I develop a new mechanism for risk allocation and information speculation called a dynamic pari-mutuel market (DPM). A DPM acts as hybrid between a pari-mutuel market and a continuous double auction (CDA), inheriting some of the advantages of both. Like a pari-mutuel market, a DPM offers infinite buy-in 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 bid-ask 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 pari-mutuel (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.

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.

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.

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.

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 13-21”, 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 NP-hard 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.

We explore the striking mathematical connections that exist between market scoring rules, cost function based prediction markets, and no-regret 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.

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(q||u) ≥ D(q||u) 2. Proof of Lemma 5: Because both D(q||u) = D(1 − q||1 − u) and GF(q||u) = GF(1 − q||1 − 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(q||q) = GF(q||q) = 0. Thus, differentiating with respect to u, it is sufficient to prove that GF ′ (q||u) ≥ D ′ (q||u)/2 for all u ≥ q,u ≤ 1. We change variables by setting y = u − q. We use the notation D ′ (y) to denote D ′ (q||u)|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(q||u) = 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(q||u) = qlog(1+y 2 − 2qy)+qlog(1+y 2 + 2qy)

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 single-elimination tournaments. With n competing teams, the outcome space is of size 2 n−1. We show that the general pricing problem for tournaments is #P-hard. We derive a polynomial-time 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.

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—weighted-score wagering—that satisfies all of the properties. Moreover, we show that a single-parameter generalization of weighted-score 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

Future market conditions can be a pivotal factor in making business decisions. We present and evaluate methods used by our agent, Deep Maize, to forecast market prices in the Trading Agent Competition Supply Chain Management Game. As a guiding principle we seek to exploit as many sources of available information as possible to inform predictions. Since information comes in several different forms, we integrate well-known methods in a novel way to make predictions. The core of our predictor is a nearest-neighbors machine learning algorithm that identifies historical instances with similar economic indicators. We augment this with an online learning procedure that transforms the predictions by optimizing a scoring rule. This allows us to select more relevant historical contexts using additional information available during individual games. We also explore the advantages of two different representations for predicting price distributions. One uses absolute prices, and the other uses statistics of price time series to exploit local stability. We evaluate these methods using both data from the 2005 tournament final round and additional simulations. We compare several variations of our predictor to one another and a baseline predictor similar to those used by many other tournament agents. We show substantial improvements over the baseline predictor, and demonstrate that each element of our predictor contributes to improved performance.