[Paper Review] An Axiomatic Characterization of CFMMs and Equivalence to Prediction Markets

Rafael Frongillo, Maneesha Papireddygari+1 more|arXiv (Cornell University)|Feb 1, 2023

Sports Analytics and Performance4 citations

TL;DR

This paper establishes a formal, axiomatic equivalence between constant-function market makers (CFMMs) like Uniswap and cost-function prediction markets, showing that CFMMs with concave potential functions satisfy the same axioms as proper scoring rules for belief elicitation. The key contribution is a bidirectional transformation between CFMMs and prediction markets, proving that good market-making properties imply good information-elicitation properties and vice versa.

ABSTRACT

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.

Motivation & Objective

• To formally characterize desirable CFMMs using a set of axioms grounded in market-making efficiency and incentive compatibility.
• To establish a rigorous, one-to-one technical equivalence between CFMMs and cost-function prediction markets.
• To demonstrate that every CFMM implicitly defines a proper scoring rule for truthful belief elicitation.
• To enable cross-pollination of design techniques between decentralized finance and prediction market literature, particularly in liquidity adaptation.

Proposed method

• Proposes a set of axioms (path independence, incentive compatibility, etc.) that define 'good' automated market makers.
• Introduces a transformation from any cost function C in prediction markets to a concave, increasing potential function φ for CFMMs.
• Develops a reverse transformation from any concave, increasing φ to a convex cost function C satisfying prediction market axioms.
• Applies perspective transform techniques from convex analysis to construct reserves-aware CFMMs with non-parallel level sets.
• Demonstrates that 1-homogeneous CFMMs—common in practice—can be derived via the proposed construction.
• Proposes a practical on-chain verification method for implicit potential functions using known cost functions C, enabling fee-based liquidity growth without closed-form φ.

Experimental results

Research questions

• RQ1 Can CFMMs be formally characterized by axioms that also define good prediction markets?
• RQ2 Is there a one-to-one correspondence between CFMMs with concave potential functions and cost-function prediction markets?
• RQ3 Do CFMMs inherently support proper scoring rules for belief elicitation?
• RQ4 Can liquidity adaptation mechanisms from prediction markets be transferred to CFMMs, and vice versa?
• RQ5 How can implicit potential functions in CFMMs be verified on-chain without closed-form expressions?

Key findings

• CFMMs with concave, increasing potential functions satisfy the same axioms as proper scoring rules in prediction markets, establishing a formal duality.
• Every CFMM implicitly defines a proper scoring rule; for Uniswap, this corresponds to a known, unusual scoring rule.
• A bidirectional transformation exists between any cost function C in prediction markets and a CFMM potential φ, preserving trade validity and market properties.
• The construction yields a 1-homogeneous CFMM from the log market scoring rule (LMSR), demonstrating a new market design with curvature-adapting level sets.
• Transaction fees in CFMMs can be modeled via a γ-discounted trade condition, which increases liquidity over time without depleting reserves.
• On-chain verification of trades in CFMMs with implicit φ is feasible by checking the cost function C at scaled reserve levels, enabling practical deployment without closed-form φ.