An Axiomatic Characterization of CFMMs and Equivalence to Prediction Markets

Authors Rafael Frongillo , Maneesha Papireddygari , Bo Waggoner

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Rafael Frongillo
  • University of Colorado, Boulder, CO, USA
Maneesha Papireddygari
  • University of Colorado, Boulder, CO, USA
Bo Waggoner
  • University of Colorado, Boulder, CO, USA


We thank David Pennock, Daniel Reeves, Anson Kahng, and Manifold Markets for collaboration in working out the cost function corresponding to Uniswap. We also thank Scott Kominers, Ciamac Moallemi, Abe Othman, Tim Roughgarden, and Christoph Schlegel for helpful discussions.

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Rafael Frongillo, Maneesha Papireddygari, and Bo Waggoner. An Axiomatic Characterization of CFMMs and Equivalence to Prediction Markets. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 51:1-51:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


Constant-function market makers (CFMMs), such as Uniswap, are automated exchanges offering trades among a set of assets. We study their technical relationship to another class of automated market makers, cost-function prediction markets. We first introduce axioms for market makers and show that CFMMs with concave potential functions characterize "good" market makers according to these axioms. We then show that every such CFMM on n assets is equivalent to a cost-function prediction market for events with n outcomes. Our construction directly converts a CFMM into a prediction market, and vice versa. Using this equivalence, we give another construction which can produce any 1-homogenous, increasing, and concave CFMM, as are typically used in practice, from a cost function. Conceptually, our results show that desirable market-making axioms are equivalent to desirable information-elicitation axioms, i.e., markets are good at facilitating trade if and only if they are good at revealing beliefs. For example, we show that every CFMM implicitly defines a proper scoring rule for eliciting beliefs; the scoring rule for Uniswap is unusual, but known. From a technical standpoint, our results show how tools for prediction markets and CFMMs can interoperate. We illustrate this interoperability by showing how liquidity strategies from both literatures transfer to the other, yielding new market designs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithmic game theory
  • Mathematics of computing → Information theory
  • Convex analysis
  • Equivalence result
  • Axiomatic characterization
  • Market Makers
  • Prediction markets
  • Scoring rules
  • Cost-functions


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