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 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

Abstract. A potential downside of prediction markets is that they may incentivize agents to take undesirable actions in the real world. For example, a prediction market for whether a terrorist attack will happen may incentivize terrorism, and an in-house prediction market for whether a product will be successfully released may incentivize sabotage. In this paper, we study principal-aligned prediction mechanisms– mechanisms that do not incentivize undesirable actions. We characterize all principal-aligned proper scoring rules, and we show an “overpayment” result, which roughly states that with n agents, any prediction mechanism that is principal-aligned will, in the worst case, require the principal to pay Θ(n) times as much as a mechanism that is not. We extend our model to allow uncertainties about the principal’s utility and restrictions on agents ’ actions, showing a richer characterization and a similar “overpayment ” result.

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.

We study principles and methods for task routing that aim to harness people’s abilities to jointly contribute to a task and to route tasks to others who can provide further contributions. In the particular context of prediction tasks, the goal is to efficiently obtain accurate probability assessments for an event of interest. We introduce routing scoring rules for promoting collaborative behavior, that bring truthfully contributing information and optimally routing tasks into a Perfect Bayesian Equilibrium under common knowledge about agents ’ abilities. However, for networks where agents only have local knowledge about other agents ’ abilities, optimal routing requires complex reasoning over the history and future routing decisions of users outside of local neighborhoods. Avoiding this, we introduce a class of local routing rules that isolate simple routing decisions in equilibrium, while still promoting effective routing decisions. We present simulation results that show that following routing decisions induced by local routing rules lead to efficient information aggregation.

Abstract 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 non-myopic strategies and examine what information structures lead to truthful betting by traders. Specifically, we analyze the behavior of risk-neutral traders with incomplete information playing in a dynamic game. We consider finite-stage and infinite-stage 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 finite-stage 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

First and foremost, to my advisor, Michael Kearns. At the risk of sounding cliche, Michael has been a near ideal advisor to me. He helped me cultivate a taste in research problems, taught me how to recognize and design good models and algorithms, and was a constant source of invaluable advice on navigating the academic world. Perhaps most importantly, he helped me to develop confidence in myself as a researcher — no easy feat, I’m sure. I feel truly lucky to have had the chance to learn so much from him at the start of my career, and I sincerely hope that we will remain both collaborators and friends for many years to come. To my thesis committee, Sanjeev Khanna, Yishay Mansour, Fernando Pereira, and Ben Taskar, for their feedback and advice, and for being so supportive of me and of this work. To Kevin Leyton-Brown, Eugene Nudelman, Yoav Shoham, and the rest of the Multiagent Group at Stanford circa 2003, for giving me my first opportunity to get involved in research. It was because of Kevin’s encouragement and contagious enthusiasm for research that I decided to go on for the Ph.D., so the credit (or blame) for me being here in the first place should go to him. To Eyal Even-Dar and (again) Yishay Mansour, for acting as informal mentors and always giving me valuable career advice, whether I wanted to hear it or not. To my other collaborators, colleagues, and teachers, from whom I have learned so much. I am

Abstract. We study the strategic behavior of risk-neutral non-myopic agents in Dynamic Parimutuel Markets (DPM). In a DPM, agents buy or sell shares of contracts, whose future payoff in a particular state depends on aggregated trades of all agents. A forward-looking agent hence takes into consideration of possible future trades of other agents when making its trading decision. In this paper, we analyze non-myopic strategies in a two-outcome DPM under a simple model of incomplete information and examine whether an agent will truthfully reveal its information in the market. Specifically, we first characterize a single agent’s optimal trading strategy given the payoff uncertainty. Then, we use a two-player game to examine whether an agent will truthfully reveal its information when it only participates in the market once. We prove that truthful betting is a Nash equilibrium of the two-stage game in our simple setting for uniform initial market probabilities. However, we show that there exists some initial market probabilities at which the first player has incentives to mislead the other agent in the two-stage game. Finally, we briefly discuss when an agent can participate more than once in the market whether it will truthfully reveal its information at its first play in a three-stage game. We find that in some occasions truthful betting is not a Nash equilibrium of the three-stage game even for uniform initial market probabilities. 1