On the Significance of the Absolute Margin

Abstract

Abstract.Consider the hypothesis H that a defendant is guilty (a patient has condition C), and the evidence E that a majority of h out of n independent jurors (diagnostic tests) have voted for H and a minority of k:=n-h against H. How likely is the majority verdict to be correct? By a formula of Condorcet, the probability that H is true given E depends only on each juror's competence and on the absolute margin between the majority and the minority h-k, but neither on the number n, nor on the proportion h/n. This paper reassesses that result and explores its implications. First, using the classical Condorcet jury model, I derive a more general version of Condorcet's formula, confirming the significance of the absolute margin, but showing that the probability that H is true given E depends also on an additional parameter: the prior probability that H is true. Second, I show that a related result holds when we consider not the degree of belief we attach to H given E, but the degree of support E gives to H. Third, I address the implications for the definition of special majority voting, a procedure used to capture the asymmetry between false positive and false negative decisions. I argue that the standard definition of special majority voting in terms of a required proportion of the jury is epistemically questionable, and that the classical Condorcet jury model leads to an alternative definition in terms of a required absolute margin between the majority and the minority. Finally, I show that the results on the significance of the absolute margin can be resisted if the socalled assumption of symmetrical juror competence is relaxed.

Epistemic aspects of representative government

“…They might thereby end up electing trustee-style representatives who are much better than electors themselves at complex decisions. 30 As List (2003) has shown, group competence depends on the absolute margin (and not the proportion).…”

An Epistemic Free-Riding Problem?

“…4 The Condorcet jury theorem is robust to certain relaxations of the assumptions. A version of it still holds in certain cases where different jurors have different competence levels, but where the average competence r =  , r + (1-r) exp(-m log(p / (1-p))) r =  , r + (1-r) ( (1-p) / p ) m which is exactly Condorcet's formula (see List 2003). 17 Compare our previous note on the idealizing character of this assumption.…”

“…Let us restate Condorcet's framework in terms of hypothesis testing (List 2003 Suppose our evidence is that juror i has voted for 'X = 1'. We have…”

“…If the given evidence is only that a majority among the n jurors have voted for 'X = 1' (respectively for 'X = 0'), but the evidence does not include any information about the precise size of that majority, then the posterior probability that X = 1 (respectively, that X = 0) is P n , i.e. equal to the collective competence level (List 2003).…”

“…2 We focus, more particularly, on how reflexivity may operate in a rather unlikely area. The theory we take as our target, if indeed it can be called a theory, is the jury theorem that was first identified by the eighteenth century French thinker, the Marquis de Condorcet (1976), and that has recently attracted renewed attention (Grofman, Owen and Feld 1983, Lahda 1992, Estlund 1994, Austen-Smith and Banks 1996, Feddersen and Pesendorfer 1998, List and Goodin 2001, List 2003. We argue that Condorcet's result is subject to a reflexivity effect of broadly the kind that Popper had in mind.…”