A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries
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Combinatorial Information Market Design
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Betting boolean-style: a framework for trading in securities based on logical formulas
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A dynamic pari-mutuel market for hedging, wagering, and information aggregation
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Pricing combinatorial markets for tournaments
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Pari-mutuel markets: mechanisms and performance
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An optimization-based framework for automated market-making
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A preliminary study on EDAs for permutation problems based on marginal-based models
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Fourier-information duality in the identity management problem
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A tractable combinatorial market maker using constraint generation
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Efficient rank aggregation using partial data
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We focus on a permutation betting market under parimutuel call auction model where traders bet on final rankings of n candidates. We present a Proportional Betting mechanism for this market. Our mechanism allows traders to bet on any subset of the n 2 `candidate-rank' pairs, and rewards them proportionally to the number of pairs that appear in the final outcome. We show that market organizer's decision problem for this mechanism can be formulated as a convex program of polynomial size. Further, the formulation yields a set of n 2 unique marginal prices that are sufficient to price the bets in this mechanism, and are computable in polynomial-time. These marginal prices reflect the traders' beliefs about the marginal distributions over outcomes. More importantly, we propose techniques to compute the joint distribution over n ! permutations from these marginal distributions. We show that using a maximum entropy criterion, we can obtain a concise parametric form (with only n 2 parameters) for the joint distribution which is defined over an exponentially large state space. We then present an approximation algorithm for computing the parameters of this distribution. In fact, our algorithm addresses a generic problem of finding the maximum entropy distribution over permutations that has a given mean, and is of independent interest.