which can be concretised by instantiating the economic mechanisms. We exploit our theory to formally prove a set of fundamental properties of AMMs, characterizing both structural and economic aspects. We do this by abstracting from the actual economic mechanisms used in implementations, and identifying sufficient conditions which ensure the relevant properties. Notably, we devise a general solution to the arbitrage problem, the main game-theoretic foundation behind the economic mechanisms of AMMs.