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32
A utility framework for boundedloss market makers
 In Proceedings of the 23rd Conference on Uncertainty in Artificial Intelligence
, 2007
"... We introduce a class of utilitybased market makers that always accept orders at their riskneutral prices. We derive necessary and sufficient conditions for such market makers to have bounded loss. We prove that hyperbolic absolute risk aversion utility market makers are equivalent to weighted pseu ..."
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Cited by 70 (26 self)
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We introduce a class of utilitybased market makers that always accept orders at their riskneutral prices. We derive necessary and sufficient conditions for such market makers to have bounded loss. We prove that hyperbolic absolute risk aversion utility market makers are equivalent to weighted pseudospherical scoring rule market makers. In particular, Hanson’s logarithmic scoring rule market maker corresponds to a negative exponential utility market maker in our framework. We describe a third equivalent formulation based on maintaining a cost function that seems most natural for implementation purposes, and we illustrate how to translate among the three equivalent formulations. We examine the tradeoff between the market’s liquidity and the market maker’s worstcase loss. For a fixed bound on worstcase loss, some market makers exhibit greater liquidity near uniform prices and some exhibit greater liquidity near extreme prices, but no market maker can exhibit uniformly greater liquidity in all regimes. For a fixed minimum liquidity level, we give the lower bound of market maker’s worstcase loss under some regularity conditions. 1
Complexity of Combinatorial Market Makers ∗
"... 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 ex ..."
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Cited by 36 (18 self)
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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 #Phard, even when the same language admits polynomialtime 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.
Pricing combinatorial markets for tournaments
 In Proc. of STOC
, 2008
"... In a prediction market, agents trade assets whose value is tied to a future event, for example the outcome of the next presidential election. Asset prices determine a probability distribution over the set of possible outcomes. Typically, the outcome space is small, allowing agents to directly trade ..."
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Cited by 25 (18 self)
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In a prediction market, agents trade assets whose value is tied to a future event, for example the outcome of the next presidential election. Asset prices determine a probability distribution over the set of possible outcomes. Typically, the outcome space is small, allowing agents to directly trade in each outcome, and allowing a market maker to explicitly update asset prices. Combinatorial markets, in contrast, work to estimate a full joint distribution of dependent observations, in which case the outcome space grows exponentially. In this paper, we consider the problem of pricing combinatorial markets for singleelimination tournaments. With n competing teams, the outcome space is of size 2 n−1. We show that the general pricing problem for tournaments is #Phard. We derive a polynomialtime algorithm for a restricted betting language based on a Bayesian network representation of the probability distribution. The language is fairly natural in the context of tournaments, allowing for example bets of the form “team i wins game k”. We believe that our betting language is the first for combinatorial market makers that is both useful and tractable. We briefly discuss a heuristic approximation technique for the general case.
Inferring rankings under constrained sensing
"... Motivated by applications like elections, webpage ranking, revenue maximization etc., we consider the question of inferring popular rankings using constrained data. More specifically, we consider the problem of inferring a probability distribution over the group of permutations using its first orde ..."
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Cited by 25 (8 self)
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Motivated by applications like elections, webpage ranking, revenue maximization etc., we consider the question of inferring popular rankings using constrained data. More specifically, we consider the problem of inferring a probability distribution over the group of permutations using its first order marginals. We first prove that it is not possible to recover more than O(n) permutations over n elements with the given information. We then provide a simple and novel algorithm that can recover up to O(n) permutations under a natural stochastic model; in this sense, the algorithm is optimal. In certain applications, the interest is in recovering only the most popular (or mode) ranking. As a second result, we provide an algorithm based on the Fourier Transform over the symmetric group to recover the mode under a natural majority condition; the algorithm turns out to be a maximum weight matching on an appropriately defined weighted bipartite graph. The questions considered are also thematically related to Fourier Transforms over the symmetric group and the currently popular topic of compressed sensing. 1
An OptimizationBased Framework for Automated MarketMaking
 EC'11
, 2011
"... We propose a general framework for the design of securities markets over combinatorial or infinite state or outcome spaces. The framework enables the design of computationally efficient markets tailored to an arbitrary, yet relatively small, space of securities with bounded payoff. We prove that any ..."
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Cited by 23 (11 self)
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We propose a general framework for the design of securities markets over combinatorial or infinite state or outcome spaces. The framework enables the design of computationally efficient markets tailored to an arbitrary, yet relatively small, space of securities with bounded payoff. We prove that any market satisfying a set of intuitive conditions must price securities via a convex cost function, which is constructed via conjugate duality. Rather than deal with an exponentially large or infinite outcome space directly, our framework only requires optimization over a convex hull. By reducing the problem of automated market making to convex optimization, where many efficient algorithms exist, we arrive at a range of new polynomialtime pricing mechanisms for various problems. We demonstrate the advantages of this framework with the design of some particular markets. We also show that by relaxing the convex hull we can gain computational tractability without compromising the market institution’s bounded budget.
Designing Markets for Prediction
, 2010
"... We survey the literature on prediction mechanisms, including prediction markets and peer prediction systems. We pay particular attention to the design process, highlighting the objectives and properties that are important in the design of good prediction mechanisms. ..."
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Cited by 20 (3 self)
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We survey the literature on prediction mechanisms, including prediction markets and peer prediction systems. We pay particular attention to the design process, highlighting the objectives and properties that are important in the design of good prediction mechanisms.
Efficient market making via convex optimization, and a connection to online learning
 ACM Transactions on Economics and Computation. To Appear
, 2012
"... We propose a general framework for the design of securities markets over combinatorial or infinite state or outcome spaces. The framework enables the design of computationally efficient markets tailored to an arbitrary, yet relatively small, space of securities with bounded payoff. We prove that any ..."
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Cited by 19 (9 self)
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We propose a general framework for the design of securities markets over combinatorial or infinite state or outcome spaces. The framework enables the design of computationally efficient markets tailored to an arbitrary, yet relatively small, space of securities with bounded payoff. We prove that any market satisfying a set of intuitive conditions must price securities via a convex cost function, which is constructed via conjugate duality. Rather than deal with an exponentially large or infinite outcome space directly, our framework only requires optimization over a convex hull. By reducing the problem of automated market making to convex optimization, where many efficient algorithms exist, we arrive at a range of new polynomialtime pricing mechanisms for various problems. We demonstrate the advantages of this framework with the design of some particular markets. We also show that by relaxing the convex hull we can gain computational tractability without compromising the market institution’s bounded budget. Although our framework was designed with the goal of deriving efficient automated market makers for markets with very large outcome spaces, this framework also provides new insights into the relationship between market design and machine learning, and into the complete market setting. Using our framework, we illustrate the mathematical parallels between cost function based markets and online learning and establish a correspondence between cost function based markets and market scoring rules for complete markets. 1
Selffinanced wagering mechanisms for forecasting
 EC
"... We examine a class of wagering mechanisms designed to elicit truthful predictions from a group of people without requiring any outside subsidy. We propose a number of desirable properties for wagering mechanisms, identifying one mechanism—weightedscore wagering—that satisfies all of the properties. ..."
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Cited by 14 (7 self)
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We examine a class of wagering mechanisms designed to elicit truthful predictions from a group of people without requiring any outside subsidy. We propose a number of desirable properties for wagering mechanisms, identifying one mechanism—weightedscore wagering—that satisfies all of the properties. Moreover, we show that a singleparameter generalization of weightedscore wagering is the only mechanism that satisfies these properties. We explore some variants of the core mechanism based on practical considerations. Categories and Subject Descriptors
Betting on the Real Line
"... Abstract. We study the problem of designing prediction markets for random variables with continuous or countably infinite outcomes on the real line. Our interval betting languages allow traders to bet on any interval of their choice. Both the call market mechanism and two automated market maker mech ..."
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Cited by 10 (8 self)
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Abstract. We study the problem of designing prediction markets for random variables with continuous or countably infinite outcomes on the real line. Our interval betting languages allow traders to bet on any interval of their choice. Both the call market mechanism and two automated market maker mechanisms, logarithmic market scoring rule (LMSR) and dynamic parimutuel markets (DPM), are generalized to handle interval bets on continuous or countably infinite outcomes. We examine problems associated with operating these markets. We show that the auctioneer’s order matching problem for interval bets can be solved in polynomial time for call markets. DPM can be generalized to deal with interval bets on both countably infinite and continuous outcomes and remains to have bounded loss. However, in a continuousoutcome DPM, a trader may incur loss even if the true outcome is within her betting interval. The LMSR market maker suffers from unbounded loss for both countably infinite and continuous outcomes.
Parimutuel betting on permutations
 In International Workshop on Internet and Network Economics
, 2008
"... We focus on a permutation betting market under parimutuel call auction model where traders bet on the final ranking of n candidates. We present a Proportional Betting mechanism for this market. Our mechanism allows the traders to bet on any subset of the n 2 ‘candidaterank ’ pairs, and rewards them ..."
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Cited by 10 (0 self)
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We focus on a permutation betting market under parimutuel call auction model where traders bet on the final ranking of n candidates. We present a Proportional Betting mechanism for this market. Our mechanism allows the traders to bet on any subset of the n 2 ‘candidaterank ’ 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. More importantly, 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 polynomialtime. The marginal prices reflect the traders ’ beliefs about the marginal distributions over outcomes. We also 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, the algorithm addresses the generic problem of finding the maximum entropy distribution over permutations that has a given mean, and may be of independent interest. 1