Optimal decision rules for the discursive dilemma Aureli Alabert Department of Mathematics

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Optimal decision rules for the discursive dilemma

Aureli Alabert

Department of Mathematics

Universitat Autònoma de Barcelona

08193 Bellaterra, Catalonia

Aureli.Alabert@uab.cat

Mercè Farré

Department of Mathematics

Universitat Autònoma de Barcelona

08193 Bellaterra, Catalonia

farre@mat.uab.cat

Rubén Montes

Department of Mathematics

Universitat Autònoma de Barcelona

08193 Bellaterra, Catalonia

October 25, 2022

Abstract

We study the classical discursive dilemma from the point of view of ﬁnding the best

decision rule according to a quantitative criterion, under very mild restrictions on the set

of admissible rules. The members of the deciding committee are assumed to have a certain

probability to assess correctly the truth or falsity of the premisses, and the best rule is the

one that minimises a combination of the probabilities of false positives and false negatives

on the conclusion.

Keywords: Discursive dilemma, doctrinal paradox, judgment aggregation, truth tracking.

1 Introduction

1.1 Statement of the problem

Nowadays, the so-called doctrinal paradox is a classical problem in judgment aggregation, in

which diﬀerent reasonable majority-type voting rules may lead to diﬀerent conclusions: A group

of people must assess the simultaneous truth or not of a set of premisses, and voting ﬁrst on each

premiss or voting directly on the conclusion not necessarily yield the same result. A slightly

diﬀerent formulation of the paradox have received the name of discursive dilemma.

In practice, the situation may appear when a court is deciding if a defendant is guilty (a set of

evidences are all veriﬁed), or not-guilty (at least one of the evidences is false), and this is the

origin of the name (Kornhauser [20]). But obviously it is present beyond legal cases. A prize

or a job position can be awarded if and only if several debatable conditions concur; or several

subjective medical indicators determine the presence of an illness or the need of a treatment;

and so on. As soon as three people join to make a decision on a compound question, the paradox

is potentially present.

To precise, suppose there are two clauses Pand Qand each member of a committee has to

decide between Pand its negation ¬Pand between Qand its negation ¬Q; and that the ﬁnal

goal is to assess if C:= P∧Qis true or its negation ¬Cis true. In the court example, the jury

has to decide if the defendant is guilty or not, and it is agreed beforehand that the guilty verdict

is logically equivalent to the truth of both premisses Pand Q.

arXiv:2210.13100v1 [math.OC] 24 Oct 2022

A. Alabert – M. Farré – R. Montes Optimal decision rules for the discursive dilemma

Suppose the voters ﬁrst decide by simple majority between Pand ¬P, and separately between

Qand ¬Q. If both Pand Qget the majority, then the conclusion is C, and otherwise it is

¬C. This decision rule is called premiss-based. Suppose on the other hand that each voter

decides directly on Cor ¬C, and then the collective decision is taken by simple majority on

these alternatives. This rule is called conclusion-based.

There are cases where the premiss-based rule leads to C, while the conclusion-based yields ¬C.

For instance, for a 3-member committee, this happens when one of the members thinks both

Pand Qare true, the second one thinks Pis true and Qis false, and the last one thinks Pis

false and Qis true. Sometimes it is said that the conclusion-based rule is more “conservative”

than the premiss-based rule (or that the latter is more “liberal” than the former), because the

positive conclusion Cis frequently the “risky one”.

These two rules are quite natural, and both can be justiﬁed on intuitive or philosophical grounds;

see for example, Mongin [32, section 2]. In particular, the conclusion-based rule respects the

deliberation of the individual judges; in the premiss-based rule the decision can be fully justiﬁed

in legal terms. Others rules can be proposed. In [1], we introduced a new rule, which stands in

some sense midway between the premiss-based and the conclusion-based rules.

In this paper we study general decision rules for the situation given. We will consider the set

of all possible decision rules, subject only to a very mild rationality requirement. They will be

called admissible rules. We want to study this set as a whole, and ﬁnd the best rule according to

some objective criterion, disregarding whether that rule can or cannot be explained on “logical”

or “intuitive” grounds, it is a consequence of some political or sociological idea, or it satisﬁes

some other desirable property.

The mentioned rationality requirement states only that if a member of the committee changes

their1opinion on a clause in some direction, the conclusion can only eventually change in the

same direction.

We take the epistemic point of view that there is an actual truth that we want to guess with the

highest possible conﬁdence. This is diﬀerent from the aggregation of preferences as in elections,

or in taking decisions on the course of actions, where there is not an absolute truth.

Our objective criterion is related to the minimisation of the combined chances to incur in false

positives (deciding Cwhen the reality is ¬C) and in false negatives (deciding ¬Cwhen the

reality is C). This is explained in detail in Section 3 and of course involves a mathematical

(probabilistic) setting where all elements have to be precisely deﬁned. Our point of view is thus

“conclusion-centric”, in the sense that we do not care about the correct guessing of the premisses.

We emphasise that the adoption of a particular criterion is a modelling choice, and it is what

confers the rationale to the best rule under it. The criterion proposed here can be replaced by

another one, if deemed better for the situation at hand, and the philosophy of ﬁnding the best

under the chosen criterion can be applied as well. We consider here, in fact, a family of criteria,

parametrised by the relative weight put on false positives and false negatives.

With this optimisation approach, we do not need to talk about majorities. The votes of the n

members of the committee will be split into four slots: P∧Q,P∧ ¬Q,¬P∧Q, and ¬P∧¬Q,

and the aggregated number of votes for each possibility will be non-negative integers x, y, z, t,

respectively, with x+y+z+t=n, being nthe number of voters. In the case of three premisses,

there will be 8 slots, and in general ppremisses would give 2pdiﬀerent possible votes of each

member. A decision rule states, for each possible values of x, y, z, t which decision, Cor ¬C, is

taken.

1The singular they/their will be used to avoid gender bias.

A. Alabert – M. Farré – R. Montes Optimal decision rules for the discursive dilemma

The number of rules grows exponentially with n. The admissible rules are much less, and they

can be implicitly enumerated so that all computations needed to ﬁnd the optimal rule or a

ranking of rules are relatively eﬃcient.

If each committee member could infallibly guess the truth or falsity of each premiss, then the

correct truth or falsity of the conclusion will be reached without diﬃculty. In fact, a single-

member committee would suﬃce. The whole point of having multi-member committees is to

alleviate the possibility that the ﬁnal conclusion be wrong. It is therefore quite natural to use

a probabilistic model that starts with the (estimated) probability that the committee members

make the correct guessing on each premiss. We call this probability their competence, and we

assume that it is greater that 1

2, and that is the same for all members of the committee and for

all premisses, although this is easily relaxed, as we will see in the ﬁnal section.

The collective decision guesses correctly or incorrectly with some probability that depends on

the voters’ competence and on the real truth value of the premisses. Only the conclusion

matters, and only the premisses are voted. One may think, as pointed out by Mongin [32],

that an external judge has to decide on the conclusions after the committee has sent them their

individual opinions.

1.2 Related literature

Doctrinal paradox. The term doctrinal paradox appears ﬁrst in the works of Kornhauser [21],

and Kornhauser and Sager [22]. They were interested in legal court cases, so that they spoke of

issue-by-issue and case-by-case majority voting.

Pettit [38] and List and Pettit [26] formulate the problem in terms of propositional logic, and

called it the discursive dilemma. The simple example of the three-member committee cited

above, can be summarised in Table 1: In the Kornhauser–Sager formulation, the commit-

Voter Proposition PProposition QProposition C

1P Q C

2P¬Q¬C

3¬P Q ¬C

majority P Q ¬C

Table 1: The discursive dilemma: The collective majority voting in the three premisses is

inconsistent with the doctrine C⇔P∧Q.

tee votes either on the ﬁrst two propositions (premiss-based/issue-by-issue), or on the third

(conclusion-based/case-by-case), and the two results are diﬀerent. In the List–Pettit formula-

tion, the committee votes on the three propositions, and this leads to a logical inconsistency.

The inconsistency comes from the constraint C⇔P∧Q(the “doctrine” to which there is a pre-

vious agreement). We see that the individual members of the committee adhere to the doctrine;

however the committee as a whole does not.

The advantage of the formulation in terms of propositional logic is that it can be generalised

to any set of propositions, to the point that the distinction between premisses and conclusions

may be unnecessary. In general, an agenda is a logically consistent set of propositions, closed

under negation, on which judgments have to be made, and that can be entangled by logical

constraints. In our case the agenda is P, ¬P, Q, ¬Q, C, ¬C, with the constraint C⇔P∧Q.

In this setting, the doctrinal paradox (or more properly, the discursive dilemma) reads: If all

pairs of formulae in the agenda are decided by majority, the resulting set of propositions can be

inconsistent.

A. Alabert – M. Farré – R. Montes Optimal decision rules for the discursive dilemma

Judgment aggregation. The body of knowledge that has been developed from List–Pettit

formulation is known as Judgment Aggregation Theory (or Logical Aggregation Theory, as

proposed by Mongin [32]). In a quite natural way, the backbone of the theory is formed by

(im)possibility results on the existence of aggregation rules satisfying certain desirable axioms.

List and Pettit [26], [27] already proved results of this kind, extended very soon by Pauly and

van Hees [36], Dietrich [8], and Nehring and Puppe [34].

The aggregation problem is described in full generality for example in the preliminaries of

Nehring and Pivato [33] and Lang et al. [23], and in the complete surveys by Mongin [32],

List and Puppe [29], List and Polak [28], and List [25]. A judgment is deﬁned as a mapping

from the agenda to the doubleton {True,False}; a feasible judgment respects moreover the under-

lying logical constraints of the propositions2. The judgment aggregation problem is then deﬁned

as the construction of a feasible reasonable collective judgment from the voters’ individual judg-

ments. Formally, an aggregation rule Fis a mapping that assigns to every proﬁle (J1, . . . , Jn)

of individual judgments Jiof the nvoters, a collective judgment J=F(J1, . . . , Jn). A feasible

aggregation rule must assign a feasible judgment to any input of feasible proﬁles. Feasible rules

trivially exist; for instance J≡Jifor some iis such a rule (called a dictatorship, for obvious

reasons). Banning dictatorships and imposing other mild desirable conditions leads very quickly

to non-existence of feasible aggregation rules. The range of possible voting paradoxes is the set

of non-feasible mappings. The classical doctrinal paradox case, despite its simplicity, already

features one such non-feasible mapping, namely P7→ True, Q7→ True, C7→ False.

In a truth-functional agenda, the propositions are split into a set of premisses, and a set of

conclusions. Assigning a Boolean value to all premisses, and applying the logical constraints,

the value of all conclusions is determined. This is clearly the case in the doctrinal paradox

setting, where moreover the premisses consist of mutually independent (not linked by constraints)

proposition-negation pairs. The truth-functional case in general has been studied mainly in

Nehring and Puppe [34], for independent as well as interdependent premisses, and in Dokow

and Holzman [13] (see also Miller and Osherson [30]).

Distance-based methods and truth-tracking. From 2006 (Pigozzi [39], Dietrich and List

[9]) another point of view emerged, in which speciﬁc judgment rules are proposed, and their

properties studied. See Lang et al. [23] for a partial survey, and the references therein. Most of

these rules can be deﬁned as some sort of optimisation with respect to a criterion, i.e. the rule

is deﬁned as the one(s) that maximises or minimises a certain quantity, usually a distance or

pseudo-distance to the individual proﬁles, while providing a consistent consensus judgment set.

There are two diﬀerent situations to which they can be applied. Either the collective judgment

set is a decision on the course of actions (as in the adoption of public policies), or there is an

underlying objective truth of each proposition under scrutiny that one would like to guess (as

in court cases). The latter is called truth-tracking (or epistemic) judgment aggregation, and it

is where the present work belongs.

Thus the goal is to get the right values of this pre-existent “state of nature”, or at least the

right values of the set of conclusions in the truth functional case. In this context, the concept of

competence of the voters arise naturally: how likely is that a voter guess the correct answer to

an issue? And it is also natural to model this likelihood as a probability. Actually, this approach

dates back to Condorcet and his celebrated Jury Theorem.

The competence as a parameter has been studied, for example, in Bovens and Rabinowicz [4]

and in Grofman et al. [17] for the one-issue case. The latter extends the Condorcet theorem in

several directions, particularly for the case of unequal competences among voters.

List [24] computed the probability of appearance of the doctrinal paradox, and the probability

2Lang et al. [23] call it consistent judgment, in the sense that it is logically consistent (not a contradiction)

when the logical constraints are added.

A. Alabert – M. Farré – R. Montes Optimal decision rules for the discursive dilemma

of correct truth-tracking as a function of the diﬀerent states of nature, allowing for diﬀerent

competences in judging both premisses but the same across individuals. Fallis [14] also observed

that the premiss-based rule is better or not than the conclusion-based rule depending on the

competence and on the “scenario” (state of nature).

The fact that the probability to guess correctly the truth depend on the unknown true state of

nature leads to a modelling choice: Either we specify an a priori probability distribution on the

possible states of nature (for the doctrinal paradox, the four states P∧Q,P∧ ¬Q,¬P∧Q

and ¬P∧ ¬Q), or we have to resort to conservative estimations, as in classical (non-Bayesian)

statistics. The main approach in this paper is the second, but most of the related literature

assumes the ﬁrst. Notably:

The cited paper by Bovens and Rabinowicz [4] compares the premiss-based and the conclusion-

based rules under the assumption of same competence for both premisses and their negations,

and independence of voters, as in the present paper. They impose a Bernoulli prior on each

premiss, the same for all.

Hartmann et al. [19] aims at generalising [24] and [4] with a conjunctive truth-functional agenda

allowing more than two premisses. The authors propose a continuum of distance-based rules,

parametrised by the weight of the conclusion relative to the premisses, and containing the

premiss- and conclusion-based procedures as extreme cases. The hypotheses are essentially

the same as in [4]. Miller and Osherson [30] also propose a variety of distance metrics, and

distinguish between “underlying metric” and “solution method”. Each solution method chooses

a loss function to minimise (based on the metric) and a set of eligible rules.

The point of view of Pivato [41] is that the votes are observations of the ‘truth plus noise’.

This allows to think of the proﬁle of individual judgments as a statistical sample (at least under

the hypothesis of the same noise distribution for all voters), and study the decision rules as

statistical estimators.

Truth-functional agendas and truth-tracking. The abovementioned papers List [24], Fallis

[14], Bovens and Rabinowicz [4], Hartman et al. [19] and, partially, Miller and Osheron [30], deal

with truth-functional agendas. Nehring and Puppe [34] focusses on truth-functional agendas

with logically interdependent premisses.

Combined with a truth-functional agenda, the truth-tracking setting can still be concerned either

with guessing the truth of all propositions (‘getting the right answer for the right reasons’) or only

on the conclusion (‘getting the right answer for whatever reasons’). Bovens and Rabinowicz [4]

discuss the merits of premiss- and conclusion-based procedures for both goals. Bozbay et al. [6]

and Bozbay [5] also study both aims, for independent and for interrelated issues, respectively.

The cited work by Hartmann et al. [19] is conclusion-centric (“whatever reasons”), while Pigozzi

et al. [40], being conclusion-centric, applies later a procedure based on Bayesian networks to get

the premisses that “interpret” the previously decided conclusions.

Distance-based methods are nothing else that the minimisation of a loss function that measures

the dissatisfaction with every possible consistent outcome. Equivalently, one may maximise util-

ity functions. Both are capable to account for the consequences of the decisions, and thus allow

to set up more complete models, in line with Statistical Decision Theory (Berger [2]). Diﬀerent

loss functions or utilities give rise to possibly diﬀerent optimal rules, and it is a modelling task

to choose the right loss function for the problem at hand.

In this sense, Fallis [14] writes about the ‘epistemic value’, highlighting that guessing correctly

a proposition may have a diﬀerent value that guessing correctly its contrary; Bozbay [5] uses

a simple 0-1 utility function to indicate incorrect-correct guessing (of all propositions or of the

conclusions alone); Hartmann et al. [19] tries giving diﬀerent utilities to false positives and false

negatives on the conclusion to assess the performance of their continuum of metrics; ﬁnally,

Bovens and Rabinowicz [4], in the discussion section, suggest introducing diﬀerent utilities to

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OptimaldecisionrulesforthediscursivedilemmaAureliAlabertDepartmentofMathematicsUniversitatAutònomadeBarcelona08193Bellaterra,CataloniaAureli.Alabert@uab.catMercèFarréDepartmentofMathematicsUniversitatAutònomadeBarcelona08193Bellaterra,Cataloniafarre@mat.uab.catRubénMontesDepartmentofMathematicsUnive...

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Optimal decision rules for the discursive dilemma Aureli Alabert Department of Mathematics

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