We study the problem of eliciting and aggregating probabilistic information from multiple agents. In order to successfully aggregate the predictions of agents, the principal needs to elicit some notion of confidence from agents, capturing how much experience or knowledge led to their predictions. To formalize this, we consider a principal who wishes to learn the distribution of a random variable. A group of Bayesian agents has each privately observed some independent samples of the random variable. The principal wishes to elicit enough information from each agent, so that her posterior is the same as if she had directly received all of the samples herself. Leveraging techniques from Bayesian statistics, we represent confidence as the number of samples an agent has observed, which is quantified by a hyperparameter from a conjugate family of prior distributions. This then allows us to show that if the principal has access to a few samples, she can achieve her aggregation goal by eliciting predictions from agents using proper scoring rules. In particular, with access to one sample, she can successfully aggregate the agents' predictions if and only if every posterior predictive distribution corresponds to a unique value of the hyperparameter, a property which holds for many common distributions of interest. When this uniqueness property does not hold, we construct a novel and intuitive mechanism where a principal with two samples can elicit and optimally aggregate the agents' predictions.
The modern web critically depends on aggregation of information from self-interested agents, for example opinion polls, product ratings, or crowdsourcing. We consider a setting where multiple objects (questions, products, tasks) are evaluated by a group of agents. We first construct a minimal peer prediction mechanism that elicits honest evaluations from a homogeneous population of agents with different private beliefs. Second, we show that it is impossible to strictly elicit honest evaluations from a heterogeneous group of agents with different private beliefs. Nevertheless, we provide a modified version of a divergence-based Bayesian Truth Serum that incentivizes agents to report consistently, making truthful reporting a weak equilibrium of the mechanism.
We study minimal single-task peer prediction mechanisms that have limited knowledge about agents' beliefs. Without knowing what agents' beliefs are or eliciting additional information, it is not possible to design a truthful mechanism in a Bayesian-Nash sense. We go beyond truthfulness and explore equilibrium strategy profiles that are only partially truthful. Using the results from the multi-armed bandit literature, we give a characterization of how inefficient these equilibria are comparing to truthful reporting. We measure the inefficiency of such strategies by counting the number of dishonest reports that any minimal knowledge-bounded mechanism must have. We show that the order of this number is θ(log n), where n is the number of agents, and we provide a peer prediction mechanism that achieves this bound in expectation.
We pay particular attention to the design process, highlighting the objectives and properties that are important in the design of good prediction mechanisms. 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?