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Abstract: We analyze the computational complexity of market maker pricing algorithmsfor combinatorial prediction markets. We focus on Hanson-s popular logarithmicmarket scoring rule market maker LMSR. Our goal is to implicitly maintaincorrect LMSR prices across an exponentially large outcome space. We examineboth permutation combinatorics, where outcomes are permutations of objects, andBoolean combinatorics, where outcomes are combinations of binary events. Welook at three restrictive languages that limit what traders can bet on. Evenwith severely limited languages, we find that LMSR pricing is $\SP$-hard, evenwhen the same language admits polynomial-time matching without the marketmaker. We then propose an approximation technique for pricing permutationmarkets based on a recent algorithm for online permutation learning. Theconnections we draw between LMSR pricing and the vast literature on onlinelearning with expert advice may be of independent interest.



Author: Yiling Chen, Lance Fortnow, Nicolas Lambert, David M. Pennock, Jennifer Wortman

Source: https://arxiv.org/



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