The AI Journal

When a public inquiry or an antitrust panel expands from five members to eleven, the change is sometimes defended with a piece of eighteenth-century mathematics rather than any institutional argument. The Marquis de Condorcet proved in 1785 that if each of n voters independently has better than even odds of picking the correct one of two options, the probability that a majority of them picks correctly rises monotonically toward one as n grows. Double the jury, the reasoning goes, and certainty comes closer for free. The theorem is real and its result is uncontroversial. What is not uncontroversial is how casually the inference travels from the theorem to an actual jury, panel or committee, because the proof holds only under a condition its intended targets are specifically built to violate.

Condorcet’s own model assumes independence: each voter’s probability of error is uncorrelated with every other voter’s, as though each cast a ballot alone before hearing what anyone else thought. Bernard Grofman, Guillermo Owen and Scott Feld’s 1983 survey of the theorem’s variants treats this as the load-bearing assumption, not a technical footnote: relax it and the convergence result can fail outright. Condorcet was modelling a vote on a proposition already put to an assembly, not a body that talks its way toward a verdict. Juries, expert panels and parliamentary committees do the opposite. Letting members change one another’s minds before the vote is their entire institutional point, and it is exactly the mechanism the proof excludes.

Philip Boland’s 1989 extension in The Statistician modelled the mechanism directly. Each juror, instead of voting independently, adopts the opinion of a designated leader with some fixed probability, and otherwise votes on their own competence. Boland showed the classical result survives only below a specific threshold on that deference probability; above it, adding more jurors no longer pushes the group’s accuracy toward one. It pushes accuracy toward the leader’s own error rate, converging on whatever that single member happens to get wrong. A committee that defers to its most persuasive voice does not become more reliable by growing. It becomes a slower, more expensive way of asking that voice again.

Krishna Ladha’s 1992 treatment generalises the point beyond a single leader to any pattern of correlated error, deriving an upper bound on the average pairwise correlation between voters beyond which enlarging the group actively lowers its reliability rather than raising it. The bound is not an exotic case. Shared information sources, a common expert witness heard first, and the sequential order in which people speak in a meeting are all ordinary ways of pushing correlation past it. A body that reads the same briefing pack or watches a confident early speaker before the vote has already spent some of the independence the size guarantee requires.

None of this makes deliberation worthless. Christian List and Robert Goodin’s 2001 generalisation of the theorem beyond two options, and Franz Dietrich and Kai Spiekermann’s 2013 reconstruction under weaker premises, both keep a version of the size guarantee alive by replacing raw independence with a narrower requirement: that whatever correlation deliberation introduces come from a genuine exchange of evidence rather than conformity to status or confidence. The strongest reply to the whole argument is that deference need not be a defect. If an opinion leader is simply more accurate than the panel’s average voter, following them can raise collective competence rather than lower it, correlation notwithstanding. That rescue depends on an empirical fact about who tends to lead deliberation, not on any property deliberation itself supplies, and no institution gets to assume in advance that its most persuasive member is also its best-informed one.

The number that should worry a panel designer is not how many seats to fill but how many of the people in them would still disagree if they had never spoken to one another. A jury selected for size and breadth of background can still fail the independence test in the place it is never checked: the seating chart, the order of speakers, and the identity of whoever talks first.