Efficiency of Constant Log Utility Market Makers

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.