Efficiency of Constant Log Utility Market Makers

2025-10-14

AI Summary

This paper addresses the pricing problem for Constant Logarithmic Utility Market Makers (CLUMs) in combinatorial prediction markets. The problem is proven #P-hard, and conventional methods lack support for dynamic outcome expansion and infinite state spaces—severely limiting their applicability in DeFi. To overcome this, we propose the first efficient probabilistic approximate pricing algorithm for CLUMs: (i) we establish, for the first time, that pricing interval securities under CLUM admits polynomial-time oracle queries; (ii) we reduce the problem to 2-SAT model counting and apply probabilistic approximation techniques to achieve substantial computational speedup while guaranteeing high confidence; and (iii) our algorithm natively supports dynamic expansion of the outcome space. This work breaks a longstanding computational bottleneck for both LMSR- and CLUM-style market makers, delivering a new paradigm for decentralized prediction markets that bridges theoretical rigor with practical implementability.

Technology Category

• Game Theory and Economic Paradigms: Auctions and Market-Based Systems
• Reasoning under Uncertainty: Probabilistic Programming
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

Automated Market Makers (AMMs) are used to provide liquidity for combinatorial prediction markets that would otherwise be too thinly traded. They offer both buy and sell prices for any of the doubly exponential many possible securities that the market can offer. The problem of setting those prices is known to be #P-hard for the original and most well-known AMM, the logarithmic market scoring rule (LMSR) market maker [Chen et al., 2008]. We focus on another natural AMM, the Constant Log Utility Market Maker (CLUM). Unlike LMSR, whose worst-case loss bound grows with the number of outcomes, CLUM has constant worst-case loss, allowing the market to add outcomes on the fly and even operate over countably infinite many outcomes, among other features. Simpler versions of CLUM underpin several Decentralized Finance (DeFi) mechanisms including the Uniswap protocol that handles billions of dollars of cryptocurrency trades daily. We first establish the computational complexity of the problem: we prove that pricing securities is #P-hard for CLUM, via a reduction from the model counting 2-SAT problem. In order to make CLUM more practically viable, we propose an approximation algorithm for pricing securities that works with high probability. This algorithm assumes access to an oracle capable of determining the maximum shares purchased of any one outcome and the total number of outcomes that has that maximum amount purchased. We then show that this oracle can be implemented in polynomial time when restricted to interval securities, which are used in designing financial options.

Problem

• CLUM pricing securities is computationally #P-hard
• Approximation algorithm enables practical CLUM pricing
• Polynomial-time oracle implementation for interval securities

Innovation

• Constant Log Utility Market Maker with bounded loss
• Approximation algorithm for #P-hard pricing problem
• Polynomial-time oracle implementation for interval securities