Jury theorems are mathematical theorems about the ability of collectives to make correct decisions. Several jury theorems carry the optimistic message that, in suitable circumstances, ‘crowds are wise’: many individuals together (using, for instance, majority voting) tend to make good decisions, outperforming fewer or just one individual. Jury theorems form the technical core of epistemic arguments for democracy, and provide probabilistic tools for reasoning about the epistemic quality of collective decisions. The popularity of (...) jury theorems spans across various disciplines such as economics, political science, philosophy, and computer science. This entry reviews and critically assesses a variety of jury theorems. It first discusses Condorcet's initial jury theorem, and then progressively introduces jury theorems with more appropriate premises and conclusions. It explains the philosophical foundations, and relates jury theorems to diversity, deliberation, shared evidence, shared perspectives, and other phenomena. It finally connects jury theorems to their historical background and to democratic theory, social epistemology, and social choice theory. (shrink)

We give a review and critique of jury theorems from a social-epistemology perspective, covering Condorcet’s (1785) classic theorem and several later refinements and departures. We assess the plausibility of the conclusions and premises featuring in jury theorems and evaluate the potential of such theorems to serve as formal arguments for the ‘wisdom of crowds’. In particular, we argue (i) that there is a fundamental tension between voters’ independence and voters’ competence, hence between the two premises (...) of most jury theorems; (ii) that the (asymptotic) conclusion that ‘huge groups are infallible’, reached by many jury theorems, is an artifact of unjustified premises; and (iii) that the (nonasymptotic) conclusion that ‘larger groups are more reliable’, also reached by many jury theorems, is not an artifact and should be regarded as the more adequate formal rendition of the ‘wisdom of crowds’. (shrink)

Peer review is often taken to be the main form of quality control on academic research. Usually journals carry this out. However, parts of maths and physics appear to have a parallel, crowd-sourced model of peer review, where papers are posted on the arXiv to be publicly discussed. In this paper we argue that crowd-sourced peer review is likely to do better than journal-solicited peer review at sorting papers by quality. Our argument rests on two key claims. First, crowd-sourced peer (...) review will lead on average to more reviewers per paper than journal-solicited peer review. Second, due to the wisdom of the crowds, more reviewers will tend to make better judgments than fewer. We make the second claim precise by looking at the Condorcet Jury Theorem as well as two related jury theorems developed specifically to apply to peer review. (shrink)

Panels, boards, and committees throughout society evaluate all manner of things by grading them, first individually and then collectively. Thus risks are prioritized, research proposals are funded, and candidates are shortlisted for jobs. It is not usual to pick winners in political elections by grading the candidates, but there are examples in history. This article takes up a question about the quality of judgments and decisions made by grading: under which conditions are they likely to be right? An answer comes (...) in the form of a jury theorem for median grading. Here, the collective grade for a thing is the median of its individually assigned grades—the one in the middle, when all of them are listed from "top" to "bottom." A second objective of this article is to suggest a solution to problems of voter ignorance in democracies. The idea is for democratic assemblies to use voting methods that make more of people's limited knowledge than do commonly used methods, such as majority voting. It turns out that in theory anyway, and perhaps also in practice, median grading can enable unenlightened assemblies to “track the truth”—even as majority voting would run them off the rails. (shrink)

This chapter explores the relationship between Goldman's thesis and the classical jury theorem. It identifies the minimal modification needed in order to recover Goldman's thesis in a Condorcetian framework. Goldman's thesis can be recast as a generalization of the classical Condorcet jury theorem. The central move needed to recover Goldman's thesis from a generalized jury theorem is to replace Condorcet's assumption that there is a single truth to be tracked with the assumption of multiple such truths: one (...) for each voter. In Goldman's picture, there is still an indirect, derivative sense in which there exists an 'overall' truth about the correct outcome for the electorate in aggregate. This 'overall' truth is determined, of course, by the profile of voter‐specific truths and therefore depends on the identity of the electorate. The chapter also provides some numerical examples to illustrate the implications of the generalized jury theorem. (shrink)

Suppose that a majority of jurors decide that a defendant is guilty (or not), and we want to know the likelihood that they reached the correct verdict. The French philosopher Marquis de Condorcet (1743-1794) showed that we can get a mathematically precise answer, a result known as the “Condorcet Jury Theorem.” Condorcet’s theorem isn’t just about juries, though; it’s about collective decision-making in general. As a result, some philosophers have used his theorem to argue for democratic forms of government. (...) This essay explains Condorcet’s theorem and how philosophers have used it as an argument for democracy. (shrink)

Condorcet's famous jury theorem reaches an optimistic conclusion on the correctness of majority decisions, based on two controversial premises about voters: they are competent and vote independently, in a technical sense. I carefully analyse these premises and show that: whether a premise is justi…ed depends on the notion of probability considered; none of the notions renders both premises simultaneously justi…ed. Under the perhaps most interesting notions, the independence assumption should be weakened.

Many political theorists and philosophers use Condorcet's Jury Theorem to defend democracy. This paper illustrates an uncomfortable implication of Condorcet's Jury Theorem. Realistically, when the conditions of Condorcet’s Jury Theorem hold, even in very high stakes elections, having more than 100,000 citizens vote does no significant good in securing good political outcomes. On the Condorcet model, unless voters enjoy voting, or unless they produce some other value by voting, then the cost to most voters of voting exceeds (...) the expected epistemic benefits to the common good of their casting a vote. Anyone who is committed to democracy on the basis of the Jury Theorem ought also to hold that widespread voting is wasteful, at least unless she can provide some further justification of mass democratic participation. (shrink)

The Condorcet jury theorem (CJT) is based on the assumption of homogeneous voters who imperfectly know the correct policy. We reassess the validity of the CJT when voters are homogeneous and each knows the correct decision with an average probability of more than a half.

The purpose of this paper is to illustrate, formally, an ambiguity in the exercise of political influence. To wit: A voter might exert influence with an eye toward maximizing the probability that the political system (1) obtains the correct (e.g. just) outcome, or (2) obtains the outcome that he judges to be correct (just). And these are two very different things. A variant of Condorcet's Jury Theorem which incorporates the effect of influence on group competence and interdependence is developed. (...) Analytic and numerical results are obtained, the most important of which is that it is never optimal--from the point-of-view of collective accuracy--for a voter to exert influence without limit. He ought to either refrain from influencing other voters or else exert a finite amount of influence, depending on circumstance. Philosophical lessons are drawn from the model, to include a solution to Wollheim's "Paradox in the Theory of Democracy". (shrink)

This paper generalises the classical Condorcet jury theorem from majority voting over two options to plurality voting over multiple options. The paper further discusses the debate between epistemic and procedural democracy and situates its formal results in that debate. The paper finally compares a number of different social choice procedures for many-option choices in terms of their epistemic merits. An appendix explores the implications of some of the present mathematical results for the question of how probable majority cycles (as (...) in Condorcet's paradox) are in large electorates. (shrink)

In Ruling Majorities and Reasoning Pluralities, Professor Saul Levmore explores the “division of labor” between the various thresholds of agreement required for collective action—supermajority, simple majority, or plurality rule. His particular emphasis is on the choice between the last two options. To improve our understanding of this choice in various settings, Professor Levmore considers the relationship between two well-known contributions to the study of group decisionmaking, namely, the Condorcet Jury Theorem and the Condorcet Criterion, which have not generally been (...) treated together. This essay explores the relationship between these two insights in the context of judicial decisionmaking. Counterintuitively, the essay demonstrates that while the Condorcet Criterion continues to hold great promise as a tool of decision in en banc appellate courts, the Condorcet Jury Theorem is most appealing in judicial contexts that appear less collegial and thus less like a jury. In contexts in which jurists engage in deliberation and in which they anticipate that their outputs will have precedential effect, the Jury Theorem has little if any role to play. In contrast, the Theorem might prove more useful in evaluating low-level judicial outputs or decisions that otherwise are not expected to be treated as precedent. Thus, common law trial court decisions and civil law court decisions are more likely to be useful benchmarks in a Condorcet Jury Theorem analysis. It is because such courts do not anticipate that their decisions will have the force of precedent that they are more likely to be derived from governing legal texts, and thus more prone to characterizations of accuracy, than are judicial outputs that are driven by policy concerns and are generated with the understanding that they will be treated as precedent. (shrink)

Democratic decision-making is often defended on grounds of the ‘wisdom of crowds’: decisions are more likely to be correct if they are based on many independent opinions, so a typical argument in social epistemology. But what does it mean to have independent opinions? Opinions can be probabilistically dependent even if individuals form their opinion in causal isolation from each other. We distinguish four probabilistic notions of opinion independence. Which of them holds depends on how individuals are causally affected by environmental (...) factors such as commonly perceived evidence. In a general theorem, we identify causal conditions guaranteeing each kind of opinion independence. These results have implications for whether and how ‘wisdom of crowds’ arguments are possible, and how truth-conducive institutions can be designed. (shrink)

The agents of a jury have to decide between a good and a bad option through simple majority voting. In this paper the jury consists of N independent couples. Each couple consists of two correlated agents of the same competence level. Different couples may have different competence levels. In addition, each agent is assumed to be better than completely random guessing. We prove tight lower and upper bounds for the quality of the majority decision. The lower bound is (...) the same as the competence of majority voting of N independent agents. The upper bound cases for negatively correlated couples can be much better than the value for 2N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2 \, N$$\end{document} independent agents. (shrink)

This paper proves two theorems for homogeneous juries that arise from different solutions to the problem of aggregation of dichotomous choice. In the first theorem, negative correlation increases the competence of the jury, while positive correlation has the opposite effect. An enlargement of the jury with positive correlation can be detrimental up to a certain size, beyond which it becomes beneficial. The second theorem finds a family of distributions for which correlation has no effect on a (...) class='Hi'>jury’s competence. The approach allows us to compute the bounds on a jury’s competence as the maximum and minimum probability of it being correct for a given individual competence and dependence structure. (shrink)

One might think that if the majority of virtue signallers judge that a proposition is true, then there is significant evidence for the truth of that proposition. Given the Condorcet Jury Theorem, individual virtue signallers need not be very reliable for the majority judgment to be very likely to be correct. Thus, even people who are skeptical of the judgments of individual virtue signallers should think that if a majority of them judge that a proposition is true, then that (...) provides significant evidence that the proposition is true. We argue that this is mistaken. Various empirical studies converge on the following point: humans are very conformist in the contexts in which virtue signalling occurs. And stereotypical virtue signallers are even more conformist in such contexts. So we should be skeptical of the claim that virtue signallers are sufficiently independent for the Condorcet Jury Theorem to apply. We do not seek to decisively rule out the relevant application of the Condorcet Jury Theorem. But we do show that careful consideration of the available evidence should make us very skeptical of that application. Consequently, a defense of virtue signalling would need to engage with these findings and show that despite our strong tendencies for conformism, our judgements are sufficiently independent for the Condorcet Jury Theorem to apply. This suggests new directions for the debate about the epistemology of virtue signalling. (shrink)

This paper explores the role that difference plays in collective decision-making using the Condorcet jury theorem. Agents facing a dichotomous decision might prove biased toward one of the options facing them. That is, they may be more likely to decide correctly when one of the options is correct than when the other option is. A juror might be more likely to convict a guilty defendant than to acquit an innocent one. Agents may display opposing biases. This paper identifies the (...) optimal committee composition in the presence of such bias. This composition depends upon both the prior probability that each option is correct and the relative costs of the two types of possible error. Optimal committee composition will be expressed in terms of the absolute margin that a group with one form of bias should enjoy over the other. (shrink)

This item was published as 'Appendix 3: An Implication of the k-option Condorcet jury mechanism for the probability of cycles' in List and Goodin (2001) http://eprints.lse.ac.uk/705/. Standard results suggest that the probability of cycles should increase as the number of options increases and also as the number of individuals increases. These results are, however, premised on a so-called "impartial culture" assumption: any logically possible preference ordering is assumed to be as likely to be held by an individual as any (...) other. The present chapter shows, in the three-option case, that given suitably systematic, however slight, deviations from an impartial culture situation, the probability of a cycle converges either to zero (more typically) or to one (less typically) as the number of individuals increases. (shrink)

This article presents a generalization of the Condorcet Jury Theorem. All results to date assume a fixed value for the competence of jurors, or alternatively, a fixed probability distribution over the possible competences of jurors. In this article, we develop the idea that we can learn the competence of the jurors by the jury vote. We assume a uniform prior probability assignment over the competence parameter, and we adapt this assignment in the light of the jury vote. (...) We then compute the posterior probability, conditional on the jury vote, of the hypothesis voted over. We thereby retain the central results of Condorcet, but we also show that the posterior probability depends on the size of the jury as well as on the absolute margin of the majority. (shrink)

A jury and two valid options are given. Each agent of the jury picks exactly one of these options. The option with the most votes will be chosen by the jury. In the N-couple model of Althöfer and Thiele, the jury consisted of 2N agents. These agents form N independent couples, with dependencies within the couples. The authors assumed that the agents who form a couple have the same competence level. In this note, we relax this (...) assumption by allowing different competence levels within the couples. We show that the theoretical results of Althöfer and Thiele remain valid under this relaxation. (shrink)

In The Social Contract, Rousseau asserts his infallibility thesis: that the general will can never err, and that one's position in the minority indicates that what one took the general will to be was not actually so. Several theorists have argued that the Condorcet Jury Theorem, which states that in a sufficiently large group the majority opinion on a yes/no question is highly likely to be correct, provides a way to interpret and justify Rousseau's bold claim. I argue that (...) the Jury Theorem cannot be used to interpret the infallibility thesis, since Rousseau takes the majority opinion to be constitutive, and not merely strong evidence, of the content of the general will. Since there is no standard of correctness that is independent of the majority opinion, the Jury Theorem cannot be invoked to support Rousseau's claim. I suggest a different strategy for situating the infallibility thesis in The Social Contract. (shrink)

In a homogeneous jury, in which each vote is correct with the same probability, and each pair of votes correlates with the same correlation coefficient, there exists a correlation-robust voting quota, such that the probability of a correct verdict is independent of the correlation coefficient. For positive correlation, an increase in the correlation coefficient decreases the probability of a correct verdict for any voting rule below the correlation-robust quota, and increases that probability for any above the correlation-robust quota. The (...) jury may be less competent under the correlation-robust rule than under simple majority rule and less competent under simple majority rule than a single juror alone. The jury is always less competent than a single juror under unanimity rule. (shrink)

Under the independence and competence assumptions of Condorcet’s classical jury model, the probability of a correct majority decision converges to certainty as the jury size increases, a seemingly unrealistic result. Using Bayesian networks, we argue that the model’s independence assumption requires that the state of the world (guilty or not guilty) is the latest common cause of all jurors’ votes. But often – arguably in all courtroom cases and in many expert panels – the latest such common cause (...) is a shared ‘body of evidence’ observed by the jurors. In the corresponding Bayesian network, the votes are direct descendants not of the state of the world, but of the body of evidence, which in turn is a direct descendant of the state of the world. We develop a model of jury decisions based on this Bayesian network. Our model permits the possibility of misleading evidence, even for a maximally competent observer, which cannot easily be accommodated in the classical model. We prove that (i) the probability of a correct majority verdict converges to the probability that the body of evidence is not misleading, a value typically below 1; (ii) depending on the required threshold of ‘no reasonable doubt’, it may be impossible, even in an arbitrarily large jury, to establish guilt of a defendant ‘beyond any reasonable doubt’. (shrink)

Richard Swinburne: Introduction Elliott Sober: Bayesianism - its scopes and limits Colin Howson: Bayesianism in Statistics A P Dawid: Bayes's Theorem and Weighing Evidence by Juries John Earman: Bayes, Hume, Price, and Miracles David Miller: Propensities May Satisfy Bayes's Theorem 'An Essay Towards Solving a Problem in the Doctrine of Chances' by Thomas Bayes, presented to the Royal Society by Richard Price. Preceded by a historical introduction by G A Barnard.

Does pre-voting group deliberation improve majority outcomes? To address this question, we develop a probabilistic model of opinion formation and deliberation. Two new jury theorems, one pre-deliberation and one post-deliberation, suggest that deliberation is beneficial. Successful deliberation mitigates three voting failures: (1) overcounting widespread evidence, (2) neglecting evidential inequality, and (3) neglecting evidential complementarity. Formal results and simulations confirm this. But we identify four systematic exceptions where deliberation reduces majority competence, always by increasing Failure 1. Our analysis recommends (...) deliberation that is 'participatory', 'neutral', but not necessarily 'equal', i.e., that involves substantive sharing, privileges no evidences, but might privilege some persons. (shrink)

Procedural justifications of democracy emphasize inclusiveness and respect and by doing so come into conflict with instrumental justifications that depend on voters’ competence. This conflict raises questions about jury theorems and makes their standing in democratic theory contested. We show that a type of no-regret learning called meta-induction can help to satisfy the competence assumption without excluding voters or diverse opinion leaders on an a priori basis. Meta-induction assigns weights to opinion leaders based on their past predictive performance (...) to determine the level of their inclusion in recommendations for voters. The weighting minimizes the difference between the performance of meta-induction and the best opinion leader in hindsight. The difference represents the regret of meta-induction whose minimization ensures that the recommendations are optimal in supporting voters’ competence. Meta-induction has optimal truth-tracking properties that support voters’ competence even if it is targeted by mis/disinformation and should be considered a tool for supporting democracy in hyper-plurality. (shrink)

Epistemic democracy aims to show, often by appeal to the Condorcet Jury Theorem, that democracy has a high chance of reaching correct decisions. It has been argued that epistemic democracy is compatible with various metaethical accounts, such as moral realism, conventionalism and majoritarianism. This paper casts doubt on that thesis and reveals the following metaethical dilemma: if we adopt moral realism, it is doubtful that voters are, on average, more than 0.5 likely to track moral facts and identify the (...) correct alternative. By contrast, if we adopt conventionalism or majoritarianism, we cannot expect that voters are both competent and sincere. Either way, the conditions for the application of Condorcet’s theorem are not met. (shrink)

The Federalist, justifying the Electoral College to elect the president, claimed that a small group of more informed individuals would make a better decision than the general mass. But the Condorcet Jury Theorem tells us that the more independent, better-than-random voters there are, the more likely it will be that the majority among them will be correct. The question thus arises as to how much better, on average, members of the smaller group would have to be to compensate for (...) the epistemic costs of making decisions on the basis of that many fewer votes. This question is explored in the contexts of referendum democracy, delegate-style representative democracy, and trustee-style representative democracy. (shrink)

In his 2010 paper “Philosophical Naturalism and Intuitional Methodology”, Alvin I. Goldman invokes the Condorcet Jury Theorem in order to defend the reliability of intuitions. The present note argues that the original conditions of the theorem are all unrealistic when analysed in connection to the case of intuitions. Alternative conditions are discussed.

The contemporary theory of epistemic democracy often draws on the Condorcet Jury Theorem to formally justify the ‘wisdom of crowds’. But this theorem is inapplicable in its current form, since one of its premises – voter independence – is notoriously violated. This premise carries responsibility for the theorem's misleading conclusion that ‘large crowds are infallible’. We prove a more useful jury theorem: under defensible premises, ‘large crowds are fallible but better than small groups’. This theorem rehabilitates the importance (...) of deliberation and education, which appear inessential in the classical jury framework. Our theorem is related to Ladha's (1993) seminal jury theorem for interchangeable (‘indistinguishable’) voters based on de Finetti's Theorem. We also prove a more general and simpler such jury theorem. (shrink)

The article considers the role of truth and objectivity in democracy. Democratic decisions usually take account of empirical truths and can be regarded as objectively correct according to various metaethical accounts such as moral realism, conventionalism, and majoritarianism. Epistemic theories of democracy aim to demonstrate the high reliability of democratic decisions. One of its main arguments rests on the Condorcet Jury Theorem, which can be applied if decision makers, e. g. members of parliaments, vote to some extent sincerely, independently, (...) and competently. However, under the assumption of moral realism, it is doubtful that they are sufficiently competent to track moral facts, since democratic legislatures make widely differing decisions. But even under the assumption of conventionalism or majoritarianism, we cannot expect decision makers to vote both competently and sincerely, because they often have differing motivations for voting the way they do. This casts doubts on the assumption that democracy is highly likely to reach objectively correct decisions. (shrink)

In recent years, the receipt and the perception of information has changed in ways which have fueled fears about the fates of our democracies. However, real information on these possibilities or the direction of these changes does not exist. Into this gap, Hahn and colleagues bring the power of Condorcet's (1785) Jury Theorem to show that changes in our information networks have affected voter inter‐dependence so that it is likely that voters are now collectively more ignorant even if individual (...) voter competence remains unchanged. (shrink)

This paper defends the communitarian account of meaning against Boghossian’s (Wittgensteinian) arguments. Boghossian argues that whilst such an account might be able to accommodate the infinitary characteristic of meaning, it cannot account for its normativity: he claims that, since the dispositions of a group must mirror those of its members, the former cannot be used to evaluate the latter. However, as this paper aims to make clear, this reasoning is fallacious. Modelling the issue with four (justifiable) assumptions, it shows that (...) Condorcet’s ‘Jury Theorem’ can be used to prove that the dispositions of the majority of the members of a group can differ from those of any individual member in a way that makes it possible to use communal dispositions as a standard with which individual dispositions can be assessed. Moreover, the argument of the paper is also shown to have general implications for the use of formal methods in the explanation of the nature of certain fallacious inferences. (shrink)

It has been argued that an epistemically rational agent’s evidence is subjectively mediated through some rational epistemic standards, and that there are incompatible but equally rational epistemic standards available to agents. This supports Permissiveness, the view according to which one or multiple fully rational agents are permitted to take distinct incompatible doxastic attitudes towards P (relative to a body of evidence). In this paper, I argue that the above claims entail the existence of a unique and more reliable epistemic standard. (...) My strategy relies on Condorcet’s Jury Theorem. This gives rise to an important problem for those who argue that epistemic standards are permissive, since the reliability criterion is incompatible with such a type of Permissiveness. (shrink)

In response to widespread voter ignorance, Jason Brennan argues for a voting ethics that can be summarized as one negative duty: do not vote badly. The implication that abstaining is always permissible entails no incentive for citizens to become competent voters or to vote once competent. Following the Condorcet Jury Theorem, this can lead to suboptimal outcomes, suggesting that voter turnout should concern instrumentalist epistemic accounts of democratic legitimacy. This could be addressed by adding two positive voting duties: to (...) make an effort to become a competent voter; and, once competent, to vote. (shrink)

This short paper proposes an application of Condorcet's jury theorem to animal group decisions.

(This is for the Cambridge Handbook of Analytic Philosophy, edited by Marcus Rossberg) In this handbook entry, I survey the different ways in which formal mathematical methods have been applied to philosophical questions throughout the history of analytic philosophy. I consider: formalization in symbolic logic, with examples such as Aquinas’ third way and Anselm’s ontological argument; Bayesian confirmation theory, with examples such as the fine-tuning argument for God and the paradox of the ravens; foundations of mathematics, with examples such as (...) Hilbert’s programme and Gödel’s incompleteness theorems; social choice theory, with examples such as Condorcet’s paradox and Arrow’s theorem; ‘how possibly’ results, with examples such as Condorcet’s jury theorem and recent work on intersectionality theory; and the application of advanced mathematics in philosophy, with examples such as accuracy-first epistemology. (shrink)

This paper explores the fact that linear opinion pooling can be represented as a Bayesian update on the opinions of others. It uses this fact to propose a new interpretation of the pooling weights. Relative to certain modelling assumptions the weights can be equated with the so-called truth-conduciveness known from the context of Condorcet's jury theorem. This suggests a novel way to elicit the weights.

In 'A Constitution of Many Minds' Cass Sunstein argues that the three major approaches to constitutional interpretation – Traditionalism, Populism and Cosmopolitanism – all rely on some variation of a ‘many-minds’ argument. Here we assess each of these claims through the lens of the Condorcet Jury Theorem. In regard to the first two approaches we explore the implications of sequential influence among courts (past and foreign, respectively). In regard to the Populist approach, we consider the influence of opinion leaders.

Can experimental philosophy help us answer central questions about the nature of moral responsibility, such as the question of whether moral responsibility is compatible with determinism? Specifically, can folk judgments in line with a particular answer to that question provide support for that answer. Based on reasoning familiar from Condorcet’s Jury Theorem, such support could be had if individual judges track the truth of the matter independently and with some modest reliability: such reliability quickly aggregates as the number of (...) judges goes up. In this chapter, however, I argue, partly based on empirical evidence, that although non-specialist judgments might on average be more likely than not to get things right, their individual likelihoods fail to aggregate because they do not track truth with sufficient independence. (shrink)

Many groups make decisions over multiple interconnected propositions. The “doctrinal paradox” or “discursive dilemma” shows that propositionwise majority voting can generate inconsistent collective sets of judgments, even when individual sets of judgments are all consistent. I develop a simple model for determining the probability of the paradox, given various assumptions about the probability distribution of individual sets of judgments, including impartial culture and impartial anonymous culture assumptions. I prove several convergence results, identifying when the probability of the paradox converges to (...) 1, and when it converges to 0, as the number of individuals increases. Drawing on the Condorcet jury theorem and work by Bovens and Rabinowicz , I use the model to assess the “truth-tracking” performance of two decision procedures, the premise- and conclusion-based procedures. I compare the present results with existing results on the probability of Condorcet’s paradox. I suggest that the doctrinal paradox is likely to occur under plausible conditions. (shrink)

Can collectives be wise? The thesis that they can has recently received a lot of attention. It has been argued that, in many judgmental or decision-making tasks, suitably organized groups can outperform their individual members. In this paper, I discuss the lessons we can learn about collective wisdom from the emerging theory of judgment aggregation, as distinct from the literature on Condorcet’s jury theorem.

The standard epistemic justification for inclusiveness in political decision making is the Condorcet Jury Theorem, which states that the probability of a correct decision using majority rule increases in group size (given certain assumptions). Informally, majority rule acts as a mechanism to pool the information contained in the judgements of individual agents. I aim to extend the explanation of how groups of political agents track the truth. Before agents can pool the information, they first need to find truth-conducive information. (...) Increasing group size is also important in the initial search for truth-conducive information. (shrink)

Suppose a committee or a jury confronts a complex question, the answer to which requires attending to several sub-questions. Two different voting procedures can be used. On one, the committee members vote on each sub-question and the voting results are used as premises for the committee’s conclusion on the main issue. This premise-based procedure can be contrasted with the conclusion-based approach, which requires the members to directly vote on the conclusion, with the vote of each member being guided by (...) her views on the relevant sub-questions. The two procedures are not equivalent: There may be a majority of voters supporting each of the premises, but if these majorities do not significantly overlap, there will be a majority against the conclusion. Pettit (2001) connects the choice between the two procedures with the discussion of deliberative democracy. The problem we want to examine instead concerns the relative advantages and disadvantages of the two procedures from the epistemic point of view. Which of them is better when it comes to tracking truth? As it turns out, the answer is not univocal. On the basis of Condorcet’s jury theorem, the premise-based procedure can be shown to be superior if the objective is reach truth for the right reasons, without making any mistakes on the way. However, if the goal instead is to reach truth for whatever reasons, right or wrong, there will be cases in which using the conclusion-based procedure turns out to be more reliable, even though, for the most part, the premise-based procedure will retain its superiority. (shrink)

The Condorcet Jury Theorem formalises the “wisdom of crowds”: binary decisions made by majority vote are asymptotically correct as the number of voters tends to infinity. This classical result assumes like-minded, expected utility maximising voters who all share a common prior belief about the right decision. Ellis : 865–895, 2016) shows that when voters have ambiguous prior beliefs—a set of priors—and follow maxmin expected utility, such wisdom requires that voters’ beliefs satisfy a “disjoint posteriors” condition: different private signals lead (...) to posterior sets with disjoint interiors. Both the original theorem and Ellis’s generalisation assume symmetric penalties for wrong decisions. If, as in the jury context, errors attract asymmetric penalties then it is natural to consider voting rules that raise the hurdle for the decision carrying the heavier penalty for error. In a classical model, Feddersen and Pesendorfer :23–35, 1998) have shown that, paradoxically, raising this hurdle may actually increase the likelihood of the more serious error. In particular, crowds are not wise under the unanimity rule: the probability of the more serious error does not vanish as the crowd size tends to infinity. We show that this “Jury Paradox” persists in the presence of ambiguity, whether or not juror beliefs satisfy Ellis’s “disjoint posteriors” condition. We also characterise the strictly mixed equilibria of this model and study their properties. Such equilibria cannot exist in the absence of ambiguity but may exist for arbitrarily large jury size when ambiguity is present. In addition to uninformative strictly mixed equilibria, analogous to those exhibited by Ellis : 865–895, 2016), there may also exist strictly mixed equilibria which are informative about voter signals. (shrink)