Groups often need to reach decisions and decisions can be complex,involving the assessment of several related issues. For example, in auniversity a hiring committee typically decides on a candidate on thebasis of her teaching and research qualities. A city councilconfronted with the decision of building a bridge, may ask its membersto state whether they are favorable or not and, at the same time, toprovide reasons for their position (like economical and environmentalimpacts, or expenditure considerations). Lastly, jurors are requiredto decide on the liability of a defendant by expressing theirjudgments on the conditions prescribed by the relevant code of law forthe case at hand. As pointed out by Kornhauser and Sager (1986)referring to real jury trials, the aggregation of individual opinionson logically interrelated propositions can lead to a paradoxicalresult, the so-calleddoctrinal paradox. Inspired by thedoctrinal paradox in jurisprudence, the problem of judgmentaggregation attracted the interest of political scientists,philosophers, logicians, economists and computer scientists. Links tosocial choice theory have shown that, similar to the problem ofpreference aggregation (Arrow 1951/1963; Sen 1970), a judgmentaggregation procedure that satisfies a number of desirable propertiesdoes not exist.

The question judgment aggregation addresses is how we can defineaggregation procedures that preserve individual rationality at thecollective level. From a philosophical point of view, such questionconcerns the nature of group attitudes such as group beliefs (Roth2011). When the city council decides to build the bridge, the decisionis taken on the basis of individual beliefs that, for example, thebridge will have a positive impact on the development of the economicactivities in the area and does not represent an environmental threat.Thus, the formal approach to judgment aggregation can serve to castlight on the dependence between individual and collective beliefs (ifthere are any). The questions tackled by judgment aggregation are alsorelevant for the testimony problem investigated in social epistemology(Goldman 1999, 2004, 2010). How shall the diverging opinions ofexperts in a panel be combined and how should a rational agent respondto such disagreement?

The problem of combining potentially conflicting pieces of informationdoes not arise only when a group of people needs to make a decision.Artificial intelligence also explores ways to aggregate conflictingsensors’ information, experts’ opinions or databases intoa consistent one (Bloch et al. 2001). The combination of informationcoming from heterogeneous sensors improves the deficiencies of theindividual sensors increasing the performances of a system. Examplesare sensors of gesture recognition, screen rotation, and accelerometerin smart phones. Distributed databases may need to be accessed andmanaged at the same time to share data, for example, a hospital mayneed to access the data collected about patients by different units.Internet users can find ratings on products provided by people whohave purchased and assessed them on different online platforms. Thetype of information to be combined can differ, and so itsrepresentation can be numerical or symbolic: numbers, linguisticvalues, statistical probability distributions, binary preferences,utility functions etc. Yet all examples mentioned above deal with theproblem of merging items coming from heterogeneous sources and withthe issue of managing conflicts. At a purely formal level, beliefmerging studies the fusion of independent and equally reliable sourcesof information expressed in propositional logic. As with judgmentaggregation, belief merging addresses the problem of fusing severalindividual bases expressed in propositional logic into a consistentone. Given the structural similarity of the problems investigated bythese two disciplines, exploring their connections can reveal howsimilar these formalisms really are. On a more practical level, theapplication of operators defined in belief merging to judgmentaggregation problems lead to the definition of a wider class ofaggregation operators for judgment aggregation, the so-calleddistance-based procedures.

The focus of this entry is to draw explicit connections betweenjudgment aggregation and the belief merging literature. Judgmentaggregation will be briefly introduced in the next section. For a morecomprehensive introduction to judgment aggregation the reader isreferred to (Grossi and Pigozzi 2014; Endriss 2016).

• 1. Judgment Aggregation
• 2. Belief Merging
  • 2.1 A framework for merging under integrity constraints
  • 2.2 The distance-based approach
• 3. Belief merging applied to the discursive dilemma
  • 3.1 Extensions and criticisms
• 4. Other topics
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Judgment Aggregation

The formal work on judgment aggregation stemmed from the“doctrinal paradox” in the jurisprudence literature(Kornhauser and Sager 1986, 1993, 2004; Kornhauser 1992). The paradoxshows that judges may face a real danger of falling into collectiveirrationality when trying to reach a common and justified verdict.Despite the recent birth of the discipline, structurally similarproblems seemed to have been first pointed out by Poisson in 1837 (aspointed out in Elster 2013), and later noted by the Italian legaltheorist Vacca in 1921 (see Spector 2009).

In Kornhauser and Sager’s court example (1993), a three-membercourt has to reach a verdict in a breach of contract case between aplaintiff and a defendant. According to the contract law, thedefendant is liable for breach of contract (propositionr) ifand only if the contract forbid the defendant to do a certain actionX (propositionp) and the defendant did actionX(propositionq). Suppose that the three judges express thejudgments as inTable 1.

Propositionr is theconclusion, whereasp andq are thepremises. The legal doctrine can thus belogically expressed as \((p\land q)\leftrightarrow r\), stating thatpremisesp andq are both necessary and sufficient forthe conclusionr.Table 1 shows that each judge respects the given legal doctrine, by declaringthe conclusion to be true if and only if she deems both premises true.If the judges aggregate their individual opinions using majority ruleon the judgments on each proposition, the resulting judgment set is{p,q, notr}, which constitutes a violation ofthe legal doctrine. This is an instance of the doctrinal paradox:despite the individuals being logically consistent, the group’sjudgment on the propositions is not consistent with the legaldoctrine. In the example above, the judges cannot declare thedefendant not liable and, at the same time, state that both conditionsfor her liability apply. Thus, the court faces a dilemma. Eitherjudges are asked to express judgments on the premises only, and thecourt’s decision onr is logically derived from themajority on the premises (thepremise-based orissue-by-issue procedure), or the verdict is decided by themajority judgment onr (theconclusion-based orcase-by-case procedure) ignoring the opinions on thepremises. Instances such as that inTable 1 illustrate that the two procedures may give opposite results. In thecourt example the issues on which the judges have to express aposition are distinguished into premises and conclusion. We shouldnote, however, that the theory of judgment aggregation does notrequire such a distinction. The court decision of declaring thedefendant not liable (despite a majority in favour of the two criteriato declare her liability) would be inconsistent with the decision rule\((p \land q) \leftrightarrow r\) even without a distinction betweenpremises and conclusion.

This was not the first time that the definition of a collectiveoutcome by majority rule resulted in a paradoxical result. Already in1785, the Marquis de Condorcet discovered what is now known as theCondorcet paradox. Given a set of individual preferences, ifwe compare each of the alternatives in pairs and apply majorityvoting, we may obtain an intransitive preference (or cycle) in thecollective outcome, of the type that alternativex is preferredtoy,y is preferred toz, andz tox, making it impossible to declare an alternative the winner.The similarities between the Condorcet paradox and the judgmentaggregation paradox were promptly noticed by Kornhauser and Sager(1986) and List and Pettit (2004). The study of the aggregation ofindividual preferences into a social preference ordering is the focusof social choice theory (List 2013). The Nobel Prize winner KennethArrow proved the landmark result by showing that the problem whichCondorcet stumbled upon is more general and not limited to majorityrule. Arrow’s impossibility theorem (Arrow 1951/1963; Morreau2014) states that, given a finite set of individual preferences overthree or more alternatives, there existsno aggregationfunction that satisfies a few plausible axioms. There are a number ofresults similar to Arrow’s theorem that demonstrate the“impossibility” of judgment aggregation. The firstimpossibility theorem of judgment aggregation (List and Pettit 2002)was followed by further generalizations (Pauly and van Hees 2006;Dietrich 2006; Mongin 2008).

Let us restrict attention to the aggregation of judgments formulatedin the languageL of propositional logic (the problem ofjudgment aggregation can be generalized to modal and conditionallogics as well as predicate logic, see Dietrich 2007; for a morein-depth discussion of non-classical logics and judgment aggregationsee Grossi 2009, Porello 2017 and Xuefeng 2018). The set of formulason which the individuals express judgments is calledagenda\((A\subseteq L\)). The agenda does not contain double negations\((\neg \neg \varphi\) is equivalent to \(\varphi\)), is closed withrespect to negation (i.e., if \(\varphi \in A\), then also \(\neg\varphi \in A\)) and is generally assumed not to contain tautologiesor contradictions. For example, the agenda of the court case is \(A =\{p, \neg p, q, \neg q, p \wedge q, \neg (p \wedge q)\}\). Dietrichand List (2007a) showed that, when the agenda is sufficiently rich(like, for instance, the agenda of the court decision or \(\{p, \negp, q, \neg q, p \to q, \neg (p \to q)\}\)), the only judgmentaggregation rules satisfying thedesiderata below aredictatorships. An aggregation function is a dictatorship when, forany input, the collective outcome is taken to be theindividual judgment of one (and the same) individual, i.e., thedictator. In a dictatorial aggregation function all individual inputsbut the dictator’s are ignored. Ajudgment set is aconsistent and complete set of formulae \(J \subseteq A\). A judgmentset is complete if, for any element \(\varphi\) of the agenda, either\(\varphi\in A\) or \(\neg \varphi\in A\) (that is, any item of theagenda has to be accepted of rejected). Given a group ofnindividuals, aprofile is an-tuple of individualjudgment sets \(\langle J_1, \ldots, J_n \rangle\). Finally, ajudgment aggregation rule F is a function that assigns toeach profile \(\langle J_1, \ldots, J_n\rangle\) a collective judgmentset \(F(J_1, \ldots, J_{n}) \subseteq A\). The conditions imposed onF are the following:

Universal Domain: All profiles of consistent andcomplete (with respect to the agenda) judgment sets are accepted asinput of the aggregation function. The profile of the doctrinalparadox inTable 1 is a legitimate input because the judges expressed acceptance orrejection on each issue in the agenda and their opinions respected therule \((p \wedge q) \leftrightarrow r\).

Collective Rationality: Only complete and consistentcollective judgments are acceptable as outputs. The collectivejudgment of the example inTable 1 is complete (the group accepts or rejects each agenda’s item)but is not consistent because it violates the rule \((p \wedge q)\leftrightarrow r\).

Independence: The collective judgment on eachproposition depends only on the individual judgments on thatproposition, and not on other (considered to be independent)propositions in the agenda. (This condition reformulates in thejudgment aggregation framework theindependence of the irrelevantalternatives condition in Arrow’s theorem for preferenceaggregation.) Proposition-wise majority rule (as in the court exampleinTable 1) satisfies the independence condition because the groupacceptance/rejection of each agenda’s item depends on whether amajority of the individuals accepted/rejected that proposition.

Unanimity Preservation: If all individuals submit thesame judgment on a propositionp\(\in\)A, this is inthe collective judgment set.

Despite the undemanding conditions, it can be shown that there existsno judgment aggregation ruleF that jointly satisfies the aboveconditions that is not a dictatorship. This impossibility result isparticularly meaningful because, when reformulated for a preferenceframework, it can be shown that Arrow’s theorem (for strictpreference orderings) is obtained as a corollary (Dietrich and List2007a). This led Dietrich and List to say that judgment aggregationcan be seen as a more general problem than preference aggregation (seeGrossi and Pigozzi 2014 for details on such reformulation).

In addition to formal connections between the two types of aggregationproblems, from a conceptual point of view judgment aggregation extendsthe problems of preference aggregation to more general decisionproblems. Although the models provided by social choice have improvedour understanding of many familiar collective decision problems suchas elections, referenda and legislative decisions, they focusprimarily on collective choices between alternative outcomes such ascandidates, policies or actions. They do not capture a whole class ofdecision problems in which a group has to form collectively endorsedbeliefs or judgments onlogically interconnectedpropositions. Yet, as the examples given in the introduction alsoshow, such decision problems are common and not limited to courtdecisions. Pettit (2001) coined the term ofdiscursivedilemma to highlight the fact that such problem can arise in allsituations in which a group of individuals needs to reach a commonstance on multiple propositions.

Impossibility results often bear a negative flavor. However, they alsoindicate possible escape routes. Consistent collective outcomes can beobtained when the universal domain condition is relaxed (considering,for example,unidimensionally aligned profiles (List 2002), aconditions similar to Black’s single-peakedness in preferenceaggregation (Black 1948)) or when the collective rationality conditionis limited to require consistent (but not complete) collectivejudgments. Possibility results are also obtained when the independencecondition is relaxed. The premise-based procedure seen in thecourt’s case is an example of an aggregation rule that violatesindependence. There, the collective position on the conclusion isderived by logical implication from the majority judgments on thepremises. More in general,sequential priority rules violateindependence and guarantee consistent group positions: the elements ofthe agenda are aggregated following a pre-fixed order, and earlierdecisions constrain later ones. The reader is referred to (List andPuppe 2009; List 2013; Grossi and Pigozzi 2014; Endriss 2016) forthorough introductions to judgment aggregation and an overview on moreimpossibility theorems as well as on escapes routes from such results.In the next section we introduce the problem of combining conflictinginformation as it has been addressed in computer science. We will seethat some operators introduced in belief merging are instances ofaggregation procedures that violate the independence condition andthat such operators can be applied to hold concrete aggregationprocedures to judgment aggregation problems.

2. Belief Merging

Computer scientists have studied the aggregation of severalindependent and potentially conflicting sources of information into aconsistent one. As mentioned in the introduction, examples are thecombination of conflicting sensors’ information received by anagent, the aggregation of multiple databases to build an expertsystem, and multi-agent systems (Borgida and Imielinski 1984; Baral etal. 1992; Chawathe et al. 1994; Elmagarmid et al. 1999; Subrahmanian1994; Kim 1995). Belief merging (orfusion) studies theaggregation of symbolic information (expressed in propositional logic)into a consistent base. As we shall see, the process of mergingseveral bases has tight links withbelief change (orbelief revision), an active discipline since the 1980s acrossformal philosophy and computer science that models how human andartificial agents change their beliefs when they encounter newinformation. Revision, expansion and contraction are the three maintypes of theory change studied by Alchourrón, Gärdenforsand Makinson (the so-called AGM theory) who also provided rationalitypostulates for each of them (Alchourrón et al. 1985;Gärdenfors 1988). In belief revision, the focus is on how onebelief base changes in face of new, completely reliable informationand this new piece may be conflicting with existing beliefs in thebase. When I learn that a new acquaintance Rob has a child, I simplyadd the piece of information and eventually derive new consequences(this is expansion). The most interesting case, though, is when thenew information conflicts with previously held beliefs. Suppose thatfrom a common friend I understood that Rob had no kids. Now I learnfrom Rob himself that he has a child. In order to accommodate the newinformation, I need to perform a revision, which consists of removingthe wrong belief that he had no kids (and all the other beliefs thatmay depend on that), add the new input that in fact he has a child,and derive possibly new consequences (see Hansson 2011 for an overviewand Fermé and Hansson 2018 for a comprehensive introduction tobelief change). As for belief revision, in merging the term“knowledge” is used in a broader sense than in theepistemological literature, such that “knowledge” refersto formulas accepted by an agent (i.e., formulas in her knowledgebase), which are not necessarily true. Then, “knowledgebase” and “belief base” are used interchangeably.For Grégoire and Konieczny (2006) belief merging operators canbe used to aggregate also other types of information than knowledgeand beliefs, such as goals, observations, and norms.

If belief revision focuses on how one base changes following theaddition of a new piece of information, belief fusion studies how toaggregate several different and potentially conflicting bases (like,for instance, different experts’ opinions, several databases,information coming from different sources etc.) to obtain a consistentbase. Different approaches have been proposed in the literature. Herewe briefly mentioncombination andarbitrationbefore moving to the merging operators as defined by Konieczny andPino Pérez, which have been applied to judgment aggregation.The first approach to the problem of dealing with aggregatingdifferent and possibly inconsistent databases (Baral et al. 1991;Baral et al. 1992) built on Ginsberg’s idea of consideringmaximally consistent subsets when facing an inconsistent theory(Ginsberg 1986), such as the one that may result from the union of theinformation coming from several self-consistent (but conflicting withone another) agents. Combination operators take the union of theknowledge bases (a finite set of logical formulas) and, if the unionis logically inconsistent, select some maximal consistent subsets. Thelogical properties of suchcombination operators have beeninvestigated in (Konieczny 2000) and compared tomergingoperators as defined in (Konieczny and Pino Pérez 1998, 1999).There are several differences between combining and merging knowledgebases. One difference is that the method by Baral et al. (1991, 1992)is syntax-dependent while merging operators obey the principle ofirrelevance of syntax. According to the principle of irrelevance ofsyntax, an operation on two equivalent knowledge bases should returntwo equivalent knowledge bases. For instance, \(K_1 = \{a, b\}\) and\(K_{2}= \{a \wedge b\}\) have the same logical consequences. So, if\(K_3 = \{\neg b\}\), merging \(K_1\) with \(K_3\) or merging \(K_2\)with \(K_3\) will give two equivalent knowledge bases. On the otherhand, the combination of \(K_1\) and \(K_3\) may not be logicallyequivalent to the combination of \(K_2\) and \(K_3\). Let \(E_1 = K_1\bigsqcup K_3\) (\(\bigsqcup\) is the union on multi-set) and \(E_2 =K_2 \bigsqcup K_3\). The maximal consistent subsets of \(E_1\) are\(\{a, b\}\) and \(\{a, \neg b\}\), and those of \(E_2\) are \(\{a\wedge b\}\) and \(\{\neg b\}\). So each maximal consistent subset of\(E_1\) implies \(a\), but this is not the case for all maximalconsistent subsets of \(E_2\) (example from Konieczny 2000). Anotherdifference is that when combination operators are used, theinformation about the source of the knowledge bases is ignored. Thismeans that, unlike merging, combination operators cannot take intoaccount cardinality considerations. Suppose, for example, that fourmedical experts advise on the effectiveness of four vaccines foradults over 65 years old. Let the propositions \(a, b, c, d\) standfor “Vaccine A (respectively, B, C, D) is effective inover-65s”. If two experts agree that vaccines A and B areeffective in over-65s, one expert esteems that vaccine D (but notvaccine A) is effective and, finally, the last expert agrees with thefirst two that vaccine A is effective and that, if B is effective inover-65s, so is vaccine C too, we can represent the four expertsopinions as four knowledge bases: \(K_1 = K_{2}= \{a, b\}\), \(K_{3}=\{\neg a, d\}\) and \(K_4 = \{a, b\rightarrow c\}\). The union ofthese four bases is \(\{a,\neg a, b, b\rightarrow c, d\}\), which isclearly logically inconsistent. Considering maximal consistent subsetsis one way to avoid inconsistency while retaining as much informationas possible. In this example, the two maximal consistent subsets are:\(\{a, b,b\rightarrow c, d\}\) and \(\{\neg a, b, b\rightarrow c,d\}\). This means that we cannot decide whether to accepta or\(\neg a\). However, a majority of knowledge bases containeda,and only one base contained \(\neg a\). It seems intuitive thata should be in the resulting knowledge base as long as allknowledge bases are treated equally. If, for whatever reason,\(K_{3}\) is more trustworthy than the other knowledge bases, then wemay prefer a combined base in which \(\neg a\) is accepted.

Arbitration is another operator to fuse knowledge bases thathas been introduced in the early Nineties (Revesz 1993, Liberatore andSchaerf 1995). In belief revision, it is commonly assumed that the newinformation is accepted and must be included in the revised base. Bycontrast, arbitration addresses situations in which two sources givecontradicting information but are equally reliable (examples are twoequally competent experts, two equally trustworthy witnesses etc.). Ifwe have no reason to dismiss one of the two sources, the solution isto fuse the two bases rather than revise one by the other. Theoperation is arbitration in the sense that, since both sources areequally reliable, the resulting base should contain as much aspossible of both sources. Liberatore and Schaerf (1995) proposedaxioms for arbitration between two belief bases, and the operatorproposed by Revesz only satisfied some of them. Their proposalsuffered from the fact of being limited to the arbitration of only twobases. This limitation is overcome in the belief merging approach,where a finite number of bases can be merged into a consistentone.

None of the above methods could take into account the popularity of aspecific information item. This meant that those operators could notcapture the view of the majority. The first to introduce a majoritypostulate for the merging of several knowledge bases were Lin andMendelzon (1999). The idea was inspired by the majority rule in socialchoice theory. However, their majority postulate includes a notion ofpartial support that captures the specificity of knowledgemerging with respect to voting, and is not limited to count the numberof bases supporting a propositiona vs. the number of basescontaining \(\neg a\). A knowledge base was defined to partiallysupport a literall if there is a propositiona thatcontains no atoms appearing inl, such that the agent believeseitherl ora is true without knowing which one. Amodel-theoretic characterization of the postulates and specificmerging operators are given in Lin and Mendelzon (1999). In the beliefmerging literature, sources of information are generally assumed to beequally reliable. One way to help to solve the conflict is to relaxthis assumption as, for example, in the extension to merging weightedknowledge bases given in (Lin 1996) or in prioritized knowledge bases(Benferhat et al. 1998; Cholvy 1998; Delgrande et al. 2006).

A new set of postulates for merging operators and the distinction (interms of axioms they satisfy) between arbitration and majorityoperators were introduced by Konieczny and Pino Pérez (1998).In subsequent works (Konieczny and Pino Pérez 1999, 2002) theyextended the framework to include merging underintegrityconstraints, that is, a set of exogenously imposed conditionsthat have to be satisfied by the merged base (Kowalski 1978; Reiter1988). In the next section we present the formal framework introducedby Konieczny and Pino Pérez, which is now the standardframework for belief merging as it overcomes the limitations of theprevious proposals.

The formal methods developed in belief merging have been exported andapplied in areas of social epistemology, like elections and preferenceaggregation (Meyer et al. 2001), group consensus (Gauwin et al. 2005),and judgment aggregation (Pigozzi 2006) to which we return inSection 2.2.

2.1 A framework for merging under integrity constraints

Konieczny and Pino Pérez consider a propositional languageL built up from a finite setAt of atomic propositionsand the usual connectives \((\neg, \land, \lor , \rightarrow,\leftrightarrow )\). An interpretation is a total function \(At\rightarrow \{0, 1\}\) that assigns 0 (false) or 1 (true) to eachatomic proposition. For example, if \(At =\{p, q, r\}\), then \((1, 0,1)\) is the interpretation that assigns true top andrand false toq.^[1] Denote the set of all interpretations by \(W = \{0, 1\}^{At}\). Forany formula \(\varphi \in L\), \(\mymod(\varphi) = \{\omega \in W |\omega \models \varphi\}\) denotes the set of models of \(\varphi\),i.e., the set of truth assignments that makes \(\varphi\) true. If wetake the formula that expressed the contractual law in the doctrinalexample, then \(\mymod((p\land q) \leftrightarrow r) = \{(1,1,1), (1,0, 0), (0, 1, 0), (0,0,0)\}.\) As usual, a formula \(\varphi\) isconsistent if it has at least a model, and a formula \(\varphi\)follows from a set of formulae \(\Phi\) if every interpretation thatmakes all formulae in \(\Phi\) true, makes also \(\varphi\) true.

Abelief base \(K_i\) is a finite set of propositionalformulas representing the explicit beliefs held by individuali. Each \(K_{i}\) is assumed to be consistent. \(\mathcal{K}\)denotes the set of all consistent belief bases. The postulates formerging consider amulti-set of belief bases (beliefprofile, orbelief set, the terminology used in earlypapers) \(E = \{K_1, \ldots , K_{n}\}\). The reason for usingmulti-sets is that an element can appear more than once, thus allowingthe representation of the fact that two or more agents can hold thesame beliefs. This is needed to take into account the popularity of apiece of information, hence to define majority operators. To mark thedistinction with the usual set union \(\cup\), the multi-set union isdenoted by \(\sqcup\) and defined as \(\{\varphi\} \sqcup \{\varphi\}= \{\varphi, \varphi\}\). Two belief profiles are equivalent\((E_1\equiv E_{2})\) if and only if there exists a bijectionffrom \(E_1\) to \(E_{2}\) such that, for any \(B\in E_1\), we havethat \(\models\land f(B)\leftrightarrow \land B\).

Integrity constraints represent extra conditions that should followfrom the merged bases. The interest of integrity constraints is toensure that the aggregation of individual pieces of informationsatisfies some problem-specific requirement. For example, suppose thatmembers of a city council have to decide what to build in a certainarea. We can have constraints on the available budget (enough to coveronly some of the projects) but also constraints on the coexistence ofdifferent projects (we may not build a parking lot and a playground inthat area, but we may build a playground and a public library). If the(possibly empty) set of integrity constraints is denoted by the beliefbaseIC, \(\Delta_{\IC}(E)\) denotes the result of mergingthe multi-setE of belief bases givenIC. Intuitively,the result will be a consistent belief base representing thecollective beliefs and implyingIC.

Konieczny and Pino Pérez (1999, 2002) put forward the followingpostulates forIC fusion operators between equally reliablesources. Let \(E\), \(E_1\), \(E_{2}\), be belief profiles, \(K_1\),\(K_{2}\) be consistent belief bases, and \(\IC\), \(\IC_1\),\(\IC_{2}\), be integrity constraints. \(\Delta\) is anICfusion operator if and only if it satisfies the following rationalitypostulates:

(IC0)

\(\Delta_{\IC}(E) \models \IC\)

(IC1)

If \(\IC\) is consistent, then \(\Delta_{\IC}(E)\) isconsistent.

(IC2)

If \(\land E\) is consistent withIC, then\(\Delta_{\IC}(E)\equiv \land E\land \IC\)

(IC3)

If \(E_1\equiv E_{2}\), and \(\IC_1\equiv \IC_{2}\), then\(\Delta_{\IC_1}(E_1)\equiv \Delta_{\IC_2}(E_{2})\)

(IC4)

If \(K_1 \models\IC\) and \(K_{2} \models\IC\), then\(\Delta_{\IC}(\{K_1, K_{2}\})\land K_1\) is consistent if and only if\(\Delta_{\IC}(\{K_1, K_{2}\})\land K_{2}\) is consistent.

(IC5)

\(\Delta_{\IC}(E_1) \land \Delta_{\IC}(E_{2})\models\Delta_{\IC}(E_1\sqcup E_{2})\)

(IC6)

If \(\Delta_{\IC}(E_1) \land \Delta_{\IC}(E_{2})\) is consistent,then \(\Delta_{\IC}(E_1\sqcup E_{2}) \models \Delta_{\IC}(E_1) \land\Delta_{\IC}(E_{2})\)

(IC7)

\(\Delta_{\IC_{1}}(E) \land \IC_{2} \models\Delta_{\IC_{1\land}\IC_{2}} (E)\)

(IC8)

If \(\Delta_{\IC_1}(E) \land \IC_{2}\) is consistent, then\(\Delta_{\IC_{1\land}\IC_{2}} (E) \models\Delta_{\IC_1}(E)\)

In order to illustrate these postulates, we consider the followingexample, due to (Konieczny and Pino Pérez 1999). A group offlats co-owners wish to improve their condominium. At the meeting, thechairman proposes to build a tennis court, a swimming pool or aprivate parking. He also points out that building two of the threeoptions will lead to a significant increase of the yearly maintenanceexpenses (this corresponds to theIC).

(IC0) ensures that the resulting merged base satisfies the integrityconstraints. This is an obvious condition to impose since these arepostulates for merging under integrity constraints, where the idea isto ensure that the result of the merging satisfies the integrityconstraints. By employing a merging operator, the chairman knows thatthe group will agree on the increase of the expenses, if they decideto build at least two of the three facilities. (IC1) states that, whenIC is consistent, then the result of the fusion operator willalso be consistent. Again, given that the interpretations of themerged bases are selected among the interpretations of the integrityconstraints, ifIC is consistent, the result will also beconsistent. (IC2) states that the result of the merging operator issimply the conjunction of the belief profile and theIC,whenever such conjunction is consistent. In our running example, ifeach person wishing to build two or more facilities endorses the riseof the expenses and the opinions given by the co-owners areconsistent, then the merging will just return the conjunction of theIC and the individual opinions. (IC3) states that if twobelief profiles \(E_1\) and \(E_{2}\) are logically equivalent and\(\IC_1\) and \(\IC_{2}\) are also equivalent, then merging the firstbelief profile with \(\IC_1\) will be equivalent as merging the secondbelief profile with \(\IC_{2}\). This postulate expresses a principlealready imposed on belief revision operators (of which, as we shallsee, merging operators are extensions), that is, theprinciple ofirrelevance of syntax, which says that the result of a mergingoperator depends only on the semantical content of the merged basesand not on their syntactical expression. (IC4) is known as thefairness postulate because it states that when merging twobelief bases \(K_1\) and \(K_{2}\), no priority should be given to oneof them. The merging is consistent with one of them if and only if itis consistent with the other. This postulate expresses a symmetriccondition that operators that give priority to one of the two baseswill not satisfy. (IC5) and (IC6) were first introduced in (Revesz1997) and together they mean that if two groups agree on at least oneitem, then the result of the fusion will coincide with those items onwhich the two groups agreed on. So, if the group of co-owners can besplit in two parties, such that one wants to build the tennis courtand the swimming pool and the other wants the swimming pool and theparking, the building of the swimming pool will be selected as thefinal group decision. Finally, (IC7) and (IC8) guarantee that if theconjunction between the merging on \(E\) under \(\IC_1\) and\(\IC_{2}\) is consistent, then \(\IC_1\) will remain satisfied if\(E\) is merged under a more restrictive condition, that is, theconjunction of \(\IC_1\) and \(\IC_{2}\). This is a naturalrequirement to impose as, less formally, (IC7) and (IC8) togetherstate that if the swimming pool is chosen among the set of threealternatives, it will still be selected if we reduce the set ofalternatives to the tennis court and swimming pool. The last twopostulates are a generalization of two postulates for revision (R5 andR6) in (Katsuno and Mendelzon 1991), who analyzed the revisionoperator from a model-theoretic point of view and gave acharacterization of revision operators satisfying the AGM rationalitypostulates (Alchourrón et al. 1985) in terms of minimal changewith respect to an ordering over interpretations. Like Katsuno andMendelzon’s postulates, (IC7) and (IC8) ensure that the notionof closeness is well behaved, in the sense that if an outcome isselected by the merging operator under \(\IC_1\), then that outcomewill also be the closest (i.e., it will be selected) to \(\IC_{2}\)within the more restrictive constraint \(\IC_1 \land \IC_{2}\)(assuming \(\Delta_{\IC_1}(E) \land \IC_{2}\) to be consistent). InKatsuno and Mendelzon model-theoretic approach, revision operatorschange the initial belief base by choosing the closest interpretationin the new information. Similarly,IC merging operatorsselect the closest interpretation in the integrity constraints to theset of belief bases. Hence, belief merging can be interpreted as ageneralization of belief revision to multiple belief bases(Grégoire and Konieczny 2006).

Two sub-classes of \(\IC\) fusion operators are defined. AnICmajority fusion operator minimizes the level of totaldissatisfaction (as introduced by Lin and Mendelzon 1996), whereas anIC arbitration operator aims at equally distributing thelevel of individual dissatisfaction among the agents. The majorityoperator is similar in spirit to the utilitarian approach in socialchoice theory, whereas the arbitration is inspired toegalitarianism.

Let, for every integer \(n\), \(E^n\) denote the multi-set containingn timesE. AnIC majority operator satisfiesthe following additional postulate:

(Maj)

\(\exists n \Delta_{\IC} (E_1\sqcup E^{n}_{2})\models\Delta_{\IC}(E_{2})\)

Thus, (Maj) states that enough repetitions of \(E_{2}\) will make\(E_{2}\) the opinion of the group. The number of repetitions neededdepends on the specific instance.

AnIC arbitration operator is characterized by the followingpostulate, in addition to (IC0)–(IC8):

(Arb)

Let IC1 and IC2 be logically independent. If \[\begin{align}\Delta_{\IC_1}(K_1) &\equiv \Delta_{\IC_2}(K_{2}), \textrm{ and}\\ \Delta_{\IC_{1} \leftrightarrow \neg \IC_{2}} (\{K_1, K_{2}\}) &\equiv (\IC_1\leftrightarrow \neg \IC_{2}), \textrm{ then} \\\Delta_{\IC_{1}\lor \IC_{2}} (\{K_1, K_{2}\}) &\equiv \Delta_{\IC_1}(K_1).\end{align}\]

Intuitively, this axiom states that the arbitration operator selectsthe median outcomes that areIC-consistent. The behavior ofsuch operator will be clearer when expressed in a model-theoreticalway, as we shall see in the next section.

An example can help to appreciate the different behavior of a majorityand an arbitration operator. Suppose three friends need to decidewhether to buy a birthday present for a common acquaintance. Supposenow that two of them want to buy her a book and invite her out fordinner, while the third friend does not want to contribute to eitherof those presents. If the group takes its decision by majority, thethree friends would resolve to buy a book and to invite her out fordinner, making the third friend very unhappy. If, on the other hand,they use an arbitration operator, they would either buy her a book orinvite her out to a restaurant, making the three members equallydissatisfied. Everyone has exactly one formula in their belief basethat is not being satisfied, so the “amount” ofdissatisfaction for each friend is the same.

The fusion operators in the literature can be divided into twoclasses:syntax-based fusion andmodel-based fusion.The first type takes the propositional formulas as the informationinput, and typically considers the maximally consistent subsets of thebelief profile. In a model-based operator, on the other hand, it isthe interpretations of the formulas that are considered as inputs tothe merging process. Hence, each belief base is seen as a set ofmodels and the syntactic representation of its formulas is irrelevant.Recall the example we used at the beginning ofSection 2 to illustrate that the combination of belief bases issyntax-dependent. We had \(K_1 = \{a, b\}\), \(K_2 = \{a \wedge b\}\)and \(K_3 = \{\neg b\}\). A syntax-based fusion would treat \(a, b, a\wedge b, \neg b\) as inputs, whereas a model-based fusion would take\(mod(K_1) = mod(K_2) = \{(1, 1)\}\) and \(mod(K_3) = \{(1,0),(0,0)\}\). Since model-based operators have been applied to theproblem of judgment aggregation, we will focus on that class ofmerging operators and refer to (Baral et al. 1992; Konieczny 2000;Grégoire and Konieczny 2006) for more on syntax-basedfusion.

2.2 The distance-based approach

AnIC model-based fusion operator selects, among the modelsof the integrity constraintsIC, those that are preferred,where the preference relation depends on the operator that is used.Thus, the collective belief set \(\Delta_{\IC}(E)\) is guaranteed tobe a set of formulas that are true in all of the selected models andto satisfy theIC. In the example of the city council seenearlier, this means that building a playground and a parking lot willnever be selected as a decision outcome. The preference informationusually takes the form of a total pre-order (to recall, a pre-order isa reflexive and transitive relation) \(\le\) on the interpretationsinduced by a notion of distanced between an interpretation\(\omega\) and the profileE, denoted by \(d(\omega,E)\).Intuitively, this is to select a collective outcome that is theclosest (with respect to some notion of distance to be specified) toall individual belief bases while satisfying the integrityconstraints. It should be noted that a distance-based fusion operatordoes not always guarantee a unique result. We will come back to thispoint when we look at the application of belief merging to judgmentaggregation.

We have seen that majority operators are characterized by trying tominimize the total dissatisfaction, whereas the arbitration operatorsaim at minimizing the local dissatisfaction. We can thus see thedistance as a way to capture the notion of dissatisfaction. Inspiredby theeconomy principle employed in belief revision, theoutcome in merging should keep as much as information as possible fromeach individual belief base \(K_i\). In other words, sincethe sources of information are assumed to be equally reliable, themerge should delete as little as possible from the sources. The ideathen is to select the interpretations that minimize the distancebetween the models ofIC and the models of the belief profileE. Formally, this can be expressed as follows:

A distanced between interpretations is a total function \(d: W\times W \rightarrow R^{+}\) such that for all \(\omega, \omega'\inW\):

1. \(d(\omega ,\omega') = d(\omega',\omega )\)
2. \(d(\omega ,\omega') = 0 \textrm{ iff } \omega =\omega'\).

The first point states that the distance is symmetric. Suppose thereare three belief bases: \(K_1 = K_3 = \{a, b, \neg c, d\}\) and \(K_2= \{\neg a, b, c, d\}\). If we denote by \(\omega_i\) theinterpretation of \(K_i\), we have \(\omega_1 = \omega_3 = (1,1,0,1)\)and \(\omega_2 = (0,1,1,1)\). The first point requires that\(d(\omega_1 ,\omega_2) = d(\omega_2 ,\omega_1)\). The second pointstates that if two interpretations are identical, the distance is 0,so \(d(\omega_1 ,\omega_3) = 0\).^[2]

Two steps are needed to find the models ofIC that minimizethe distances to the belief profile. In the first step, we calculatethe distance between each interpretation satisfyingIC (thatis, each candidate merged base) and each individual belief base. Theintuition here is to quantify how far each individual opinion is fromeach possible collective outcome (recall that the outcomes will beselected among the interpretations satisfyingIC). In thesecond step, we need to aggregate all those individual distances todefine the collective distance, that is, the distance of the beliefprofile to each model ofIC. This amounts to quantify how farthe group is from each possible outcome. Finally, the (possibly morethan one) base that minimizes such distance is selected as anoutcome.

For the first step, we need to define the distance between aninterpretation \(\omega\) and a belief baseK. This is theminimal distance between \(\omega\) and the models ofK.Formally: \(d(\omega, K) = \mymin_{\omega'\in \mymod(K)} d(\omega,\omega')\). IfK has more than one model (e.g., \(K_i = \{a\vee b\}\) has three models: \(\{(0,1) (1,0), (1,1)\}\)), \(\omega'\)will be the closest to \(\omega\).

We can now define the distance between an interpretation \(\omega\)and a belief profileE, which is needed for the second step. Weneed an aggregation function \(D: R^{+n} \rightarrow R^{+}\) thattakes the distances between the models ofIC and the beliefbases \(K_i\) calculated in the first step, and aggregate them into acollective distance. This is: \(D(\omega, E) = D(d(\omega, K_1)\),\(d(\omega, K_2)\), \(\ldots\), \(d(\omega, K_n))\). A total pre-orderover the setW of all interpretations is thus obtained. Themerging operator can now select all interpretations that minimize thedistance to the profileE.

Technically, an aggregation function \(D: R^{+n} \rightarrow R^{+}\)assigns a nonnegative real number to every n-ary tuple of nonnegativereal numbers. For any \(x_1,\ldots, x_{n}, x, y \in R^{+}, D\)satisfies the following properties:

1. if \(x\ge y\), then \(D(x_1,\ldots , x,\ldots , x_{n})\geD(x_1,\ldots , y,\ldots , x_{n})\)
2. \(D(x_1,\ldots , x,\ldots , x_{n})=0\) if and only if \(x_1=\ldots=x_{n}=0\)
3. \(D(x)=x\)

The outcome of the merging operator clearly depends on the chosendistance functionsd andD. Among the first proposals(Lin and Mendelzon 1999; Revesz 1993) it was to adopt theHammingdistance (defined below) ford and thesum or themax forD (denoted respectively \(D_{\Sigma}\) and \(D_{\mymax}\)).^[3] WhenD is the sum, the global distance is obtained by summingthe individual ones. The corresponding merging operator is a majorityoperator and is calledminisum as it will select thoseinterpretations that minimize the sum. The merging operator that uses\(D_{\mymax}\) is known asminimax and outputs the judgmentset that minimizes the maximal distance to the individual bases (Bramset al. 2007b). Intuitively,minimax aims at minimizing thedisagreement with the most dissatisfied individual. Two oppositeoutcomes may be selected when \(D_{\Sigma}\) or \(D_{\mymax}\) is used(Brams et al. 2007b; Eckert and Klamler 2007).

The Hamming distance was a commonly used distance in belief revision.The idea is simple. The Hamming distance counts the number ofpropositional letters on which two interpretations differ. So, forexample, if \(\omega = (1, 0, 0)\) and \(\omega' = (0, 1, 0)\), wehave that \(d(\omega, \omega') = 2\) as the two interpretations differon the assignment to the first and the second propositions. Also wellknown is thedrastic distance, which assigns distance 0 iftwo interpretations are the same and 1 otherwise. But the choice ofthe distance is not restricted to those options. Other distances canbe used, that still satisfy the postulates given above (Konieczny andPino Pérez 1999, 2002). The minimization of the sum of theindividual distances is an example of anIC majority mergingoperator. In the next section, we will see this operator applied tothe discursive dilemma.

The distance-based approach can clarify the distinction betweenarbitration and majority operators.Leximax is an example ofarbitration operator. Aleximax operator may taked asthe Hamming distance and, for each interpretation, the distancesbetween that interpretation and then bases \(K_i\) form alist. A pre-order over interpretations is defined by taking thelexicographical order between sequences of distances, fixing an orderover the set of agents. Finally, \(D_{\textit{leximax}}\) selects theminimum. The intuition is that, unlike a majority operator thatselects the option that minimizes thetotal disagreement (byminimizing the sum of the individual distances, for example), anarbitration operator looks at thedistribution of suchdisagreement and selects the option that is fairer to all individuals,that is, it aims at equally distributing the individualdissatisfaction with the chosen outcome (recall the birthday giftexample above). This follows from the definition of the Hammingdistance: the larger the Hamming distance, the more disagreement thereis between two interpretations (here disagreement simply means thatthe interpretations assign different truth values to the sameformula). Suppose that a belief profileE has three bases.Suppose as well that the distances from the two models ofIC(\(\omega\) and \(\omega'\)) are \(D_{\Sigma} (\omega, E) = D_{\Sigma}(\omega', E) = 6\) when we take the sum of the Hamming distances, and\(D_{\textit{leximax}}(\omega, E)= (2,2,2)\) and\(D_{\textit{leximax}}(\omega', E)= (5,1,0)\) when we take thelexicographic order on the distances. In this example, the majorityoperator cannot distinguish between \(\omega\) and \(\omega'\)(because in both cases the sum is 6), while the arbitration operatorwill prefer \(\omega\) to \(\omega'\) as \(\omega\) distributes theindividual disagreement in a fairer way than \(\omega'\).

As mentioned earlier, Liberatore and Schaerf were among the first topropose arbitration operators. However, their approach was limited toonly two bases, and the result of the merge was one of the two bases.Such operator would give questionable results in some situations, likethe one in (Konieczny and Pino Pérez 2002). Suppose that twofinancial experts give you advice regarding four sharesa, b,c andd. According to the first expert, all four sharesare going to rise (denoted by \(\varphi_1= \{(1, 1, 1, 1)\}\), whereasthe second expert deems that all four shares will fall \((\varphi_{2}=\{(0, 0, 0, 0)\})\). According to Liberatore and Schaerf’sarbitration operator, the result will be \(\{(1, 1, 1, 1), (0, 0, 0,0)\},\) which means that either the first or the second expert istotally right. If, on the other hand, we apply an arbitration operatorà la Konieczny and Pino Pérez, we obtain\(\{(0, 0, 1, 1)\), \((0, 1, 0, 1)\), \((0, 1, 1, 0)\), \((1, 0, 0,1)\), \((1, 0, 1, 0)\), \((1, 1, 0, 0)\}\). This result can beinterpreted as that—if we assume that all sources are equallyreliable—we do not have any reason to prefer one or another and,so a reasonable position is to conclude that both can be equallyright. Still, Liberatore and Schaerf’s operator may be used inall situations where the result can be only one of the bases submittedby the individuals. For example, if two doctors meet in order todecide a patient’s therapy, they likely have to decide in favourof one of the two proposals as mixing therapies may not be a feasiblenor a safe option.

A representation theorem (Konieczny and Pino Pérez 1999, 2002)ensures that to each sub-class ofIC merging operators(majority and arbitration operators) corresponds a family ofpre-orders on interpretations (mirroring a similar representationtheorem that Katsuno and Mendelzon (1991) proved for belief revision operators).^[4]

Let us now illustrate how belief merging can be applied to judgmentaggregation problems.

3. Belief merging applied to the discursive dilemma

The merging of individual belief bases into a collective one sharessimilarities to the judgment aggregation problem. In both cases, wewish to aggregate individual inputs into a group outcome, and bothdisciplines employ logic to formalize the bases’ contents. As wehave seen inSection 1, no aggregation procedure can ensure a consistent and complete groupjudgment. However, the merging operators introduced in computerscience ensure a consistent outcome because such operators do notsatisfy independence. The collective judgment on a proposition is notonly determined by the individual judgments on that proposition butalso by considerations ofall other agenda’s items. Itis natural to apply the results about merging methods to theaggregation of individual judgments (Pigozzi 2006).

How do impossibility results in judgment aggregation conciliate withthe fact thatIC merging operators can ensure a consistentcollective outcome? The reason is that merging operators violate theindependence conditions, one of the requirements imposed onaggregation functions in the impossibility theorems. Independenceturned out to be an instrumentally attractive condition because itprotects an aggregation function from strategic manipulation (Dietrich2006; Dietrich and List 2007b). This means that an individual has nointerest in submitting an insincere judgment set in order to get abetter outcome for her. However, independence has been criticized inthe literature as a not suitabledesideratum to aggregatepropositions that are logically interconnected (Chapman 2002; Mongin2008). Clearly, paradoxical results are avoided by resorting toIC, which blocks unacceptable outcome. However, it is worthnoting that in judgment aggregation the collective rationalitycondition (that requires logically consistent outputs) plays ananalogous role asIC in belief merging, that is, blocksunacceptable outcomes like inconsistent majority judgments. Moreover,judgment aggregation impossibility results persist even if weexplicitly import additional integrity constraints into the judgmentaggregation framework (see Dietrich and List 2008b; Grandi 2012).

It has been observed (Brams et al. 2007a) that majority votingminimizes the sum of Hamming distances. This means that, wheneverproposition-wise majority voting selects a consistent judgment set,the same outcome is selected by theminisum rule. Majorityvoting has credentials for being democratic. Another reason to focuson majority distance-based procedures is that the aim of theaggregation of individual judgments should be the right decisionrather than a fair distribution of individual dissatisfactions. Theepistemic link between majority voting and right decisions has beenpointed out in theCondorcet Jury Theorem. The theorem showsthat, when the voters are independent and have an equal probability ofbeing right better than random, then majority rule ensures to selectthe right decision and the probability for doing so approaches 1 asthe voters’ group size increases (see List 2013 for this andsome more formal arguments for majority rule).

Let us consider the three judges example and see what we obtain whenapplying theminisum rule. The legal doctrine corresponds to\(\IC= \{(p\land q) \leftrightarrow r\}\). The court is represented bythe profile \(E=\{K_1 , K_{2} , K_{3}\}\), which is the multi-setcontaining the judgment sets \(K_1 , K_{2} , K_3\) of the threejudges. The three judgment sets and their corresponding modelsare:

Table 2 shows the result for the majority operator that minimizes the sum ofthe Hamming distances. In the first column are all the interpretationsfor the propositional variablesp,q, andr. Theinterpretations that are not models of theIC have a shadedbackground. So, for example, \((1,0,1)\) cannot be selected as thecollective outcome because it violates the legal doctrine. The numbersin the \(d_H (\cdot,K_1)\), \(d_H(\cdot,K_{2})\), and\(d_H(\cdot,K_{3})\) columns are the Hamming distances of each \(K_i\)from the corresponding interpretation. In the last column are the sumsof the Hamming distances.

We see that, withoutIC, the distance-based majority operatorwould select the same (inconsistent) outcome as proposition-wisemajority voting, that is, \((1,1,0)\). This is the outcome that is atthe minimum distance fromE. However, the merging operatorcannot select \((1,1,0)\) as that outcome violates the IC. Neglectingthe shaded rows, only four interpretations are candidates to beselected as collective judgment sets, that is, \((1,1,1)\),\((1,0,0)\), \((0,1,0)\), and \((0,0,0)\). Of those, three are theones that minimize the distances. Thus, collective inconsistency isavoided when a distance-based aggregation is used. However, thismethod does not always guarantee a unique outcome. In the courtexample, this aggregation selects the three models \((1,1,1)\),\((1,0,0)\) and \((0,1,0)\) as group positions. Technically, this issaid to be anirresolute procedure, and a tie-breaking ruleneeds to be combined if we wish to ensure a unique result (as commonin social choice theory).

The applicability of merging techniques developed in computer scienceto judgment aggregation problems does not mean that the twodisciplines have the same objectives. As we have seen, the originalmotivation of belief merging was to define ways to aggregateinformation coming from different sources. Since the sources can havedifferent access to the information, no externally given agenda isassumed. This is a difference with the judgment aggregation framework,where individuals are required to submit their opinion on a given setof items. Belief merging and judgment aggregation do not only differin the type of inputs they aggregate. They also output differentresults: a collective base satisfying some given integrity constraintsfor belief merging, and a collective judgment on the given agenda forjudgment aggregation.

Another difference resides in the fact that judgment aggregationassumes that all members are rational, and so they all submitconsistent judgment sets. In belief merging, this is not required.Agents can submit belief bases that are inconsistent with theIC (Grégoire 2004). If an individual submits ajudgment that violates an integrity constraint, that judgment set willnot figure among the candidates to represent the group position.However, his input will not be disregarded and will be taken intoaccount in the merging process. The possibility to abstain fromexpressing an opinion on a certain item is also easily taken intoaccount in a belief merging setting. If individuals need to have theirsay onp,q andr and one agent believesqandr to be true but does not have a clear opinion onp,this will be represented as \(\mymod(K_1)= \{(1, 1, 1),\) \((0, 1,1)\}\) and the distances calculated accordingly. The completenessrequirement on judgment sets has also been weakened in the judgmentaggregation framework. List and Pettit (2002) first discussed theweakening of completeness in the context of supermajority andunanimity rules. Later, it was shown (Gärdenfors 2006; Dokow andHolzman 2010) that, when judgment sets are not assumed to be complete,any independent and unanimous aggregation function turns out to beweakly oligarchic, that is, a subset of the individuals will decidethe collective outcome. Intuitively, this is a less negative resultthan dictatorship, though it reduces to dictatorship in the case inwhich only one individual belongs to that subset. Dietrich and List(2008a) independently obtained equivalent results on oligarchic rulesto those in (Dokow and Holzman 2010).

Finally, if model-based merging approach is syntax-dependent, judgmentaggregation explicitly permits syntax-dependence. This can give riseto decision framing problems (Cariani et al. 2008) or logical agendamanipulation (Dietrich 2006) when a judgment problem can be presentedusing two logically equivalent but syntactically differentagendas.

A formal investigation of the relationships between belief merging andjudgment aggregation can be found in (Everaere et al. 2015, 2017). Aswe have seen, belief merging takes a profile of propositional beliefbases as input, where such bases represent the beliefs of a certainindividual, not restraint to a given agenda. Judgment aggregation, onthe other hand, asks people to submit their judgments on a specificset of issues. The analysis rests on the assumption that anindividual’s beliefs allow deriving her opinion on theagenda’s items. Thus, in (Everaere et al. 2015) aprojectionfunction p (assumed to be identical for all the individuals) isdefined. The role of such projection function is precisely todetermine the judgments of an agent (input of judgment aggregationoperators) starting from her beliefs (input of belief merging). So,for example, if an individual only believes \(a\land b\) and one ofthe agenda items isa, then the projection function can derivethat the person submits a “yes” as judgment ona.However, if she believes onlya and one of the agenda item is\(a\land b\), she will probably be not able to submit a judgment onthat item. For this reason, individual judgment sets in (Everaere etal. 2015) are not necessarily complete (individuals can abstain onsome agenda’s issues). Using the projectionp, two pathsalong which a collective judgment can be derived from a profile ofbelief bases are considered. Along one path (merge-then-project), theindividual belief bases are first merged using a merging operator andthen the collective judgment is computed by the projectionp.Along the other path (project-then-aggregate), starting from theindividual bases, the individual judgment sets are first computed byp and then aggregated using a judgment aggregation procedure todetermine the collective judgment on the given agenda. Thus, thequestion addressed is whether the two collective judgments obtained byfollowing the two paths coincide. Two cases are considered: thegeneral case (incomplete agendas) and the case in which the agenda iscomplete (i.e. it contains all possible interpretations). For example,the agenda \(A= \{\neg a \land \neg b, \neg a \land b, a \land \neg b,a \land b\}\) is complete whereas the agenda \(A= \{\neg a \land \negb, \neg a \land b, a \land \neg b\}\) is not. Hence, if an individualbelieves \(a\land b\), her judgment set will be \((0,0,0,1)\) in thefirst agenda and \((0,0,0)\) in the second one, which may lead todifferent collective outcomes depending on whether themerge-then-project or the project-then-merge path is used. The factthat by properly choosing the agenda one may manipulate the result hasbeen investigated in the judgment aggregation literature (Dietrich2016, Lang et al. 2016). In the general case, IC merging methods cangive results that are inconsistent with those obtained by usingjudgment aggregation operators satisfying unanimity or majority preservation^[5]. On a more positive side, when the agenda is complete, the collectivejudgments obtained by following the two paths coincide for somejudgment aggregation operators satisfying properties closed to some ICmerging postulates.

Recently, distance-based operators have been applied to a new problemof judgment aggregation: dynamically rational judgment aggregation.Traditionally, judgment aggregation focuses on static aggregation:individuals submit their judgments, and those are aggregated usingsome procedure. However, in more realistic contexts, new informationmay become available at any time. How to aggregate judgments in thelight of new information? Dietrich and List explore whether collectivejudgments can be dynamically rational, meaning that they changerationally in response to new information (Dietrich and List 2024). Ajudgment aggregation rule is said to be dynamically rational withrespect to a revision operator if individually revising the judgmentsin light of some information and then aggregating them leads to thesame result as revising the old collective judgment with the newinformation. Distance-based revision operators, for instance, allowone to minimally change a judgment while consistently including thenew information. Thus, they are rationality-preserving. Dietrich andList investigate the conditions under which judgment aggregation rulesand revision operators can achieve dynamic rationality, highlightingthe challenges and an impossibility theorem associated with it.Possible escape routes (obtained by relaxing the conditions on theaggregation rule or the revision operator) to the impossibility resultare also examined.

3.1 Extensions and criticisms

Theminisum rule applied to merge the judgment sets of thethree judges in the previous section is based on the same principlesas theKemeny rule, a well-known preference aggregation rule(Kemeny 1959). Unlike what happened in social choice, at the beginningthe literature of judgment aggregation focused on the axiomatic methodand only few concrete aggregation rules were proposed and studied.Arguably, the interest of researchers from computer science andmulti-agent systems for judgment aggregation lead to the definition ofmore concrete aggregation rules and to the investigation of theirrelations. The same idea of minimization that plays such a crucialrole in belief merging can be found as a principle in the definitionof several voting rules in social choice theory. For instance, theminisum rule turned out to be equivalent to several otherrules recently introduced in the judgment aggregation literature (Langet al. 2017).

The interest of computer scientists for aggregation methods iswitnessed by the fact that judgment aggregation is now among thetopics of computational social choice, an interdisciplinary disciplinethat promotes exchanges and interactions between computer science andsocial choice theory. Computational issues of aggregation rules areamong the interests of computational social choice. The intuitionbehind the application of complexity theory to aggregation rules is toassess the acceptability of an aggregation rule on the basis ofpragmatic considerations, that is, the algorithmic feasibility ofapplying that rule. So, an aggregation rule is acceptable when itsoutcome is ‘easy’ to compute, that is, it can be solved byan algorithm in time which grows – at worst – polynomiallywith the size of the input (only in some pathological cases we canimagine desiring a rule that will not be able to return an outcome ina foreseeable future). On the other hand, if an aggregation rule ismanipulable, then it is acceptable when it is ‘hard’ foran individual to manipulate it. Thus, the study of the computationalcomplexity of aggregation rules may reveal that, even though a rule ismanipulable, it is actually hard for an individual to act on that. Thecomputational complexity of the distance-based procedures has beenstudied (Endriss et al. 2012; Endriss and de Haan 2015). The highcomputational complexity of Hamming rule in judgment aggregationmirrors a parallel result that the Kemeny rule in preferenceaggregation is also highly computational complex, as first shown inBartholdi et al. (1989) and Hudry (1989). A new rule has been proposedto overcome the high computational complexity of distance-basedprocedures. The average-voter rule (Grandi 2012) selects the judgmentset submitted by the individuals that minimizes the sum of thedistances. Hence, the outcome has to be one of the submitted judgmentsets. This allows reducing the computational complexity and, at thesame time, selects the most representative individual.

A generalization of distance-based methods for judgment aggregationhas been given in (Miller and Osherson 2009). Besides generalizing (bytaking a general metric) the merging operator we have applied to thedoctrinal paradox, they proposed three other distance-based proceduresfor judgment aggregation. In case proposition-wise majority collapsesinto an inconsistent collective judgment set, one method(Endpoint) selects as group outcome the closest (according tosome distance metric) consistent collective judgment set. The othertwo methods (Full andOutput) look at minimal waysto change the profile in order to output a consistent proposition-wisemajority collective judgment set. The difference is thatOutput allows the individual judgment sets in the modifiedprofile to be inconsistent.

Duddy and Piggins (2012) questioned the use of Hamming distancebetween judgment sets. The problem is that, when the agenda containspropositions that are logically connected, the Hamming distance may beresponsible of double counting because it ignores suchinterdependencies. Suppose, for example, that two individuals acceptpropositionsq but disagree on \(p\land q\) (so, one individualaccepts the conjunction, while the other rejects it). This can happenonly if they disagree onp. The Hamming distance between thetwo judgment sets \(K_1 = \{\neg p, q, \neg(p\land q)\}\) and \(K_{2}= \{p, q, (p\land q)\}\) is 2. It is the disagreement onp thatimplies the disagreement over \(p\land q\), so the distance should bejust 1. The alternative distance proposed in order to address thisproblem is a distance that takes the smallest number of logicallycoherent changes needed to convert one judgment set into theother.

4. Other topics

Belief merging is an abstract theory that addresses the problem ofaggregating symbolic inputs, without specifying whether such items arebeliefs, knowledge, desires, norms etc. It is the choice of themerging operator that should best suit the type of inputs. Theframework of judgment aggregation has also been extended to includethe aggregation of other types of attitudes, as in (Dietrich and List2010).

The literature on belief merging includes the study of the strategicmanipulation problem (Evaraere et al. 2007). When an aggregationprocedure is not strategy-proof, an individual who has a preferenceover the possible outcomes can manipulate the result by lying on hertrue beliefs and thus obtain an outcome closer to her truepreferences. In general, merging operators are not strategy-proof whenHamming distance is used, whereas they are strategy-proof when thedrastic distance is employed. For a recent survey of results onstrategic behaviour in judgment aggregation, see (Baumeister et al.2017).

In those situations in which we can assume that there is a fact of thematter (for example, a defendant has—or has not—committeda murder), which each agent has a (noisy) opinion about, thetruth-tracking properties of belief merging operators can beinvestigated (Hartmann et al. 2010; Hartmann and Sprenger 2012;Cevolani 2014). The question is then whether a certain aggregationmethod selects the right decision. Williamson (2009) argues thataggregating theevidence on which the judgments are based isbest for judgment aggregation, as it would yield to the rightdecision. The three-step proposal he advocates distinguishes betweenthree types of propositions: evidence, beliefs and judgments. Evidenceis the support for an agent’s beliefs and judgments, and it isthe right candidate for merging techniques. Judgments, on the otherhand, are best dealt with decision theory that maps degrees of beliefsand utilities to judgments.

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Acknowledgments

I am indebted to Eric Pacuit and to an anonymous reviewer of theupdated entry, who provided many valuable comments and suggestionsthat greatly improved the content and readability of this entry, andto Erman Acar who pointed out some typos in the previous version.