Bayesian markets to elicit private information

Abstract

Financial markets reveal what investors think about the future, and prediction markets are used to forecast election results. Could markets also encourage people to reveal private information, such as subjective judgments (e.g., "Are you satisfied with your life?") or unverifiable facts? This paper shows how to design such markets, called Bayesian markets. People trade an asset whose value represents the proportion of affirmative answers to a question. Their trading position then reveals their own answer to the question. The results of this paper are based on a Bayesian setup in which people use their private information (their "type") as a signal. Hence, beliefs about others' types are correlated with one's own type. Bayesian markets transform this correlation into a mechanism that rewards truth telling. These markets avoid two complications of alternative methods: they need no knowledge of prior information and no elicitation of metabeliefs regarding others' signals.prediction markets | economic incentives | truth telling | mechanism design | Bayesianism W hen trading in a market or submitting a price to an auctioneer, people reveal the extent to which they value a good. In finance, option and future markets reveal investors' beliefs. Although such revelations of tastes and beliefs originally were a by-product of regular markets, they have led to prediction markets whose primary goal is to reveal beliefs. In such markets, people can buy or sell a simple asset whose value is 1 unit (e.g., $10) if a specified event occurs and is 0 otherwise. Buying a unit at a given price is a bet that the event is more likely than people think to be on average. The resulting market price reveals aggregate expectations. Prediction markets have been used by various public organizations and companies (1, 2); for example, social scientists use this type of market to predict the replicability of experiments (3, 4). Prediction markets also have been shown to outperform polls in predicting election results (5). Unfortunately, they can only be used for events whose occurrence can be objectively verified. When collecting personal data such as opinions and self-assessed measurements, objective verification often is conceptually or practically impossible. This paper introduces Bayesian markets, which are designed to elicit private information in binary settings (yes-or-no questions) when objective verification is impossible. Bayesian markets rely on the assumption that the private information that people possess influences their belief about others. Such inference is justified by Bayesian reasoning, a widely used theory of rational reasoning (6). Answering yes (Y) provides information (a signal) that can be used to update one's prior expectation about the proportion of Y answers. In Bayesian markets, the assets traded have a value determined by the proportion of Y answers to a given question, for example, "Are you satisfied with your life?." Using Bayesian reasoning, we can predict that people who actually answer Y will expect a higher as...

The Wisdom of the Crowd and Higher-Order Beliefs

“…On the latter, an important literature has attempted to understand the performance of prediction markets: see e.g. Wolfers and Zitzewitz (2004) for an early piece summarizing the issues and Baillon (2017) for an alternate design. Wolfers and Zitzewitz (2006) study when the prediction market price can be interpreted as the average traders' beliefs, while Ottaviani and Sørensen (2015) study when the market price underreacts to new information.…”

Honesty Via Choice-Matching

“…When not assuming common prior, we adjust the definition of incentive compatibility similarly as Witkowski and Parkes (2012b), Radanovic and Faltings (2014) and Baillon (2017). We call a rule R incentive compatible if:…”

“…Witkowski and Parkes, 2012a, Radanovic and Faltings, 2014, Zhang and Chen, 2014). 2 Like these approaches, ours works with small samples. Its major practical advantages are the simplicity of the scoring principle and robustness with respect to underlying assumptions.…”

“…However, using such an assumption would lead to mechanisms that depend on the knowledge of the prior, such as "penalize the respondents who declare different priors", which are useless in practice 4. For a more extensive discussion of the empirical meaning of stochastic relevance and impersonal updating as well as for supporting evidence from psychology seePrelec (2004) andBaillon (2017).…”

Incentive-Compatible Surveys via Posterior Probabilities

“…First, it is strictly incentive-compatible (IC) (that is, strict truth-telling is an equilibrium), so that the agents' "types" corresponding to their true responses are fully revealed. 2 Second, the BTS equilibrium score of a respondent is, up to a linear transformation, equal to the logarithm of his posterior probability of the true state of nature, so that BTS ranks respondents by posteriors (henceforth called posterior ranking). We say that the BTS mechanism results in logarithmic scoring.…”