¹ Sunstein, Cass R., A Constitution of Many Minds: Why the Founding Document Doesn't Mean What It Meant Before (Princeton, N.J: Princeton University Press, 2009), pp. ix–xGoogle Scholar.

² Sunstein, , A Constitution of Many Minds, pp. ix–xGoogle Scholar.

³ Sunstein, , A Constitution of Many Mind, pp. x, 8–10Google Scholar.

⁴ Sunstein, , A Constitution of Many Minds, p. 214Google Scholar.

⁵ Everything said there could also be applied with minor adjustments to the Cosmopolitan approach, which we do not here discuss separately in consequence.

⁶ This subsumes the assumption of sincerity, which is often stated separately.

⁷ Grofman, Bernard, Owen, Guillermo and Feld, Scott L., ‘Thirteen Theorems in Search of the Truth’, Theory and Decision, 15 (1983), 261–278CrossRefGoogle Scholar; Dietrich, Franz, ‘The Premises of Condorcet's Jury Theorem Are Not Simultaneously Justified’, Episteme, 58 (2008), 56–73CrossRefGoogle Scholar.

⁸ List, Christian and Goodin, Robert E., ‘Epistemic Democracy: Generalizing the Condorcet Jury Theorem’, Journal of Political Philosophy, 9 (2001), 277–306CrossRefGoogle Scholar.

⁹ The influence of opinion leaders was first studied formally by Boland, Philip J., Proschan, Frank and Tong, Y. L., ‘Modelling Dependence in Simple and Indirect Majority Systems’, Journal of Applied Probability, 26 (1989), 81–88CrossRefGoogle Scholar. Krishna Ladha has worked extensively on the effect of correlated votes, where correlation is due to shared information, particularly in Ladha, Krishna K., ‘The Condorcet Jury Theorem, Free Speech, and Correlated Votes’, American Journal of Political Science, 36 (1992), 617–634CrossRefGoogle Scholar; Ladha, Krishna K., ‘Condorcet's Jury Theorem in Light of De Finetti's Theorem’, Social Choice and Welfare, 10 (1993), 69–85CrossRefGoogle Scholar; Ladha, Krishna K., ‘Information Pooling through Majority-Rule Voting: Condorcet's Jury Theorem with Correlated Votes’, Journal of Economic Behavior & Organization, 26 (1995), 353–372CrossRefGoogle Scholar. This idea was extended and developed into a new jury theorem by Dietrich, Franz and List, Christian, ‘A Model of Jury Decisions where All Jurors Have the Same Evidence’, Synthese, 142 (2004), 175–202CrossRefGoogle Scholar, assuming that all jurors have the same evidence. Kaniovski, Serguei, ‘Aggregation of Correlated Votes and Condorcet's Jury Theorem’, Theory and Decision, 69 (2010), 453–468CrossRefGoogle Scholar, has recently shown that a (pairwise) correlation table is not sufficient to determine the existing dependence relations between jurors and that different ways to instantiate the same pairwise correlations lead to different results.

¹⁰ Dietrich, ‘The Premises of Condorcet's Jury Theorem Are Not Simultaneously Justified’.

¹¹ With the exception of highly construed examples where it is known that both assumptions hold in virtue of the construction.

¹² A general account of the causal bases of dependence leading to a new jury theorem is presented in Dietrich, Franz and Spiekermann, Kai, ‘Epistemic Democracy with Defensible Premises’ (unpublished manuscript, 2010, available at http://personal.lse.ac.uk/spiekerk/papers/DietrichSpiekermann-EpistemicDemocracy.pdf)Google Scholar.

¹³ Among other things, they are not concerned with how their current decision might influence future judgements.

¹⁴ This is what Vermeule, Adrian, Law and the Limits of Reason (Oxford: Oxford University Press, 2009), pp. 5–6Google Scholar, see also pp. 75–7, calls the ‘Burkean Paradox’: ‘Where actors defer to the information of past others, as the Burkean position would have them do, the result is a low-value “information cascade” rather than collective wisdom.’

¹⁵ Bikhchandani, Sushil, Hirshleifer, David and Welch, Ivo, ‘A Theory of Fads, Fashion, Custom, and Cultural Change as Informational Cascades’, Journal of Political Economy, 100 (1992), 992–1026CrossRefGoogle Scholar. Similarly, Banerjee, Abhijit V., ‘A Simple Model of Herd Behavior’, Quarterly Journal of Economics, 107 (1992), 797–817CrossRefGoogle Scholar; and Smith, Lones and Sorensen, Peter, ‘Pathological Outcomes of Observational Learning’, Econometrica, 68 (2000), 371–398CrossRefGoogle Scholar; a useful introduction is Chamley, Christophe P., Rational Herds: Economic Models of Social Learning (Cambridge: Cambridge University Press, 2004)Google Scholar.

¹⁶ Note the contrast between the literature on cascades and on sequential voting games, as analysed in a seminal paper by Dekel, E. and Piccione, M., ‘Sequential Voting Procedures in Symmetric Binary Elections’, Journal of Political Economy, 108 (2000), 34–55CrossRefGoogle Scholar. The cascade literature suggests that the utility of voters is derived purely from their own court's decision in the case at hand, while voting games relate individual utility to the overall outcome of all actions by all agents.

¹⁷ Of course, this could be done in a Bayesian framework by changing the utility functions of the judges. But the Bayesian treatment comes with some algebraic costs and little gain for the purposes of this article.

¹⁸ Edwards, Ward, ‘Conservatism in Human Information Processing’, in Benjamin Kleinmuntz, ed., Formal Representation of Human Judgement (New York: Wiley, 1968)Google Scholar; reprinted in Kahneman, Daniel, Slovic, Paul and Tversky, Amos, eds, Judgement under Uncertainty: Heuristics and Biases (Cambridge: Cambridge University Press, 1982), pp. 359–369CrossRefGoogle Scholar.

¹⁹ Segal, Jeffrey A. and Spaeth, Harold J., ‘The Influence of Stare Decisis on the Votes of United States Supreme Court Justices’, American Journal of Political Science, 40 (1996), 971–1003CrossRefGoogle Scholar.

²⁰ In the BHW model, one individual decides at each point in time (i.e. k = 1). The state of the world is fixed randomly at the start with equal probability, and all judges begin with the same prior of π(θ = 1) = π(θ = 0) = 0.5. The utility function of judges is defined such that a correct decision yields payoff 1, an incorrect decision 0:

The judges maximize their utility, and here this means they vote for the alternative that they consider more likely to be correct.

²¹ Different modelling choices are conceivable to break ties, especially randomizing or assuming that judges have marginally more confidence in their own signal than in other judgements.

²² Austen-Smith, David and Banks, Jeffrey S., ‘Information Aggregation, Rationality, and the Condorcet Jury Theorem’, American Political Science Review, 90 (1996), 34–45CrossRefGoogle Scholar; Feddersen, Timothy and Pesendorfer, Wolfgang, ‘Convicting the Innocent: The Inferiority of Unanimous Jury Verdicts under Strategic Voting’, American Political Science Review, 92 (1998), 23–35CrossRefGoogle Scholar.

²³ We have already reported psychological evidence that empirically w may well be larger than 1. For similar evidence that m may be limited, see, for example, Trope, Yaacov, ‘Inferences of Personal Characteristics on the Basis of Information Retrieved from One's Memory’, Journal of Personality & Social Psychology, 36 (1978), 93–106CrossRefGoogle Scholar; reprinted in Kahneman, Slovic and Tversky, Judgement under Uncertainty: Heuristics and Biases.

²⁴ In the case of multi-member courts, these calculations would have to be expressed as margins of all previous votes. See also supplementary material §1.

²⁵ This is one way of formalizing Vermeule's suggestion that ‘individual judges might adopt an intermediate approach, according to which they give some but not complete deference to the views of the past, and correlatively think for themselves to some degree or in some circumstances.’ See Vermeule, Law and the Limits of Reason, p. 76.

²⁶ Sunstein, , A Constitution of Many Minds, pp. 103–6, 171–3Google Scholar.

²⁷ Vermeule anticipates this part of our model, but not the next, when writing, ‘Perhaps some judges in the stream of precedent or tradition have contributed independently, while some have not’. See Vermeule, , Law and the Limits of Reason, p. 76Google Scholar.

²⁸ There is an analogous problem, perhaps, in determining as a Traditionalist which past precedents are relevantly similar to the case before your current court (Vermeule, Law and the Limits of Reason, pp. 71–2).

²⁹ Sunstein, , A Constitution of Many Minds, p. 136Google Scholar, chap. 7.

³⁰ Sunstein, , A Constitution of Many Minds, p. 125Google Scholar.

³¹ Sunstein, , A Constitution of Many Minds, p. 165Google Scholar, Sunstein's emphasis. We are puzzled by Sunstein's proviso that Populism should only look for ‘intense public opposition’. To see that this statement does not make much sense in a CJT context, it suffices to consider two examples: (1) assume competence and independence hold, but each single voter is not very competent (p = 0.51, for instance). In a large population, the majority is very likely to be right, but the result is also very likely to be tight (an expected 51 per cent versus 49 per cent); (2) assume the voters are heavily influenced by one opinion leader taking the incorrect stance. The incorrect stance wins by a landslide. These two examples show that the size of the majority alone should not make us more comfortable to accept an outcome. Without knowing that the independence assumption holds, the size of the majority does not tell us much.

³² Sunstein, , A Constitution of Many Minds, p. 169Google Scholar.

³³ Sunstein, , A Constitution of Many Minds, p. 172Google Scholar.

³⁴ Sunstein, , A Constitution of Many Minds, p. 175Google Scholar.

³⁵ Most worries about the competence condition, as in Estlund, David M., Democratic Authority: A Philosophical Framework (Princeton, N.J.: Princeton University Press, 2008), pp. 225–228Google Scholar, are in our view best seen as really worries about independence. All that the CJT requires by way of competence, recall, is that voters be ‘better than random’; and how could they be worse than random, except by some common influences that systematically affect many voters at once, thus violating the independence assumption?

³⁶ Boland, Proschan and Tong, ‘Modelling Dependence in Simple and Indirect Majority Systems’.

³⁷ However, the probability-of-following is not usually identical with the correlation coefficient, as we discuss in the supplementary material §2. In this regard we develop a different technical treatment from that of Boland et al. In other respects our set-up is very similar to theirs.

³⁸ Supplementary material §3 provides an analytical result for group competence in this setting.

³⁹ For comparison: the CJT result for n = 500 and p = 0.55 is 0.986.

⁴⁰ How this constellation affects group competence in general depends on p, and the group partition.

⁴¹ The results for the setting with two opinion leaders represent an anomaly that we will discuss below.

⁴² This less extreme case is the more realistic one, of course: it is hard to think why people would follow opinion leaders unless they at least believed (perhaps wrongly of course) that the opinion leader's opinion was more likely to be correct than their own.

⁴³ For another analysis of why negatively correlated opinion leaders might be good from an epistemic point of view, see Hong, Lu and Page, Scott, ‘Interpreted and Generated Signals’, Journal of Economic Theory, 144 (2009), 2174–2196CrossRefGoogle Scholar.

⁴⁴ Vermeule, Law and the Limits of Reason.