The doctrinal paradox: comparison of decision rules in a probabilistic framework

Aureli Alabert,Mercè Farré

doi:10.1007/s00355-021-01372-5

Abstract

AbstractThe doctrinal paradox is analysed from a probabilistic point of view assuming a simple parametric model for the committee’s behaviour. The well known premise-based and conclusion-based majority rules are compared in this model, by means of the concepts of false positive rate (FPR), false negative rate (FNR) and Receiver Operating Characteristics (ROC) space. We introduce also a new rule that we call path-based, which is somehow halfway between the other two. Under our model assumptions, the premise-based rule is shown to be the best of the three according to an optimality criterion based in ROC maps, for all values of the model parameters (committee size and competence of its members), when equal weight is given to FPR and FNR. We extend this result to prove that, for unequal weights of FNR and FPR, the relative goodness of the rules depends on the values of the competence and the weights, in a way which is precisely described. The results are illustrated with some numerical examples.

Highlights

The doctrinal paradox (a name introduced by Kornhauser (1992) arises in some situations when a committee or jury has to answer a compound question divided in two subquestions or premises, P and Q
The premise-based rule (Prem) rule leads to decide in favour of P ∧ Q, whereas the Conc rule leads to the contrary
R3 defined in Sect. 2 in terms of the quantities AOT and WAOTw in the Receiver Operating Characteristics (ROC) space introduced in Sect

Summary

Introduction

The Condorcet Jury Theorem (attributed to Condorcet (1785)) states that “if n jurists act independently, each with probability θ >. 1 2 of making the correct decision, the probability that the jury (deciding by majority rule) makes the correct decision increases monotonically to 1 as n tends to infinity”. The point of interest is in deciding between the acceptance of both premises P ∧ Q (P and Q) and the acceptance of the opposite ¬(P ∧ Q) = ¬P ∨ ¬Q (not P or not Q). In view of the Condorcet Jury Theorem, some kind of majority rule seems appropriate for this two-premises problem. In some cases, the same set of individual decisions leads to different collective decisions depending on the manner in which the individual opinions are aggregated

Objectives

Methods

Conclusion