## Complexity of Combinatorial Market Makers ∗

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Citations: | 31 - 17 self |

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@MISC{Chen_complexityof,

author = {Yiling Chen and David M. Pennock and Lance Fortnow and Jennifer Wortman and Nicolas Lambert},

title = {Complexity of Combinatorial Market Makers ∗},

year = {}

}

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### Abstract

We analyze the computational complexity of market maker pricing algorithms for combinatorial prediction markets. We focus on Hanson’s popular logarithmic market scoring rule market maker (LMSR). Our goal is to implicitly maintain correct LMSR prices across an exponentially large outcome space. We examine both permutation combinatorics, where outcomes are permutations of objects, and Boolean combinatorics, where outcomes are combinations of binary events. We look at three restrictive languages that limit what traders can bet on. Even with severely limited languages, we find that LMSR pricing is #P-hard, even when the same language admits polynomial-time matching without the market maker. We then propose an approximation technique for pricing permutation markets based on a recent algorithm for online permutation learning. The connections we draw between LMSR pricing and the vast literature on online learning with expert advice may be of independent interest.

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Citation Context ... On a more positive note, we examine an approximation algorithm for LMSR pricing in permutation markets that makes use of powerful techniques from the literature on online learning with expert advice =-=[3, 19, 12]-=-. We briefly review this online learning setting, and examine the parallels that exist between LMSR pricing and standard algorithms for learning with expert advice. We then show how a recent algorithm... |

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Citation Context ...espects some independence relationships. LMSR is used by a number of companies, including inklingmarkets.com, Microsoft, thewsx.com, and yoonew.com, and is the subject of a number of research studies =-=[7, 15, 8]-=-. In this paper, we analyze the computational complexity of LMSR in several combinatorial betting scenarios. We examine both permutation combinatorics and Boolean combinatorics. We show that both comp... |

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Citation Context ...y dividing traders’ attention among an exponential number of outcomes. A combinatorial matching market—the combinatorial generalization of a standard double auction—may simply fail to find any trades =-=[11, 4, 5]-=-. In contrast, an automated market maker is always willing to trade on every bet at some price. A combinatorial market maker implicitly or explicitly maintains prices across all (exponentially many) o... |

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Citation Context ...nline learning setting, and examine the parallels that exist between LMSR pricing and standard algorithms for learning with expert advice. We then show how a recent algorithm for permutation learning =-=[16]-=- can be transformed into an approximation algorithm for pricing in permutation markets in which the market maker is guaranteed to have bounded loss. 2. RELATED WORK Fortnow et al. [11] study the compu... |

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Citation Context ...y dividing traders’ attention among an exponential number of outcomes. A combinatorial matching market—the combinatorial generalization of a standard double auction—may simply fail to find any trades =-=[11, 4, 5]-=-. In contrast, an automated market maker is always willing to trade on every bet at some price. A combinatorial market maker implicitly or explicitly maintains prices across all (exponentially many) o... |

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Citation Context ...niversity Stanford, CA 94305 1. INTRODUCTION One way to elicit information is to ask people to bet on it. A prediction market is a common forum where people bet with each other or with a market maker =-=[9, 10, 23, 20, 21]-=-. A typical binary prediction market allows bets along one dimension, for example either for or against Hillary Clinton to win the 2008 US Presidential election. Thousands of such one- or small-dimens... |

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Citation Context ... solutions, including running market makers on overlapping subsets of events, allowing traders to synchronize the markets via arbitrage. The work closest to our own is that of Chen, Goel, and Pennock =-=[6]-=-, who study a special case of Boolean combinatorics in which participants bet on how far a team will advance in a single elimination tournament, for example a sports playoff like the NCAA college bask... |

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Citation Context ...espects some independence relationships. LMSR is used by a number of companies, including inklingmarkets.com, Microsoft, thewsx.com, and yoonew.com, and is the subject of a number of research studies =-=[7, 15, 8]-=-. In this paper, we analyze the computational complexity of LMSR in several combinatorial betting scenarios. We examine both permutation combinatorics and Boolean combinatorics. We show that both comp... |

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Citation Context ...espects some independence relationships. LMSR is used by a number of companies, including inklingmarkets.com, Microsoft, thewsx.com, and yoonew.com, and is the subject of a number of research studies =-=[7, 15, 8]-=-. In this paper, we analyze the computational complexity of LMSR in several combinatorial betting scenarios. We examine both permutation combinatorics and Boolean combinatorics. We show that both comp... |

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Citation Context ...which Sinkhorn balancing does not converge in finite time, many results show that the number of Sinkhorn iterations needed to scale a matrix so that row and column sums are 1 ± ǫ is polynomial in 1/ǫ =-=[1, 17, 18]-=-. The following theorem [16] bounds the cumulative loss of the PermELearn in terms of the cumulative loss of the best permutation. Theorem 9. (Helmbold and Warmuth [16]) Let A be the PermELearn algori... |

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Citation Context ...which Sinkhorn balancing does not converge in finite time, many results show that the number of Sinkhorn iterations needed to scale a matrix so that row and column sums are 1 ± ǫ is polynomial in 1/ǫ =-=[1, 17, 18]-=-. The following theorem [16] bounds the cumulative loss of the PermELearn in terms of the cumulative loss of the best permutation. Theorem 9. (Helmbold and Warmuth [16]) Let A be the PermELearn algori... |