Resolving Paradoxes In Judgment Aggregation

RESOLVING PARADOXES IN JUDGMENTAGGREGATION

BY DAVIDE RIZZA

When a law court makes a decision based on the individual deliberation of each judge, a case ofjudgment aggregation occurs. The possibility that the aggregation’s outcome be logically inconsis-tent, even though it is based on consistent individual judgments, arises relatively easily and hasbeen the subject of several investigations. In this paper I show that this paradoxical behaviour isthe effect of decision procedures that are unable to discriminate between logically consistent andlogically inconsistent individual judgments. The paradoxes can be resolved by selecting proceduresthat are not affected by this limitation.

I. NEW PROBLEMS AND OLD SOLUTIONS

It is well known that voting procedures exhibit a wide variety of patho-logical behaviours. A famous instance of this phenomenon is Condorcet’sparadox, which is easily illustrated by means of the table below, listing,for each one of three voters 1,2,3, a transitive ranking of three candidatesa,b and c.1

1 : a �1 b b �1 c a �1 c

2 : b �2 a b �2 c c �2 a

3 : a �3 b c �3 b c �3 a

The above preferences determine an election whose outcome may becomputed by applying the rule of simple majority to each column todetermine pairwise rankings for (a, b),(b, c),(c, a). The trouble with thepreferences in the table is that they give rise to a cyclic outcome in whicha is preferred to b, b to c and c to a.

1 Each one of the expressions of the form ‘x ≻i y ' occurring in the table below, withi = 1, 2, 3, is to be read as follows: ‘voter i strictly prefers candidate x to candidate y.

The Philosophical Quarterly Vol. 62, No. 247 April 2012ISSN 0031-8094 doi: 10.1111/j.1467-9213.2011.00018.x

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An immediate reaction to Condorcet’s paradox is to acknowledge itas an instance of the shortcomings of collective decision-methods. Collec-tivities do not necessarily behave like individuals: while the latter (may)exhibit regular preferences, there is no guarantee that compound prefer-ences should display the same regularity. This implies that Condorcet’sparadox is best viewed as a limitative result: it identifies the impossibilityof forcing a type of outcome (i.e., acyclic ranking) in decision-making byelection.

A closer scrutiny of the problem suggests that the above standpointshould be reversed: the reason why a cyclic outcome occurs in Condor-cet’s paradox is not the unattainability of collective rationality but ratherthe choice of a faulty election procedure. This is revealed by the factthat the pairwise preferences in the table above can be interchangedto determine a new table in which every voter has cyclic preferencesand the (cyclic) outcome of the pairwise majority election remainsunchanged.2

Pairwise majority only acts upon two candidates at a time butcompletely disregards the interlock of pairwise rankings within individualpreferences that is expressed by their transitivity. As a result, pairwisemajority is unable to discriminate cyclic preferences from transitive (andthus, if irreflexive, acyclic) ones. It no longer seems paradoxical that theelection procedure can have cyclic outcomes: its very design makes itunable to use the information that could prevent their appearance. Thisconclusion does not only show why Condorcet’s paradox arises: it alsoshows why pairwise majority must be replaced by an alternative that canmake use of the transitivity of individual preferences.

This paper is devoted to showing that the same type of analysis holdstrue of a form of collective deliberation called judgment aggregation. Atypical example of judgment aggregation is offered by multi-membercourts applying the principles of legal doctrine to resolve cases thatdepend upon several interconnected issues. Every member of the courtmakes individual judgments, i.e., clear-cut ‘yes-or-no’ decisions, on eachissue relevant to the case. The aggregation of individual judgments leadsto a conclusive deliberation, the aggregate judgment.

The problematic nature of this form of aggregation is illustrated by thesimple example of a three-judge court faced with the task of decidingwhether a defendant is liable under a charge of breach of contract. Eachjudge must decide whether the contract was valid and whether it has

2 This is shown in D. Saari, Basic Geometry of Voting (New York: Springer, 1995) on p.48.The discussion of Condorcet’s paradox offered in this section follows Saari’s very closely.

338 DAVIDE RIZZA

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been breached: exactly when both are the case is the defendant liable.Now suppose that the members of the court express the following individ-ual judgments:

Valid Contract? Breach? Liable?

judge 1 yes yes yes

judge 2 no yes no

judge 3 yes no no

Let us assume that the court decision, i.e., the aggregation of individualjudgments, is based on simple majority. Then, if the individual liabilitydecisions of the court (given in the third column) are aggregated, thedefendant is not liable. If, on the other hand, the individual decisions onthe conditions for liability (given in the first and second columns) areaggregated, then the defendant is liable. Two decision procedures thatseem equally acceptable give rise to opposite results: this fact, known asthe doctrinal paradox,3 is a special instance of a general problem affectingjudgment aggregation, which has been recently studied by Christian Listand Philip Pettit.4

As in the case of Condorcet’s paradox, we may think that the doctrinalparadox (as well as the general result offered by List and Pettit) describesan inherent limitation of collective decision-making or we may argue thatit actually involves the application of a defective procedure. The secondpossibility is supported by the fact that the doctrinal paradox is a relabel-ing of Condorcet’s paradox.5

To see this, assign the pairs (a, b),(b, c),(a, c) to the issues ‘Valid Con-tract’, ‘Breach’ and ‘Liable’ respectively. Then, for any pair (x, y), assignthe ranking x ≻i y to the relevant occurrences of ‘yes’ and the reverseranking to the relevant occurrences of ‘no’. Under the assignment justdescribed, the last table becomes Condorcet’s paradox.

Because of this, it is plausible to think that the doctrinal paradox isaffected by the same problem that gives rise to Condorcet’s paradox.More generally, one may think that impossibility results in judgmentaggregation, insofar as they generalise the problem affecting the doctrinalparadox, should be understood in the same way and, if possible, resolvedin the same way too. The remainder of this paper will confirm this con-jecture.

3 The paradox has originally been discussed in L.A. Kornhauser and L.G. Sager,‘Unpacking the Court’, The Yale Law Journal, 96 (1986), pp. 82–117.

4 See C. List and P. Pettit, ‘Aggregating Sets of Judgments: An Impossibility Result’,Economics and Philosophy, 18 (2002), pp. 89–110.

5 This has been noticed in D. Saari, Dethroning Dictators, Demystifying Paradoxes: Social ChoiceAnalysis, (New York: Cambridge UP, 2008), at p.32.

RESOLVING PARADOXES IN JUDGMENT AGGREGATION 339


II. BASIC FORMAL SETUP

In order to discuss judgment aggregation in a general setting, I need tointroduce a simple formal description of it, which is easily presented byreferring again to the doctrinal paradox. In that context the judges wereasked to make a decision on three issues, i.e., the validity of a contract, itsbreach and the liability of the defendant. We may take liability to consistin the breach of a valid contract. Thus, using two propositional variablesp, q to represent validity and breach respectively, the defendant’s liabilitycan be symbolised by the conjunction p ∧ q.

This suggests that propositional logic may be employed to formallyportray the space of issues to which judgment aggregation applies. In par-ticular, the basic issues that a deliberation must resolve6, which are finitein number, may be identified with propositional variables p, q, r, …, z.The logical relations between issues are then represented by the formulaeof a propositional language based on the connectives ¬, ∨, ∧.

Because each issue is to be decided in the affirmative or the negative,an individual act of judgment may be seen as a single assignment ofexactly one of the truth-values 0,1 to each issue. Given the truth-tables ofpropositional logic, an assignment does not treat all issues independently:for example, whenever issue p is assigned 1, issue p ∨ q must be assigned 1as well.

In general, if an issue w is a logical consequence of an issue u, thenthe assignment of 1 to u constrains the same assignment to w: simple as itis, this fact provides the key to assessing and resolving a variety of para-doxes in judgment aggregation, as I am going to show.

If two issues are each a logical consequence of the other, we say thatthey are equivalent. It is best, for present purposes, to identify equivalentissues. There are two reasons to do so: first, it would be unrealistic toassume that a judgment act that has already assigned a truth-value to pshould be followed by separate acts that assign truth-values to the equiva-lent issues p ∨ p, p ∨ p ∨ p,…. Secondly, if there are finitely many basicissues, there are infinitely many propositional formulae that representcomplex issues but every one of these formulae is equivalent to some oneof a finite set of formulae.

With an abuse of language, I will use symbols like ‘u’, ‘w’ to referto the classes of issues that are equivalent to u, w. I will treat the con-nectives in a similar manner: thus, for example the symbol ‘∧’ in theexpression ‘p ∧ q’ denotes an operation that applies to the classes of

6 Validity and breach of contract in the doctrinal paradox.

340 DAVIDE RIZZA


issues equivalent to p, q respectively and sends them into the class ofissues equivalent to the single issue p ∧ q. Similar stipulations hold for¬ and ∨: it follows that the space of issues is a finite Boolean algebraB.7

With this apparatus in place, each individual judgment is a valuationover B, i.e., a function that assigns exactly one of 0,1 to each element ofB. If there are n agents, the individual judgment of the i-th agent isdenoted by mi. It is assumed that individual judgments preserve logicalconsequence, i.e., that, for any i (i = 1, …, n), if u,w are two elements ofB and w follows from u, then mi(u) = 1 implies mi(w) = 1.

A list of all individual judgments, which may be called a profile, is anordered n-tuple of the form:

v ¼ hmi ,. . .,mniIf V is the set of all profiles, an aggregation a of individual judgments isin general a function that sends each profile in V into a valuation m overB. This machinery suffices to provide a satisfactory analysis of variousimpossibility results in judgment aggregation, as I will show starting fromthe doctrinal paradox.

III. THE DOCTRINAL PARADOX

Stated in terms of valuations, the doctrinal paradox gives rise to the tablebelow:

Valid Contract? Breach? Liable?

judge 1 1 1 1

judge 2 0 1 0

judge 3 1 0 0

The paradox arises from the fact that two aggregation procedures basedon simple majority produce opposite results. The first procedure, call itprocedure A, uses simple majority to determine a partial, aggregate valua-tion over the issues of validity and breach of contract: once this partial

7 More general models of the space of issues and of judgment aggregation appear in P.Gardenfors, ‘A Representation Theorem for Voting with Logical Consequences’, Economicsand Philosophy, 22 (2006), pp.181–90; M. Pauly and M. Van Hees, ‘Logical Constraints forJudgment Aggregation’, Journal of Philosophical Logic, 35 (2006), pp.569–85; and F. Dietrich,‘A Generalised Model of Judgment Aggregation’, Social Choice and Welfare, 28 (2007), pp.529–65. These papers extend the negative result of List and Pettit, whereas my purpose isto resolve that result and analogous ones using as elementary an apparatus as possible.



valuation is available, the aggregate verdict concerning liability iscomputed from it by means of a truth-table. Procedure A assigns 1 toaggregate validity and aggregate breach of contract: as a consequence, italso assigns 1 to aggregate liability.

The second procedure, call it procedure B, applies simple majority tothe individual verdicts concerning liability in order to determine a finalaggregate verdict. Procedure B assigns 0 to liability, i.e., contradicts theresult of the application of procedure A.

The asymmetry between the outcomes of A and B constitutes a prob-lem only if A and B are deemed to be equally acceptable. To find outwhether this is the case, note that the assignment to liability is uniquelydetermined by the assignments to validity and breach respectively. Theseissues may therefore be represented by two propositional variables p, q,in which case liability is represented by p ∧ q, a consequence of the pair{p, q}.

When B is used, the outcome of the doctrinal paradox can be obtainedequally well from the above table and from the following one, in whichthe judges unanimously violate consequence:

p q p ∧ q

judge 1 1 1 0

judge 2 1 1 0

judge 3 1 1 0

Procedure B cannot discriminate between the last table of judgments andthe table that generates the doctrinal paradox: its outcome is the same ineither case. Exactly as pairwise majority failed to take the transitivity ofindividual preferences into account, simple majority under B fails to takeinto account the individual ‘rationality’ (i.e., the preservation of logicalconsequence) of the judges’ valuations.

Things stand otherwise with procedure A. Procedure A disregardsthe rightmost column and only works on the two columns on its left:the issues in these columns are independent of each other in the sensethat any assignment of 0,1 to one of these issues is compatible withan arbitrary assignment to the other issue. Due to the independence ofp,q, it is reasonable to aggregate their corresponding assignmentsseparately. Once the aggregate assignments are available, a uniqueassignment to p ∧ q can be computed using the truth-table of conjunc-tion.

Procedure B neglects the fact that individual judgments aggregatedunder simple majority generate aggregate judgments that depend on ashifting majority, i.e., a family of agents that may vary from one issue to

342 DAVIDE RIZZA


another. It cannot be expected that such a shifting majority should retainthe internal consistency of one rational agent: in fact, it does not. For thisreason, if the objective is to retain the consistency of a rational agent, onemight think of amending procedure B.

In particular, one may use simple majority to determine aggregateassignments over a set of independent issues while truth-tables are used todefine a unique extension of this valuation to every other issue. No viola-tion of consequence can be generated by this amendment of procedureB, which is exemplified by procedure A. It may be concluded that thedoctrinal paradox is the result of taking procedure B to be on a par withone possible amendment of it.

List and Pettit (pp. 95–6) interpret the paradox in a different way,namely as an illustration of the conflicting nature of two natural require-ments that any deliberation procedure should satisfy. One requirement isthat collective judgments should have the same structure as individualjudgments: this imposes a rationality constraint on aggregation, i.e. thecondition that aggregate judgments respect logical consequence. The sec-ond requirement is that individual judgments should be aggregated oneach issue separately in order for the aggregation procedure to beresponsive to each individual judgment. The doctrinal paradox wouldthen show an inherent conflict between the rationality constraint andindividual responsiveness.

In the light of the previous analysis, it may be argued that this diag-nosis is not correct: the form of individual responsiveness that is atwork in the doctrinal paradox is described by procedure B, which sup-presses individual rationality. If individual rationality is suppressed, col-lective rationality may emerge only accidentally. The conflict that Listand Pettit detect does not involve two distinct conditions constrainingthe same aggregation procedure but rather two distinct aggregation pro-cedures, one of which can produce paradoxical results precisely becauseit does not discriminate between rational and irrational individual judg-ments.

These remarks can be generalised: it is possible to identify axiomati-cally types of judgment aggregation that cannot discern the logicalstructure of individual judgments and, a fortiori, cannot enforce thatstructure on the aggregate judgments. This suggests that the types ofaggregations that give rise to impossibility results do so by the way theyare designed and not because of fundamental difficulties underlying col-lective decision-making. The next section provides a simple examplewhose critical discussion will prepare my analysis of the generaliseddoctrinal paradox.



IV. EXPERTS VERSUS COLLECTIVES

Several forms of decision-making involve the use of experts, typically tosettle certain issues on which they have special competences. In such casesthe experts’ judgments on the relevant issues are identified with the aggre-gate judgments on those issues.8 This constraint of expertise on judgmentaggregation can be represented within the formal framework of section IIby means of the following definition:

Definition 1. An agent i is an expert on issue u if and only if mi(u) = m(u)in every profile.

It is not difficult to obtain a negative result that affects any unanimousaggregation that involves experts. To this end, define a judgment aggre-gation for n agents over a finite issue space B to be Sen-type if and only ifit satisfies the following two properties:

1 Pareto: If mi(u) = 1 (i = 1,…,n), then m(u) = 1.2 Experts: There are two basic issues p, q and two agents i, j such that i

is an expert on p and j is an expert on q.

Then:Lemma IV.1. NoSen-type logical aggregation respects logical consequence.9

Proof. Let i be an expert on p and j an expert on q. We may choose aprofile in which miðpÞ ¼ 0 ¼ mjðqÞ and miðqÞ ¼ 1 ¼ mjðpÞ. If, in addition,mk(p ∨ q) = 1(k = 1…n), we obtain the table below:

p q … p ∨ q

1: … 1

… 1

i: 0 1 … 1

j: 1 0 … 1

… 1

n: … … … 1

8 A more detailed discussion of this situation may be found in F. Dietrich and C.List,‘A Liberal Paradox for Judgment Aggregation’, Social Choice and Welfare, 31 (2008), pp. 59–78.

9 This result translates in the context of judgment aggregation a result originally provedin A. Sen, ‘The Impossibility of a Paretial Liberal’, The Journal of Political Economy, 78 (1970),pp. 152–7. Dietrich and List (see previous footnote) have also proved variants of Sen’stheorem.

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The aggregate valuation m must assign 1 to ¬ p and to ¬ q by thedecisiveness of i, j over p, q. Since ¬ (p ∨ q) is a logical consequence of theset of issues over { ¬ p, ¬ q}, m should assign 1 to it and, thus, 0 to p ∨ q.By Pareto, m assigns 1 to p ∨ q: thus, m violates logical consequence.

Before discussing the formal structure of the above result, let me pro-vide an interpretation for it that may clarify what kind of situation thelemma is meant to describe. Under this interpretation, the decision-makers are members of an academic panel in charge of evaluating thecompetences of a job candidate x in the history of philosophy.

If i and j are the panel experts on ancient and early modern philoso-phy, we may take p to represent the issue whether x has some strengthsin ancient philosophy and q the issue whether x has some strengths inearly modern philosophy.

The panel’s deliberation, as portrayed in the last table, describes a situ-ation in which x is deemed by the experts to have no strengths in ancientor early modern philosophy, while there is a unanimous judgment that xhas some strengths in at least one of these areas.

This example illustrates the fact that lemma 4.1 might be interpreted asrepresenting a conflict between unreliable experts and other panel mem-bers, rather than an inherent limitation of rational judgment aggregation.The key to understanding why this is the case lies in the observation that,if individual valuations were allowed to violate logical consequence, theabove lemma would be entirely trivial. In particular, the paradoxicalresult would follow from an application of the Pareto condition to a pro-file like the one below:

This table, however, is not just illustrative of a trivially irrational situa-tion: it can be used to recover the one exploited in the proof of thelemma.10 It suffices to delete from it the (irrational) valuations of everynon-decisive agent on the relevant issues p, q and replace at least one ofthese valuations with its alternative. This can be done in general: any

p q … p ∨ q

i: 0 0 … 1

j: 0 0 … 1

… 1

n: 0 0 … 1

10 The argument that follows adapts to judgment aggregation the analysis of Sen’s theo-rem developed in A.Petron and D.Saari, ‘Negative Externalities and Sen’s ImpossibilityTheorem’, Economic Theory, 28 (2006), pp. 265–81.



violation of consequence produced by agents that are decisive on atomscan be obtained from a unanimous violation of consequence, by fixingthe decisive agents and adjusting the valuations of the non-decisive agentson the relevant issues as appropriate.

The possibility of reasoning in this way highlights the fact that theassumption of Experts makes the aggregation unable to discriminatebetween individual valuations that respect logical consequence and onesthat violate it. Equivalently, any Paretian aggregation with Experts doesnot make use of the fact that individual decision-makers respect logicalconsequence: this piece of information is suppressed by the aggregationprocedure.

The suppression of individual rationality leads to a conflict that isgenerated by the fact that every agent is opposed to at least one of theissues on which the experts impose their deliberation. As I mentionedearlier, such a tension between experts and non-experts makes it natu-ral to think that what is being modelled is not a ‘normal’ conflictbetween collective and individual judgment, but rather a dysfunctionalsituation in which unreliable experts seek to impose their judgment onthe non-experts. If experts were considered reliable, they should not becountered by every non-expert on the issues on which they are compe-tent.

On the other hand, when experts are trusted, a satisfactory judg-ment aggregation should let them fix the aggregate judgment on whatfollows from the issues on which they were asked to express themselvesin the first place. Equivalently, reliable experts can correctly operateprecisely when the aggregation preserves the logical dependence oftheir judgments. Thus, the lemma in this section illustrates a negativeresult only in the sense that it identifies a family of aggregation proce-dures that fail to do what is required when expert judgments cansafely be used.

Because the design of the procedures involved is causing problems, asolution to these problems can be obtained by a change of design: thiscan be done e.g. by applying an aggregation rule like A from the previoussection or the rule that will be described in section VI below.11

The same approach can be pursued with respect to the generaliseddoctrinal paradox obtained by List and Pettit: thanks to the formal setupof section II and the remarks in this section, their result can be intro-duced in a simple way that clarifies its structure, as I will now show.

11 The application of either rule to the panel example above yields a deliberation inwhich x is deemed to have no strengths in either ancient or early modern philosophy, asthe experts concluded.

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V. FROM SYSTEMATICITY TO ASYSTEMATICITY

List and Pettit present their impossibility result as the consequence ofthree assumptions on judgment aggregation. One of them, called universaldomain, is a part of the definition of aggregation from section II andamounts to the fact that an aggregation for n individuals is defined overthe family V of all profiles, i.e., the ordered n-tuples of individual valua-tions that respect logical consequence. Thus, in the present context, thetheorem of List and Pettit follows from two assumptions on aggregations.

In order to state these assumptions, let a be a judgment aggregationfor n agents over the space of issues B. Because there are finitely manyissues, they may be listed as a sequence u1, …, uk for some positive inte-ger k. For any v, I will call its restriction to a single element of B a subpro-file. Thus, for an arbitrary ui in B, the sequence mn(ui) of n individualtruth-value assignments to ui is a subprofile. List and Pettit require anaggregation a to satisfy the following two properties:

1 a is decomposable, i.e., a(v) = (b(mn(u1)), …, b(mn(uk))), for some func-tion b from ordered n-tuples of truth-values into truth-values.

2 b(mn(ui)) = m(ui) only depends on the number of individual valuationsmj ( j = 1, …n) such that mj (ui) = 1.

The first condition, called systematicity by List and Pettit, says that theaggregation of a profile is obtained by applying the same rule b to eachsubprofile in isolation and then putting together the results that have beenindependently obtained. The second condition, which is equivalent, inpresence of systematicity, to the anonymity axiom used by List and Pettit,says that, whatever b is, it only makes use of the number of occurrencesof 1 in every subprofile. Thus, if two subprofiles have the same number ofoccurrences of ‘1’, b assigns them the same truth-value. It can be shownthat:

Lemma V.1. (List and Pettit). Any aggregation satisfying 1 and 2 violateslogical consequence.

I do not wish to re-prove exactly this result but rather to show how it canbe established without using the assumption that individual valuationsrespect logical consequence. In other words, even if this assumption isdropped, it is possible to construct profiles of rational valuations that sup-port paradoxical outcomes. The construction is easily illustrated by takingas a starting point an irration profile, which is then turned into a rationalprofile.



Lemma V.2. (General Doctrinal Paradox). Any aggregation satisfying 1and 2 over unrestricted (i.e., rational or irrational) profiles violates logicalconsequence over at least one rational profile.

Proof. Because B contains at least two atoms p, q, it must also contain(p ∧ q). Then it is possible to construct a profile under which any aggrega-tion satisfying 1 and 2 assigns the same truth-value (whatever it is) to p, qand ¬ (p ∧ q). To find the relevant profile, let us start from a list of valua-tions that includes some inconsistent ones, i.e.,:

It may be assumed that in this profile every agent other than the first threehas the same consistent valuation. Thus, the columns in the above matrixare all equal. This means that a hypothetical, decomposable aggregationthat only depends upon the number of occurrences of ‘1’ in each columnwould assign the same truth-value to p, q and (¬ p ∧ ¬ q). But now it sufficesto interchange the 1’s and 0’s in the first three rows to obtain:

This profile has been obtained from the previous one by a transforma-tion that has not altered the number of occurrences of 1 in each column.Thus, any aggregation satisfying 1 and 2 must assign the same truth-valueto p,q and ¬ (p ∧ q) when operating on this rational profile.

The last proof essentially shows that systematicity and anonymity pre-serve the outcome of the aggregation under arbitrary permutations of theentries in any column of the profile matrix: as a result, the relation of logicalconsequence in the rows of the profile matrix does not generate informationthat can used by the aggregation procedure. Thus, the rationality of individ-

p q … ¬ (p ∧ q)

1: 1 1 … 1

2: 1 1 … 1

3: 0 0 … 0

n: …

p q … ¬ p ∧ q

1: 0 1 … 1

2: 1 0 … 1

3: 1 1 … 0

n: …

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ual judgments does not impose a constraint on the aggregation but canbe combinatorially generated by searching the unrestricted space ofprofiles.

List and Pettit take the outcome of their negative result to highlight aconflict between rationality and individual responsiveness: this interpreta-tion, however, would be warranted only if the aggregation proceduresdescribed by their result could at least make use of individual rationality.This is not the case because systematicity in particular requires thatlogically dependent issues in different columns of the judgment matrix betreated as if they were independent. One might say that systematicity gen-erates a disconnected or asystematic deliberation, which is not responsiveto the logical interrelatedness of distinct issues.

The problematic character of systematicity has already been pointed outby Chapman, who has described this assumption as a condition thatreduces every judgment to an unreasoned (i.e., atomic) judgment.12 Chap-man’s qualification amounts to the fact that systematicity treats the elementsof B as if they were all atoms, i.e., logically independent of each other.

The remarks of this section do not merely confirm the correctness ofChapman’s observation, but also illustrate one general condition underwhich judgment aggregation procedures are able to generate impossibilityresults: this condition is simply their invariance under the transition fromrational to irrational profiles of individual valuations.

It follows that, in order to avoid paradoxes, one should look for proce-dures that violate the invariance condition. The next section describessome simple ways of doing it.

VI. REDESIGNING AGGREGATION

In view of the formal setup of section II, an aggregation procedure maybe regarded as a method to amalgamate a finite family of valuationsrespecting logical consequence and defined over an issue-space B into asingle valuation m. The paradoxes of judgment aggregations amount tothe fact that, in presence of certain assumptions, the amalgamation doesnot preserve the relation of logical consequence. As far as the result ofList and Pettit is concerned, this is essentially due to the difficulties pro-duced by the assumption of systematicity. Because, moreover, List andPettit generalise a difficulty that emerges when simple majority is

12 See B. Chapman, ‘Rational Aggregation’, Politics, Philosophy and Economics, 1 (2002),pp. 337–54, at p.344.



employed in the aggregation, it is natural to think of solving the paradoxthat they have constructed by looking for aggregations that do not giverise to violations of logical consequence, make use of simple majority andviolate systematicity.

A procedure that trivially satisfies these desiderata is well-known fromthe literature by the name of a premise-based rule. A premise-based rule thatrelies on simple majority applies this voting method to generate aggregateassignments for the atoms of a given issue space B and then uniquelyextends this partial valuation to the whole of B by means of truth-tables.

A premise-based rule trivially avoids violations of logical consequencebut may look unattractive because it does not make any use of the indi-vidual valuations on non-atomic elements of B in order to determine theiraggregate valuation. From this point of view, a premise-based rule seemsto impose on judgment aggregation a mechanical computational structurethat bypasses the expression of individual judgments on non-atomic issues.Such an impression can work as an argument against this type of ruleonly if one denies that individual and aggregate valuations should bestructured in the same way. To see this, just note that, when individualvaluations satisfy logical consequence, they are uniquely determined bytheir atomic assignments alone. Individuals thus need but to compute,rather than express, their own judgments on non-atomic issues. If this isto be the case for aggregate valuations as well, it suffices to specify themon atomic issues alone, by means of a premise-based rule.

This, however, is not enough to establish the general viability of premise-based rules: the problem is that these rules ignore so much individual infor-mation that they are indifferent as to whether individual valuations do ordo not violate logical consequence. As any aggregation that works only onatomic issues, they must neglect individual judgments on issues that are notindependent. Thus, one may say that, while rules like the ones studied byList and Pettit are too weak to discriminate between consistent and inconsis-tent profiles of individual valuations, premise-based rules are too strong tocare whether profiles are consistent or not.

It follows that an improvement on premise-based rules must be sought,which is more sensitive to the logical structure of individual valuations. Afirst improvement based of the required kind, which is again based onsimple majority, can be easily described by considering, to fix ideas, a sit-uation in which there are only three agents 1, 2, 3. One may associate tothe family of agents a set of majority coalitions C whose elements are{1,2},{2,3},{3,1}, {1,2,3}. Each element of C describes a group of agentsthat, if unanimous on the judgment of an issue, can impose the sameaggregate judgment under simple majority. Now take an arbitrary issue u

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in B. If m(u) is the aggregate valuation of u, consider the aggregation ruledefined by the following condition:

mðuÞ ¼ i if ff j : mjðuÞ ¼ ig 2 C ; with i = 0, 1 and j = 1, 2, 3.

This rule, call it M, says that the aggregate truth-value of u is the truth-value on which some majority coalition is unanimous. The aggregationmethod that I am interested in is an amendment of M. In order to seewhy an amendment is needed, it is necessary to look at the way M workson B. The elements of B may be thought of as disjunctions of conjunc-tions of B’s atoms and negated atoms. Technically, this amounts to sayingthat each element of B is presented in disjunctive normal form. Then wemay think of an aggregation as a procedure that evaluates first the atomsof B and their negations, then their conjunctions and finally the disjunc-tions of these conjunctions.

As far as the preservation of logical consequence is concerned, M exhib-its certain desirable properties. For instance, if, under M, u ⊧ w and m (u)= 1, then m (w) = 1. This is because there is an element C of C that unani-mously assigns 1 to u: since every agent in C respects logical consequence,every agent in C assigns 1 to w as well, i.e., m(w) = 1 by the definition of M.

In addition, the rejection of conjunctions and the acceptance ofdisjunctions are dealt with as expected. By this I mean, respectively, that,given the conjunction u1 ^ . . . ^ un; if mðukÞ ¼ 0 for some k ∊ {1, …, n},then m (u1 ∧ … ∧ un) = 0 and that, given the disjunction u1 ∨ … ∨ un ifm(uk) = 1 for some k ∊ {1.…, n}, then m(u1 ∨ … ∨ un) = 1.

The above two facts prevent certain violations of logical consequencefrom a finite set of premises or to a finite set of conclusions but not anyviolation of logical consequence. Fortunately, though, it is possible to pindown exactly the potentially problematic cases, in which violations mayoccur, since only two possibilities are left. These are the situations inwhich the aggregate valuations of every issue in a conjunction is 1 whilethe aggregate valuation of the conjunction is 0, and the dual situations inwhich the aggregate valuation of a disjunction is 1 even though the aggre-gate valuation of every disjunct is 0.13

In the light of these remarks, an easy amendment N of M can be intro-duced, which prevents the violation of logical consequence. It suffices torequire that N should work exactly like M except possibly for the casesdescribed by the following two clauses:

13 To see why the former case may arise, suppose that every agent in {1, 2} assign 1 top while 1 assigns 0 to q and that every agent in {2, 3} assigns 1 to q while 3 assigns 0 to p.In this case M yields m(p ) = m(q) = 1 while m(p∧q) = 0 because {3, 1} rejects the disjunction.The dual situation involving disjunctions can be constructed in a similar manner.



1 For an arbitrary conjunction c of atoms or negated atoms, m(c) shouldbe 1 if every conjunct’s aggregate valuation is 1;

2 For an arbitrary disjunction d of conjunctions, m(d) should be 0 if everydisjunct’s aggregate valuation is 0.

The last two clauses impose amendments to M exactly when and if viola-tions of logical consequence may arise. Thus, N is an aggregation of valu-ations that respects logical consequence, uses the simple majority rule butviolates systematicity, in the light of 1 and 2 above. Note that N works foran arbitrary odd number of agents and, when supplemented with a treat-ment of ties, also extends to any even number of agents.14 The advantageof rule N over a premise-based rule is that N does not neglect the individ-ual judgments on non-atomic issues, although it may correct their aggre-gate in the light of a rationality constraint.

VII. ALTERNATIVE STRATEGIES

The particular approach that I have adopted to deal with the paradoxesof judgment aggregation is modelled after the ideas that have been devel-oped by Donald Saari in order to resolve several paradoxes of socialchoice theory and, in particular, voting theory15. Saari has emphasisedthe importance of studying how paradoxical results depend on the designof the procedures that generate them. Once the relation between theemergence of paradoxes and the properties of the type of design thatbrings them about is understood, it becomes possible to avoid paradoxesby choosing a more appropriate, alternative design.

At least two other important approaches to the resolution of paradoxeshave been developed in the context of social choice theory: in this sectionI will explain in which sense their application to judgment aggregation isless promising than the application of Saari’s strategy.

One of these strategies has been originally articulated by AmartyaSen16. For profiles of preferences, which were the objects studies by Sen,his approach requires enriching their structure through a suitable restric-

14 There are various ways to deal with ties. The easiest one is perhaps to assign anyone set of aggregate values to the tied issues that is consistent with the aggregate values ofthe remaining issues.

15 A comprehensive, elementary exposition of this approach is contained in D.Saari,Geometry of Voting, (New York: Springer 1994).

16 Especially in A. Sen, ‘On Weights and Measures: Informational Constraints in SocialWelfare Analysis’, Econometrica, 45 (1977), pp. 1539–72 and A. Sen, ‘Personal Utilities andPublic Judgements: Or What’s Wrong With Welfare Economics’, The Economic Journal, 89(1979), pp. 537–58.

352 DAVIDE RIZZA


tion of their utility representations. Sen actually showed that sufficientlyrichly structured profiles do not give rise to paradoxes. This particularstrategy could in principle be transferred to judgment aggregation,although it is not easy to see how valuations or the issue space should beenriched in order for paradoxes to stop arising. At the same time, Sen’sapproach has one obvious drawback, in that it can deal with paradoxesonly under certain strong structural assumptions, i.e., not in general. Anapplication of Sen’s approach to judgment aggregation would thereforenot resolve the paradoxes in general, unlike an application of Saari’sapproach, which does not require any structural enrichments.

The other important route to avoiding paradoxes in social choice the-ory has been to look for logically weaker aggregation procedures fromwhich unwanted outcomes cannot arise. This approach works in judg-ment aggregation as well but is not devoid of problems, as I will show bymeans of two examples.

First, it possible to avoid the generalised doctrinal paradox by keepingsystematicity and anonymity in place while restricting the family of pro-files over which the aggregation is allowed to be applied.17 Such a result,however, does not follow from the fact a family of problematic aggrega-tions is being replaced by a less problematic one. It rather follows fromthe fact that the a family of problematic aggregations is not given enoughscope to be able to generate troubles that depend on inherent shortcom-ings of the aggregations themselves.

In other words, the search for logically weaker assumptions per se, evenwhen it leads to conditions that are exempt from paradoxes, may be lessthan satisfactory because it leaves the problematic features of aggregationprocedures undisturbed while restricting their scope. This is certainly trueof the conditions that restrict the family of profiles over which a proce-dure unable to discriminate between individually rational and individuallyirrational profiles is allowed to act. In other words, the difficulty withreplacing the condition of universal domain with weaker restricteddomain conditions is that, instead of amending problematic procedures,one retains them unchanged while shrinking the space of possibilities towhich the apply. Because it is not the character of the space but the char-acter of the procedures that gives rise to paradoxes, a restriction of thespace appears less urgent than a revision of the procedures.

In fact, there are circumstances in which the isolation of weaker conditionson aggregation procedures proves even counterproductive: this is clearly

17 In other words, the assumption of unrestricted domain is weakened. This is done inC. List, ‘Possibility Theorem on Aggregation over Multiple Interconnected Propositions’,Mathematical Social Sciences, 45 (2003), pp. 1–13.



illustrated by an immediate generalisation of systematicity called indepen-dence.18 If the decomposability condition 1 of section V is weakened by drop-ping the requirement that the same aggregation rule b should be applied toevery subprofile, one obtains independence, which therefore allows the possi-bility that distinct rules apply to distinct subprofiles.

While every procedure that satisfies systematicity and anonymity alsosatisfies independence, there are pathological procedures that do not sat-isfy systematicity but satisfy independence and anonymity: in other words,the family of troublesome procedures is inflated by replacing systematicitywith independence. To see this, consider a procedure that applies an arbi-trary rule satisfying unanimity to every element of B except for an atom q,to which an arbitrary procedure that satisfies anti-unanimity19 is applied.In this case, if every agent assigns 1 to p,q and p∧q the aggregate valuationcomputed using any rule of the type just described is inconsistent. Theprocedure just defined is independent and anonymous but not systematic,because it does not apply a uniform rule to the aggregation of every issue.It follows that, in presence of independence, the family of anonymous pro-cedures that give rise to paradoxical outcomes contains members that sy-stematicity would rule out. Because of this, the problems posed by theaggregations that satisfy anonymity and systematicity do not decreasewhen systematicity is replaced with the weaker postulate of independence.

The upshot of these simple remarks is that using weaker conditions onjudgment aggregation to avoid paradoxes is not an unproblematic moveand may even reveal undesirable, if it leaves the shortcomings of thestronger procedures intact. In order to choose conditions under whichaggregation procedures do not generate paradoxes, one needs to haveclarified beforehand the reason why paradoxes arise: if this is known, thenit is possible to see what must be changed in the relevant aggregation rulein order to eliminate the paradoxes. The possibility, illustrated in the pre-vious sections, of identifying and amending the shortcomings of aggrega-tion procedures suggests that the paradoxes originated by them shouldnot be regarded as signs of inherent limitations of coordinated reason butonly as problems with the design of decision procedures. While we wouldnot be able to transcend the limits of reason, we can improve the rulesthat we ourselves construct to coordinate our deliberations.

University of East Anglia

18 For a statement of this condition and a discussion of its relation to systematicity, seeDietrich (p. 537 and 554).

19 This means that, whenever every individual valuation of q is i, its aggregate valuationis 1�i, with i = 0,1.

354 DAVIDE RIZZA


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Resolving Paradoxes In Judgment Aggregation