An Algorithm That Finds Truth Even If Most People Are Wrong

Effective use of subjective judgment is essential in all fields of knowledge. We present a method for finding truth when the subjective judgments of multiple respondents are the only evidence available, and majority opinion may be wrong. Respondents are scored for their own judgments and for their metaknowledge of others’ judgments. In a probabilistic model of belief formation, their scores converge to the subjective probabilities they implicitly assign to the truth. Hence, the judgments of the highest scoring respondents reveal truth, irrespective of majority opinion. An experiment demonstrates that the method outperforms majority rule in selecting truth, and in identifying experts. * Acknowledgments: Support by NSF SES-0519141, Institute for Advanced Study and John Simon Guggenheim Foundation (Prelec). Hospitality of Aspen Center for Physics, Zagreb Math Dept, Tom Palfrey and Princeton Lab for Exp Soc Sci. Comments and discussion: Dilip Abreu, Shane Frederick, Michael Kearns, Eric Maskin, Danica Mijovic-Prelec, Whitman Richards, Rebecca Saxe, Ray Weaver. Subjective judgment is an irreducible component of pure and applied knowledge (1). Experts estimate the likelihood of rare accidents, assess the evolutionary significance of bone fragments, pronounce on aesthetic value and give interpretation on points of law. Even the touchstone of scientific practice — peer review of articles and proposals — rests on referees’ subjective assessments of originality and significance. But experts often disagree, and can be wrong (2, 3). The lack of accepted procedures for reconciling opinions feeds skepticism about subjective judgment, and promotes controversy when expertise is claimed but difficult to prove (4). New Web-based technologies have made it easier to tap the knowledge of dispersed individuals, raising hopes that automated opinion aggregation mechanisms might provide an impartial, efficient alternative to expert judgment (2, 5). Indeed, electronic information (prediction) markets have proved remarkably accurate in some domains (6, 7). However, markets suffer from a fundamental limitation: They can only trade claims whose final value is defined by a public event. For example, one can organize a market to predict the outcome of a jury trial, which is of course a public event, but not to determine the defendant’s true guilt or innocence (5). We present a method for finding truth when subjective judgment remains the only source of evidence and there is a possibility that most people are wrong. The method selects the judgments of respondents who reveal superior metaknowledge — knowledge of others’ opinions. The explicit metaknowledge criterion distinguishes our method from other approaches, such as voting algorithms (8-13), multi-item analysis (14, 15), or Bayesian updating (1, 16-18). The instrument for measuring metaknowledge is a mathematical formula called the Bayesian Truth Serum (BTS) (19). As input to the formula, respondents provide their judgments and also their predictions of the distribution of judgments in a sample of peers. These dual reports then undergo competitive, zero-sum scoring as a function of the sample statistics. Previously, it was shown that BTS scoring provides incentives for respondents to answer truthfully (19), but the problem of identifying judgments that are closer to objective, impersonal truth was left open. Here, the same formula is shown to select judgments that are objectively true, under fairly mild assumptions about how truth and belief are related. Hence, when these assumptions hold, the BTS scoring system provides an integrated solution to both problems — rewarding truthfulness, and aggregating judgments to discover truth (20). Metaknowledge is an effective truth diagnostic whenever information is unevenly distributed among respondents, so that some pieces are widely shared but other pieces are known only by a minority (16, 21). As an elementary example, consider the (false) proposition that Chicago is the capital of the state of Illinois. Respondents might form different opinions about the truth of the proposition, depending on whether they knew: (a) that Chicago is a large city, (b) that it is located in the state of Illinois, (c) that Springfield is the actual capital of Illinois, and so on. If the typical person is aware of (a) and (b) but not of (c), then the majority of those queried might vote for the incorrect answer, that the proposition is True. A democratic poll ignores the asymmetry in metaknowledge between respondents who know the right answer and those who do not. Those who know that Chicago is not the capital of Illinois can imagine that many others will be misled. A comparable insight into the opinions of others is not available to those who falsely believe the answer is Yes (22). Our scoring method in effect reweights the votes so as to reflect different levels of metaknowledge associated with each possible answer. If the method works as claimed, the true answer should emerge as the winner, regardless of how many respondents endorse it. Later on, this claim will be justified within a Bayesian model of belief formation, using the mathematical theorem about the BTS formula given in Box 1. But first we give a description of the method itself. The scoring works at the level of a single question with m answers, which is posed to n respondents, indexed by r, s,.. Œ {1,2,..,n}. The question should admit only one right answer, and it should be reasonable to believe that if one could somehow pool information from all respondents, then that collective knowledge would determine the right answer. For example, (a) Chicago is the state capital of Illinois (m=2: True, False) (b) The best current estimate of the global temperature increase between now and 2100 is _____ (m=5: < 2°C, 2° 4°C, 4° 6°C, 6° 8°C, > 8°C) (c) On current evidence, the probability of intelligent extraterrestrial life in the universe is ____ (m=10: numerical probability scale, divided into 10 categories) The respondent is asked to endorse the answer mostly likely to be true, and to predict the proportion of the sample that will endorse each possible answer. Let † xk r Œ 0,1 { } indicate whether respondent r has endorsed answer k, and † y = (y1 ,..,ym r ) her prediction of the sample proportions (