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84
A dynamic parimutuel market for hedging, wagering, and information aggregation
 In Proceedings of the Fifth ACM Conference on Electronic Commerce (EC’04
, 2004
"... I develop a new mechanism for risk allocation and information speculation called a dynamic parimutuel market (DPM). A DPM acts as hybrid between a parimutuel market and a continuous double auction (CDA), inheriting some of the advantages of both. Like a parimutuel market, a DPM offers infinite bu ..."
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Cited by 34 (7 self)
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I develop a new mechanism for risk allocation and information speculation called a dynamic parimutuel market (DPM). A DPM acts as hybrid between a parimutuel market and a continuous double auction (CDA), inheriting some of the advantages of both. Like a parimutuel market, a DPM offers infinite buyin liquidity and zero risk for the market institution; like a CDA, a DPM can continuously react to new information, dynamically incorporate information into prices, and allow traders to lock in gains or limit losses by selling prior to event resolution. The trader interface can be designed to mimic the familiar double auction format with bidask queues, though with an addition variable called the payoff per share. The DPM price function can be viewed as an automated market maker always offering to sell at some price, and moving the price appropriately according to demand. Since the mechanism is parimutuel (i.e., redistributive), it is guaranteed to pay out exactly the amount of money taken in. I explore a number of variations on the basic DPM, analyzing the properties of each, and solving in closed form for their respective price functions.
Betting BooleanStyle: A Framework for Trading in Securities Based on Logical Formulas
, 2003
"... We develop a framework for trading in compound securities: financial instruments that pay off contingent on the outcomes of arbitrary statements in propositional logic. Buying or selling securities  which can be thought of as betting on or against a particular future outcome  allows agents both ..."
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Cited by 30 (17 self)
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We develop a framework for trading in compound securities: financial instruments that pay off contingent on the outcomes of arbitrary statements in propositional logic. Buying or selling securities  which can be thought of as betting on or against a particular future outcome  allows agents both to hedge risk and to profit (in expectation) on subjective predictions. A compound securities market allows agents to place bets on arbitrary boolean combinations of events, enabling them to more closely achieve their optimal risk exposure, and enabling the market as a whole to more closely achieve the social optimum. The tradeoff for allowing such expressivity is in the complexity of the agents' and auctioneer's optimization problems.
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 30 (17 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.
A new understanding of prediction markets via noregret learning
 In ACM EC
, 2010
"... We explore the striking mathematical connections that exist between market scoring rules, cost function based prediction markets, and noregret learning. We first show that any cost function based prediction market can be interpreted as an algorithm for the commonly studied problem of learning from ..."
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Cited by 30 (10 self)
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We explore the striking mathematical connections that exist between market scoring rules, cost function based prediction markets, and noregret learning. We first show that any cost function based prediction market can be interpreted as an algorithm for the commonly studied problem of learning from expert advice by equating the set of outcomes on which bets are placed in the market with the set of experts in the learning setting, and equating trades made in the market with losses observed by the learning algorithm. If the loss of the market organizer is bounded, this bound can be used to derive an O ( √ T) regret bound for the corresponding learning algorithm. We then show that the class of markets with convex cost functions exactly corresponds to the class of Follow the Regularized Leader learning algorithms, with the choice of a cost function in the market corresponding to the choice of a regularizer in the learning problem. Finally, we show an equivalence between market scoring rules and prediction markets with convex cost functions. This implies both that any market scoring rule can be implemented as a cost function based market maker, and that market scoring rules can be interpreted naturally as Follow the Regularized Leader algorithms. These connections provide new insight into how it is that commonly studied markets, such as the Logarithmic Market Scoring Rule, can aggregate opinions into accurate estimates of the likelihood of future events.
Betting on permutations
 In ACM Conference on Electronic Commerce
, 2007
"... We consider a permutation betting scenario, where people wager on the final ordering of n candidates: for example, the outcome of a horse race. We examine the auctioneer problem of risklessly matching up wagers or, equivalently, finding arbitrage opportunities among the proposed wagers. Requiring bi ..."
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Cited by 27 (19 self)
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We consider a permutation betting scenario, where people wager on the final ordering of n candidates: for example, the outcome of a horse race. We examine the auctioneer problem of risklessly matching up wagers or, equivalently, finding arbitrage opportunities among the proposed wagers. Requiring bidders to explicitly list the orderings that they’d like to bet on is both unnatural and intractable, because the number of orderings is n! and the number of subsets of orderings is 2 n!. We propose two expressive betting languages that seem natural for bidders, and examine the computational complexity of the auctioneer problem in each case. Subset betting allows traders to bet either that a candidate will end up ranked among some subset of positions in the final ordering, for example, “horse A will finish in positions 4, 9, or 1321”, or that a position will be taken by some subset of candidates, for example “horse A, B, or D will finish in position 2”. For subset betting, we show that the auctioneer problem can be solved in polynomial time if orders are divisible. Pair betting allows traders to bet on whether one candidate will end up ranked higher than another candidate, for example “horse A will beat horse B”. We prove that the auctioneer problem becomes NPhard for pair betting. We identify a sufficient condition for the existence of a pair betting match that can be verified in polynomial time. We also show that a natural greedy algorithm gives a poor approximation for indivisible orders.
Computation in a Distributed Information Market
, 2003
"... According to economic theory, supported by empirical and laboratory evidence, the equilibrium price of a financial security reflects all of the information regarding the security's value. We investigate the dynamics of the computational process on the path toward equilibrium, where information dis ..."
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Cited by 22 (4 self)
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According to economic theory, supported by empirical and laboratory evidence, the equilibrium price of a financial security reflects all of the information regarding the security's value. We investigate the dynamics of the computational process on the path toward equilibrium, where information distributed among traders is revealed stepby step over time and incorporated into the market price. We develop a simplified model of an information market, along with trading strategies, in order to formalize the computational properties of the process. We show that securities whose payoffs cannot be expressed as a weighted threshold function of distributed input bits are not guaranteed to converge to the proper equilibrium predicted by economic theory. On the other hand, securities whose payoffs are threshold functions are guaranteed to converge, for all prior probability distributions. Moreover, these threshold securities converge in at most n rounds, where n is the number of bits of distributed information. We also prove a lower bound, showing a type of threshold security that requires at least n/2 rounds to converge in the worst case.
Gaming Prediction Markets: Equilibrium Strategies with a Market Maker
 ALGORITHMICA
, 2008
"... We study the equilibrium behavior of informed traders interacting with market scoring rule (MSR) market makers. One attractive feature of MSR is that it is myopically incentive compatible: it is optimal for traders to report their true beliefs about the likelihood of an event outcome provided that ..."
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Cited by 21 (10 self)
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We study the equilibrium behavior of informed traders interacting with market scoring rule (MSR) market makers. One attractive feature of MSR is that it is myopically incentive compatible: it is optimal for traders to report their true beliefs about the likelihood of an event outcome provided that they ignore the impact of their reports on the profit they might garner from future trades. In this paper, we analyze nonmyopic strategies and examine what information structures lead to truthful betting by traders. Specifically, we analyze the behavior of riskneutral traders with incomplete information playing in a dynamic game. We consider finitestage and infinitestage game models. For each model, we study the logarithmic market scoring rule (LMSR) with two different information structures: conditionally independent signals and (unconditionally) independent signals. In the finitestage model, when signals of traders are independent conditional on the state of the world, truthful betting is a Perfect Bayesian Equilibrium (PBE). Moreover, it is the unique Weak Perfect Bayesian Equilibrium (WPBE) of the game. In contrast, when signals of traders are unconditionally independent, truthful betting
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 20 (15 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.
The influence limiter: Provably manipulationresistant recommender systems
 In To appear in Proceedings of the ACM Recommender Systems Conference (RecSys07
, 2007
"... This appendix should be read in conjunction with the article by Resnick and Sami [1]. Here, we include the proofs that were omitted from the main article due to shortage of space. A.1 Lemma 5 Lemma 5: For the quadratic scoring rule (MSE) loss, for all q,u ∈ [0,1], GF(qu) ≥ D(qu) 2. Proof of Lem ..."
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Cited by 19 (8 self)
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This appendix should be read in conjunction with the article by Resnick and Sami [1]. Here, we include the proofs that were omitted from the main article due to shortage of space. A.1 Lemma 5 Lemma 5: For the quadratic scoring rule (MSE) loss, for all q,u ∈ [0,1], GF(qu) ≥ D(qu) 2. Proof of Lemma 5: Because both D(qu) = D(1 − q1 − u) and GF(qu) = GF(1 − q1 − u), we can assume u ≥ q without loss of generality. Keeping q fixed, we want to show that the result holds for all u. Note that D(qq) = GF(qq) = 0. Thus, differentiating with respect to u, it is sufficient to prove that GF ′ (qu) ≥ D ′ (qu)/2 for all u ≥ q,u ≤ 1. We change variables by setting y = u − q. We use the notation D ′ (y) to denote D ′ (qu)u=q+y, treating q as fixed and implicit. Likewise, we use the notation GF ′ (y). For brevity, we use q to denote (1 − q). D(qu) = q[(q − y) 2 − q 2]+q[(q+y) 2 − q 2] = q[y 2 − 2yq]+q[y 2 + 2qy] = y 2 ⇒ D ′ (y) = 2y 1 GF(qu) = qlog(1+y 2 − 2qy)+qlog(1+y 2 + 2qy)