Counting linear extensions is #P-complete
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
PP is as hard as the polynomial-time hierarchy
SIAM Journal on Computing
The weighted majority algorithm
Information and Computation
A decision-theoretic generalization of on-line learning and an application to boosting
Journal of Computer and System Sciences - Special issue: 26th annual ACM symposium on the theory of computing & STOC'94, May 23–25, 1994, and second annual Europe an conference on computational learning theory (EuroCOLT'95), March 13–15, 1995
A game of prediction with expert advice
Journal of Computer and System Sciences - Special issue on the eighth annual workshop on computational learning theory, July 5–8, 1995
Combinatorial Information Market Design
Information Systems Frontiers
Betting Boolean-style: a framework for trading in securities based on logical formulas
Decision Support Systems - Special issue: The fourth ACM conference on electronic commerce
Prediction, Learning, and Games
Prediction, Learning, and Games
Proceedings of the 8th ACM conference on Electronic commerce
ACM SIGecom Exchanges
Pricing combinatorial markets for tournaments
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Learning permutations with exponential weights
COLT'07 Proceedings of the 20th annual conference on Learning theory
Bluffing and strategic reticence in prediction markets
WINE'07 Proceedings of the 3rd international conference on Internet and network economics
Computational challenges in e-commerce
Communications of the ACM - Rural engineering development
Combinatorial prediction markets for event hierarchies
Proceedings of The 8th International Conference on Autonomous Agents and Multiagent Systems - Volume 1
WINE '09 Proceedings of the 5th International Workshop on Internet and Network Economics
Learning Permutations with Exponential Weights
The Journal of Machine Learning Research
Prediction markets, mechanism design, and cooperative game theory
UAI '09 Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence
A new understanding of prediction markets via no-regret learning
Proceedings of the 11th ACM conference on Electronic commerce
Automated market-making in the large: the gates hillman prediction market
Proceedings of the 11th ACM conference on Electronic commerce
An axiomatic characterization of continuous-outcome market makers
WINE'10 Proceedings of the 6th international conference on Internet and network economics
Connections between markets and learning
ACM SIGecom Exchanges
An optimization-based framework for automated market-making
Proceedings of the 12th ACM conference on Electronic commerce
Liquidity-sensitive automated market makers via homogeneous risk measures
WINE'11 Proceedings of the 7th international conference on Internet and Network Economics
A tractable combinatorial market maker using constraint generation
Proceedings of the 13th ACM Conference on Electronic Commerce
An efficient Monte-Carlo algorithm for pricing combinatorial prediction markets for tournaments
IJCAI'11 Proceedings of the Twenty-Second international joint conference on Artificial Intelligence - Volume Volume One
Efficient Market Making via Convex Optimization, and a Connection to Online Learning
ACM Transactions on Economics and Computation - Special Issue on Algorithmic Game Theory
A combinatorial prediction market for the U.S. elections
Proceedings of the fourteenth ACM conference on Electronic commerce
A Practical Liquidity-Sensitive Automated Market Maker
ACM Transactions on Economics and Computation
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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 an algorithm for online permutation learning. The connections we draw between LMSR pricing and the literature on online learning with expert advice may be of independent interest.