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.

Current automated market makers over binary events suffer from two problems that make them impractical. First, they are unable to adapt to liquidity, so trades cause prices to move the same amount in both thick and thin markets. Second, under normal circumstances, the market maker runs at a deficit. In this paper, we construct a market maker that is both sensitive to liquidity and can run at a profit. Our market maker has bounded loss for any initial level of liquidity and, as the initial level of liquidity approaches zero, worstcase loss approaches zero. For any level of initial liquidity we can establish a boundary in market state space such that, if the market terminates within that boundary, the market maker books a profit regardless of the realized outcome. Furthermore, we provide guidance as to how our market maker can be implemented over very large event spaces through a novel cost-function-based sampling method.

� 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. Mechanism design has been described as “inverse game theory. ” Whereas game theorists ask what outcome results from a game, mechanism designers ask what game produces a desired outcome. In this sense, game theorists act like scientists and mechanism designers like engineers. In this article, we survey a number of mechanisms created to elicit predictions, many newly proposed within the last decade. We focus on the engineering questions: How do they work and why? What factors and goals are most important in their

Recent years have seen extensive investigation of the information aggregation properties of prediction markets. However, relatively little is known about conditions under which a market will aggregate the private information of rational risk averse traders who optimize their portfolios over time. We consider a market model involving finitely many informed risk-averse traders interacting with a market maker. Our main result identifies a basic smoothness condition on the price in the market that ensures information will be aggregated. We give conditions under which cost function market makers (or, equivalently, market makers based on market scoring rules) satisfy the smoothness requirement. We further show that regardless of the level of risk aversion of the traders, the final allocation and prices together constitute a competitive equilibrium; thus, in particular, the final portfolios of the traders are ex post Pareto efficient. 1.

We study information revelation in scoring rule and prediction market mechanisms in settings in which traders have conflicting incentives due to opportunities to profit from the market operator’s subsequent actions. In our canonical model, an agent Alice is offered an incentive-compatible scoring rule to reveal her beliefs about a future event, but can also profit from misleading another trader Bob about her information and then making money off Bob’s error in a subsequent market. We show that, in any weak Perfect Bayesian Equilibrium of this sequence of two markets, Alice and Bob earn payoffs that are consistent with a minimax strategy of a related game. We can then characterize the equilibria in terms of an information channel: the outcome of the first scoring rule is as if Alice had only observed a noisy version of her initial signal, with the degree of noise indicating the adverse effect of the second market on the first. We provide a partial constructive characterization of when this channel will be noiseless. We show that our results on the canonical model yield insights into other settings of information extraction with conflicting incentives.

Summary. Automated market makers are algorithmic agents that provide liquidity in electronic markets. We construct two new automated market makers that each solve an open problem of theoretical and practical interest. First, we formulate a market maker that has bounded loss over separable measure spaces. This opens up an exciting new set of domains for prediction markets, including markets on locations and markets where events correspond to the natural numbers. Second, by shifting profits into liquidity, we create a market maker that has bounded loss in addition to a bid/ask spread that gets arbitrarily small with trading volume. This market maker matches important attributes of real human market makers and suggests a path forward for integrating automated market making agents into markets with real money. 1.1

Options are a basic, widely-traded form of financial derivative that offer payouts based on the future price of an underlying asset. The finance literature gives us option-trading algorithms that take into consideration information about how prices move over time but do not explicitly involve the trades the agent made in the past. In contrast, the prediction market literature gives us automated market-making agents (like the popular LMSR) that are event-independent and price trades based only on the inventories the agent holds. We simulate the performance of five trading agents inspired by these literatures on a large database of recent historical option prices. We find that a combination of the two approaches produced the best results in our experiments: a trading agent that keeps track of previously-made trades combined with a good prior distribution on how prices move over time. The experimental success of this synthesized trader has implications for agent design in both financial and prediction markets. 1

Abstract. In this paper we study the design and characterization of prediction markets in the presence of traders with unknown risk-aversion. We formulate a series of desirable properties for any “market-like ” forecasting mechanism. We present a randomized mechanism that satisfies all these properties while guaranteeing that it is myopically optimal for each trader to trade honestly, regardless of her degree of risk aversion. We observe, however, that the mechanism has an undesirable side effect: the traders ’ expected reward, normalized against the inherent value of their private information, decreases exponentially with the number of traders. We prove that this is unavoidable: any mechanism that is myopically strategyproof for traders of all risk types, while also satisfying other natural properties of “market-like ” mechanisms, must sometimes result in a player getting an exponentially small normalized expected reward. 1 Introduction and Related Work Prediction markets are markets designed and deployed to aggregate information about future events by having agents with private beliefs trade in these markets. One market

Abstract. Most existing market maker mechanisms for prediction markets are designed for events with a finite number of outcomes. All known attempts on designing market makers for forecasting continuous-outcome events resulted in mechanisms with undesirable properties. In this paper, we take an axiomatic approach to study whether it is possible for continuous-outcome market makers to satisfy certain desirable properties simultaneously. We define a general class of continuous-outcome market makers, which allows traders to express their information on any continuous subspace of their choice. We characterize desirable properties of these market makers using formal axioms. Our main result is an impossibility theorem showing that if a market maker offers binary-payoff contracts, either the market maker has unbounded worst case loss or the contract prices will stop being responsive, making future trades no longer profitable. In addition, we analyze a mechanism that does not belong to our framework. This mechanism has a worst case loss linear in the number of submitted orders, but encourages some undesirable strategic behavior.

Automated market makers are algorithmic agents that price fixed-odds bets with traders. Four key qualities for automated market makers have appeared in the artificial intelligence literature: (1) bounded loss, (2) the ability to make a profit, (3) a vanishing bid/ask spread, and (4) unlimited market depth. Intriguingly, market makers that satisfy any three of these desiderata have appeared in the literature. However, it is an open question as to whether there exist market makers which can simultaneously satisfy all four of these properties; the issue is not simple to resolve because several of the qualities are oppositional, particularly in tandem. In this paper, we answer the open question in the affirmative by constructing market makers that satisfy all four qualities.