The Logic of Group Decisions: Judgment Aggregation

Abstract

Judgment aggregation studies how individual opinions on a given set of propositions can be aggregated to form a consistent group judgment on the same propositions. Despite the simplicity of the problem, seemingly natural aggregation procedures fail to return consistent collective outcomes, leading to what is now known as the doctrinal paradox. The first occurrences of the paradox were discovered in the legal realm. However, the interest of judgment aggregation is much broader and extends to political philosophy, epistemology, social choice theory, and computer science. The aim of this paper is to provide a concise survey of the discipline and to outline some of the most pressing questions and future lines of research.

1 Introduction

In a telephone interview with Vanity Fair, the U.S. Deputy Secretary of Defense Paul Wolfowitz declared that the issue of weapons of mass destructions was the reason of greatest agreement inside Bush’s team to invade Iraq:

The truth is that for reasons that have a lot to do with the U.S. government bureaucracy, we settled on the one issue that everyone could agree on, which was weapons of mass destruction as the core reason, but, there have always been three fundamental concerns. One is weapons of mass destruction, the second is support for terrorism, the third is the criminal treatment of the Iraqi people. Actually I guess you could say there’s a fourth overriding one which is the connection between the first two [67].

As we know, weapons of mass destruction were never found and this fact led to severe criticisms to the American and the British governments. Leaving all political considerations aside, what matters here is that the Bush administration had to provide reasons to justify the 2003 invasion of Iraq. Once evidence to the existence of weapons of mass destruction appeared to be lacking, the decision itself was open to criticisms.

Decisions as the one above are the object of judgment aggregation: how can a group of rational individuals reach a common decision on a proposition (the conclusion) and - at the same time - provide reasons in support of such decision? If every individual gives her Boolean evaluation on a given set of propositions, how can we determine the group decision on the same propositions? Surprisingly, the answer is not straightforward because seemingly reasonable procedures cannot guarantee a consistent group outcome.

The paradox of a group of rational individuals collapsing into collective inconsistency made its first appearance in the legal literature, where constitutional courts must provide reasoned decisions. Until recently, the discovery of the paradox was attributed to Kornhauser and Sager’s 1986 paper [38]. However, Spector [65] found out that what is now known as the doctrinal paradox [10, 37, 39] was discovered already in 1921 by the Italian legal theorist Vacca [68], who consequently raised severe criticisms to the possibility of drawing collective judgments from individual ones. Consider a three-member court that has to reach a verdict about condemning (or not) a defendant for a certain act. According to the criminal law, the defendant is liable (the conclusion, here denoted by proposition r) if and only if the defendant committed the act and that act is punished by the law (the two premises, here denoted by propositions p and q respectively). This can be formally expressed as (p∧q)⇔r. Suppose that the three judges cast their votes as in Table 1.

Table 1 An example of the doctrinal paradox

There are two procedures the court can follow. Under the conclusion-based procedure, the court can rule on the case by taking the majority vote on the conclusion r. This procedure only provides the group’s decision on the conclusion. Were the judges in the above example following this procedures, the defendant is to be acquitted. Under the other (premise-based) procedure, majority voting is applied to the judges’ recommendations on the premises and the court’s decision on r is inferred via the rule (p∧q)⇔r that formalizes the law. Under the premise-based procedure, the defendant would be convicted. Thus, it can happen that the verdict under the premise-based procedure is opposite to the one reached via the conclusion-based, that is, the court decision depends on the aggregation procedure. We can say that such a situation represents either a dilemma in the choice of the aggregation method to use, or a paradox because rational individuals may form an irrational group. That the group is logically inconsistent can be seen in the last line of Table 1, where a majority accepts both p and q but rejects r, contravening the rule (p∧q)⇔r.

The realisation that aggregations of individual judgments (hence, the possibility of the above paradox to appear) are not bound to justice courts produces further discomfort. Other examples of groups needing to make decisions based on reasons are expert panels, boards of companies, committees in organisations, or informal groups. Pettit [57] introduced the term of discursive dilemma to indicate all situations in which an aggregation of individual evaluations may lead to a choice of the aggregation method.¹

The scope of judgment aggregation is to study formally this kind of aggregation problems. Even though judgment aggregation is a young discipline, we can distinguish two phases in the literature. The early phase saw the flourishing of impossibility theorems, which typically single out few desirable properties for an aggregation procedure and prove that there exists no aggregation rule that jointly satisfies those properties. It should be noted that impossibility results are not entirely negative. They implicitly define the space for possibility ones, as escape routes may be found by relaxing one of the considered conditions. The second (current) phase focuses on the definition and the investigation of concrete aggregation rules. Here, methods to aggregate individual judgments are proposed and their properties are studied.

The aim of this paper is to give a concise and non technical introduction to the field of judgment aggregation.² Section 2 presents the original impossibility result that well illustrates the spirit of the first phase in judgment aggregation. Section 3 summarizes the more recent work on feasible aggregation procedures. In Section 4 we outline three lines of research that we hope will be part of the future research agenda of judgment aggregation. Those topics attest of the inter-disciplinary character of judgment aggregation, as they are at the crossing with philosophy, social choice theory and computer science. Section 5 concludes the paper.

2 The Early Phase: “It’s Impossible”

Majority voting is the exemplary democratic voting rule. But, as we have seen in Section 1, there are situations in which proposition-wise majority rule cannot ensure a consistent social outcome. Is majority rule to be blamed for the untenable outcome? Or, are the logical relations between the propositions the source of the trouble? These are the questions that were first addressed in the discipline. The baseline of the first findings is that rather undemanding conditions on the aggregation functions and on the set of propositions on which individuals have to vote make group inconsistency possible.

The aggregation of individual inputs into a collective outcome has been the subject of investigation before judgment aggregation. Social choice theory has long studied the aggregation of individual preferences into a collective preference order [1, 64]. The analogy between the discursive dilemma and the famous Condorcet paradox ³ in preference aggregation [13] was evident to Kornhaueser and Sager [37, 38] and to List and Pettit [47]. For the formal model of judgment aggregation [46, 47] it must have then appeared a natural choice to employ propositional logic combined with an axiomatic approach in the social choice tradition. Yet, the precise relations between preference aggregation and judgment aggregation have started to be investigated only later, when Dietrich and List [17] showed that the landmark theorem in social choice, i.e. Arrow’s impossibility theorem for strict and complete preferences, can be derived from an impossibility result in judgment aggregation.⁴

In 1951, the future Nobel prize winner Kenneth Arrow [1] showed that the risk of running into the Condorcet paradox is not limited to pairwise majority voting. He proved that no social welfare function can satisfy few desirable conditions. Drawing on the similarities between the two aggregation problems, Arrow’s conditions have inspired similar ones in the new framework of judgment aggregation, and led List and Pettit to the first impossibility result in judgment aggregation [46].

Let us first introduce some terminology and notation. Let N be a finite and non-empty set of agents. The set of propositions (and their negations) on which individuals are called to express a yes/no judgment is called agenda, and is denoted by X.⁵ An individual (resp. collective) judgment set J is the set of propositions an agent (resp. the group) accepts. Formally, J⊆X. Let J _i denote the judgment set of individual i. It is assumed that individual judgment sets are consistent (i.e. they have a model) and complete (i.e. for any proposition in the agenda, they contain either the proposition or its negation). The set of all judgment sets is denoted by $\mathcal {J}$. For example, the agenda of the court example is X={p,¬p,q,¬q,r,¬r}, and (p∧q)⇔r is a constraint expressing the logical relations among the agenda items. A judgment profile P is a |N|-tuple of judgment sets (J ₁,…,J _n ), and the set of all judgment profiles is denoted by $\mathcal {P}$. Finally, an aggregation rule is a function $f: \mathcal {P} \to \mathcal {J}$.

The first impossibility result showed that, given an agenda with at least two atomic propositions (e.g. p and q) and at least one suitable composite proposition (e.g. p∧q, p∨q or p→q) (and their negations), there exists no judgment aggregation rule f that satisfies universal domain, collective rationality, anonymity and systematicity. The conditions imposed on the agenda are rather undemanding. Let us now look at the conditions on the aggregation rule:

Universal domain::

The domain of f is the set of all profiles of consistent and complete judgment sets on X.

Collective rationality::

For any profile $(J_{1}, \ldots , J_{n}) \in \mathcal {P}$, the collective judgment set f(J ₁,…,J _n ) is a consistent and complete judgment set on X.

Anonymity::

For any two profiles $(J_{1}, \ldots , J_{n}), ~(J^{\prime }_{1}, \ldots , J^{\prime }_{n})$ in the domain of f that are permutations of each other, $f(J_{1}, \ldots , J_{n})=f(J^{\prime }_{1}, \ldots , J^{\prime }_{n})$.This means that all agents have equal weight in determining the result of the aggregation.

Systematicity::

$\forall p,q \in X, ~\forall (J_{1}, \ldots , J_{n}), (J^{\prime }_{1}, \ldots , J^{\prime }_{n}) \in \mathcal {P}$, if [$\forall i \in N: p\in J_{i} \leftrightarrow q \in J^{\prime }_{i}$] then [$p\in f(J_{1}, \ldots , J_{n}) \leftrightarrow q \in f(J^{\prime }_{1}, \ldots J^{\prime }_{n})$].Informally, this condition states that the collective judgment on each proposition depends only on the individual judgments on that proposition, and the pattern of dependence is the same for all propositions.

Proposition-wise majority rule satisfies universal domain, anonymity and systematicity, but fails to ensure a consistent group outcome, as the jury example illustrates. Yet, what this theorem says is that the problem is not limited to majority rule and that an aggregation rule that satisfies all the conditions above does not exist. Subsequent theorems have generalized and extended this first result. The landscape of impossibilities has been explored and different classes of agendas and different conditions on the aggregation rules now can be linked to different impossibility results. For example, Pauly and van Hees proved that a stronger result can be obtained just by weakening the anonymity condition to non-dictatorship [56]:

Non-dictatorship::

There exists no individual i∈N such that, for any profile in the domain of f, the collective judgment set is the same as i’s judgment set.

Whereas universal domain, anonymity and collective rationality seem reasonable properties, systematicity soon appeared to be a strong requirement and indeed, for many agendas, the weaker condition of independence is sufficient for an impossibility result to occur [47]:

Independence::

$\forall p \in X, ~\forall (J_{1}, \ldots , J_{n}), (J^{\prime }_{1}, \ldots , J^{\prime }_{n})\in \mathcal {P}$, if [$\forall i \in N: p\in J_{i} \leftrightarrow p \in J^{\prime }_{i}$], then [$p \in f(J_{1}, \ldots , J_{n}) \leftrightarrow p \in f(J^{\prime }_{1}, \ldots J^{\prime }_{n})$].Like systematicity, independence requires that the collective judgment on any proposition in the agenda depends on the individual judgments on that proposition only. The difference with systematicity is that independence does not require that the pattern of dependence is the same across all propositions.

Another result shows that for more restrictive classes of agendas (called non-simple and evenly negatable), the aggregation conditions of systematicity and unanimity⁶ are equivalent to dictatorship [17]. Dokow and Holzman [21] demonstrated that we are forced into dictatorship also when systematicity is weakened to independence and non-simple agendas are replaced by path-connected ones. More results have been proved. We refer to [30, 45] for comprehensive introductions.

In the second part of this section, we want to focus on what is arguably the most controversial condition among those imposed on the aggregation rules, namely independence.

We have seen that the problem faced by judgment aggregation is how to aggregate individual evaluations on logically connected propositions. Yet, a condition that guarantees that the collective position on each proposition depends exclusively on the individual views on that proposition is forced to neglect the connections between the agenda items [11, 50].⁷ Back to 1921, the possibility that the separate aggregations of individual judgments collapse to irrational group positions did not bewilder Vacca [68]. For him, discussion (rather then mere aggregation) is the key to reach group agreements. He compared the separate aggregation of different agenda items to someone who, desiring a beautiful statue, commission the head to a sculptor, the arms to another artist, the legs to a third sculptor, and then pretends to assemble the different parts to form a magnificent statue.

The main justification for independence is that it is a key condition to ensure that an aggregation function is non-manipulable. When an aggregation rule is manipulable, an individual may vote strategically, i.e. upon learning the evaluations of the other agents, she can submit an insincere judgment set in order to ensure a collective outcome that is better for her than if she voted sincerely. Dietrich and List [14, 19] showed that, under universal domain, an aggregation rule is non-manipulable if and only if it is independent and monotonic.⁸ Since voting strategically is commonly seen as an undesirable property, independence finds in its role of preventing manipulability a justification.⁹

We see three criticisms to such argument. First, one may argue that conditions ought to be imposed on aggregation functions because they are appropriate to the problem at hand, and not because they are merely instrumental. The peculiarity of judgment aggregation with respect to social choice theory is that it captures decision problems in which a group needs to form collectively endorsed views on connected propositions rather than focusing on group choices between alternatives (such as candidates, policies and actions). One of the most interesting challenges of this discipline is how a group of individuals may agree on a certain decision (the conclusion) and, at the same time, offer the reasons (the premises) that support that decision. Agenda items are dependent and so we need aggregation procedures that relax the independence condition in reasonable ways. We see in Section 3 that some of the proposed aggregation rule fail to satisfy independence. However, there is no consensus yet on how to relax this condition.

Second, strategic voting is not necessarily a vice. In order to manipulate, individuals need to gather information about other people preferences and to understand the mechanisms of the voting system. This, Dowding and van Hees [22] argue, provides incentives for people to understand democratic procedures and accounts for one of their arguments in favour of manipulation. More specifically related to judgment aggregation, Grossi et al. [31] introduce a form of virtuous manipulation. This happens when - in order to avoid an inconsistent social outcome - agents in a group start a coordinated action and declare a non-truthful, less preferred judgment set.

Third, even if we disagree with Dowding and van Hees, and we still aim at aggregation functions that are robust against strategic voting, we need to consider the computational complexity required to manipulate a specific voting rule. This is one of the reasons of interests in computational social choice [12], a recent and lively discipline stemming from the interactions between computer science and social choice theory. Even if a voting rule is manipulable, it may well be that its manipulation is a computationally intractable problem. These considerations are relevant to the discussions on manipulation. There is some initial work on complexity-theoretic study of judgment aggregation problems [2, 3, 27] and more results will surely be available in the near future.

To conclude, independence plays a key role in the impossibility results in judgment aggregation. Independence has been criticised and the main justification for its maintenance is that it ensures strategy-proofness. We have given three counter-arguments against this position. As we see in the next section, giving up independence is one way to define specific aggregation rules. Our hope is that alternatives to independence will be explored in future research.

3 The Current Phase: “Yes, we can”

We have seen that the early phase of the research in judgment aggregation has been dominated by impossibility results. The current phase is more focused on the construction of specific judgment aggregation rules (or families of rules), bringing the area closer to voting theory and computational social choice, where a large body of work concentrates on specific voting rules, studying their social choice theoretic properties, the computational complexity of manipulability; etc. As observed in [16], such shift became evident at the 2011 Workshop “Judgment Aggregation and Voting Theory” in Freudenstadt. Even though until recently the focus has been on impossibility results, there are a few exceptions that should be mentioned:

• Premise- and conclusion-based rules [20, 39, 46, 50]: If the agenda is partitioned in two distinct subsets (premises and conclusions) and if propositions in each of the two subsets are logically independent, aggregations can be performed on either the premises or the conclusions, as seen in Section 1. As we know, the two procedures may lead to opposite outcomes.¹⁰ This is even more troublesome once we are aware of the possibility of situations like the Paretian dilemma [52], where the premise-based procedure violates a unanimous vote on the conclusion.¹¹

• If we generalise the idea of considering agenda items sequentially, following a fixed linear order over the agenda (e.g. temporal precedence or priority) and that earlier decisions constrain later ones, we obtain sequential procedures [18, 42]. Such procedures represent another way of relaxing independence and of ensuring collective consistency. The problem however is that, in the general case, the result depends on the order followed (i.e. the aggregation is path-dependent) and path-dependence is tightly linked to manipulability [18].

• Quota rules accept a proposition only if it is accepted by a number of individuals greater than a prefixed threshold. The intuition is that some decisions may require a stronger support than simple majority, for instance a supermajority with a threshold of 2/3. The conditions under which quota rules may produce consistent social outcomes are analysed in [18, 54].

• Distance-based rules [49, 59]: Distance-based judgment aggregation rules have been originally derived from distance-based merging operators for belief bases introduced in computer science [36]. Unlike sequential rules, distance-based rules take the individual judgments on the entire agenda into account. Intuitively, distance-based rules work as follows: a predefined distance between judgment sets is assumed. A commonly used distance is Hamming distance, which corresponds to the number of issues on which two judgment sets differ. The consistent judgment sets which are closest (for some notion of closeness) to the individual evaluations are selected as collective outcomes. The procedure violates independence. It is also irresolute, as it may result in multiple outcomes: we can say that group rationality is ensured at price of possible indecision. The solution is to combine the procedure with a tie-breacking rule. Duddy and Piggins [24] proposed an alternative metric to overcome the fact that Hamming distance neglects that propositions are logically connected, thus in some cases leading to double count.

Even if a few families of judgment aggregation rules have been introduced and investigated in the early phase, the literature was dominated by the search of impossibilities, which contrasts with preference aggregation theory, where voting rules are defined and studied per se. Yet, more recent work testifies of a shift toward the construction of concrete judgment aggregation rules.

The first example are scoring rules, a class of standard aggregation rules in preference aggregation, like the plurality rule or Borda rule [5]. Translated into the framework of judgment aggregation, scoring rules select those collective judgments that maximize the total score [16]. Surprisingly, Dietrich found out that several existing judgment aggregation rules (e.g. distance-based and premise-based) can be re-modelled as scoring rules.

Nonetheless, not all aggregation rules fall inside the class of scoring rules. An example is the Condorcet admissible rule introduced by Nehring et al. [53], which approximate the majority judgments when these are inconsistent. Interestingly, several rules in [53] have been independently introduced by Lang et al. [40]. The motivation behind [40] is to define judgment aggregation voting rules based on some minimization process, as it is the case for several voting rules (like Kemeny, Dodgson, Slater, etc.). The intuition is to minimize the change required to obtain a social consistent outcome. There are several ways to implement such minimization principle. Four families of rules are introduced. The rules of the first family restore consistency of the majoritarian judgment, for example, by removing the minimum number of propositions. The rules of the second family take into account the strength of the majority on each issue and restore consistency at the group level by removing the issues with the minimal support. Those of the third family restore the consistency of the majoritarian judgment by removing or changing some individual judgments in a minimal way. Finally, those of the fourth family are based on some predefined distance between judgment sets, and look for a consistent collective judgment minimizing the overall distance to the individual judgment sets, thus extending the class of distance-based rules of [49, 59].

These proposals testify the initial attempts to propose concrete and feasible procedures to the problem of judgment aggregation, and surely do not exhaust the space of concrete judgment aggregation rules. Besides new rules, future developments may lead to complete axiomatic foundations of the newly defined procedures, to study their computational complexity as well as the complexity required to manipulate them.

4 The Research Agenda

In addition to new concrete aggregation rules, what is in the research agenda of judgment aggregation for the near future? What are some of the most pressing and exciting lines of research? It is not easy to predict the future evolution of the discipline. We have already indicated possible developments in the investigation of the complexity of the manipulation problem, in the proposal of new aggregation rules, and in the exploration of proper ways to relax independence. In this section we outline three research questions that have started to be addressed but require further investigation. The selection is influenced by the author’s own research interests. These three topics illustrate also the interdisciplinary character of judgment aggregation with disciplines like philosophy, social choice theory and computer science.

4.1 The Nature of Group Beliefs and of Collective Actions

From a philosophical point of view, the formal analysis of judgment aggregation may help to cast some light on the long debated question of the nature of joint (or collective) actions [62, 63]. A rational agent makes decisions based on her beliefs and desires. But how do groups engage in joint actions? What is the nature of group beliefs? If individual beliefs determine group beliefs and group actions, what are the features of such dependance? Does a group believe that p only when all or most of its members believe that p (the so-called summative view of group beliefs), or can a group have attitudes that diverge from those of its members (the non-summative view)? Addressing the question of whether we can legitimately say that groups bear epistemic states like beliefs is relevant for social epistemology [28].¹²

Pettit [58] argues that some groups have “minds on their own” (in a non-summative way), and hence should be considered subjects of intentional attitudes. When groups need to make reasoned decisions, they tend to follow the premise-based procedure to avoid the risk of incurring into an inconsistent group outcome. By doing so, they take the risk of endorsing a conclusion that was unanimously rejected by the group members or endorsed only by a minority. This, Pettit claims, shows that such groups constitute a unity that satisfies the constraints of rationality, and so they are genuine intentional subjects. Concerns about this argument have been expressed. Roth [62], for example, observes that the fact that individuals follow the premise-based procedure in order to preserve the group rationality does not prove that a group mind exists. A different critique has been moved by Elster [26], who opposes the attempts to assimilate individual and collective actions. In the premise-based procedure, the individuals determining the majority on each premise can be different: in Table 1, for instance, Judges 1 and 3 determine the majority for p and Judges 2 and 3 that of q. To see each of such majority as an expression of the group’s will — Elster continues — we should accept Rousseau’s stance that the general will is revealed by majority voting, thus assuming that it pre-existed the political process. Elster’s critique can be linked to the distinction between aggregative and deliberative models of group decision-making, to which we turn now.

4.2 Group Decisions by Aggregation and Deliberation

As advocated by deliberative democrats, individuals should debate, share information and eventually revise their own opinions rather than simply aggregate their judgments on a pre-fixed agenda in a one-step vote. Pettit examined the implications of the discursive dilemma for deliberative democracy and concluded that the dilemma highlights an ambiguity in the deliberation process [57]. He argued that deliberative democrats need to clarify whether people should be disciplined by reason at the individual or at the collective level. Under the first option, the group may endorse an inconsistent position, whereas under the second option (corresponding to the premise-based procedure) the group may endorse a conclusion that only a minority (or none) of them individually accepted. The analogy that Pettit draws between discursive dilemma and deliberative democracy has been criticised by Ottonelli [55]. The debate in deliberative democracy is by definition unstructured. To imagine a pre-fixed agenda, with reasons and conclusions, and a pre-determined rule connecting them means to misunderstand the meaning of democratic deliberation, where everything can be questioned and members can revise their previously held views.

A reconciliation between social choice theory and deliberative democracy has been promoted by Dryzek and List [23]. They argue that one approach does not have to exclude the other: a deliberation phase may precede aggregation. The advantage being that deliberation facilitates the search of agreement and thus assists the aggregation phase.¹³ However, unlike aggregation, little formal work exists on deliberative models of group decision-making. The relevance of formal models of deliberation becomes clear when two considerations are taken into account. On the one hand, there is empirical evidence to support that deliberation increases homogeneity in a group [66]. On the other hand, in judgment aggregation a possibility result is obtained when the domain of the aggregation function is restricted to accept only profiles with a certain structure. The condition that a profile has to satisfy is called unidimensional alignment [41], and requires that the individuals can be ordered from left to right in such a way that, for every item in the agenda, the individuals accepting it are either to the left or to the right of all the individuals rejecting it.¹⁴ Thus, a topic of investigation is the effect that deliberation can have on individual views. More precisely, it would be interesting to understand to which extent the debate that precedes a vote can help the individuals to revise their positions so that the resulting profile is more homogeneous and, in particular, under which conditions deliberation may lead to a unidimensionally aligned profile. A first social choice theoretic model to revise individual judgments in deliberation has been proposed by List [44], which resulted in an impossibility result. A plausible escape route to such impossibility consists in requiring that each individual should not revise her judgments on a single proposition at a time but should consider the interdependencies among propositions.

4.3 Judgment Aggregation in Abstract Argumentation

Starting from Dung’s seminal paper [25], argumentation theory provides a logic-based formalism for the treatment of defeasible reasoning, conflict resolution, negotiation, and argumentation-based dialogues. Abstract argumentation theory studies the positions (acceptance, rejection, abstention) that a rational agent can take in presence of a given set of arguments, where some arguments are in conflict with others. Roughly, if two arguments are in conflict, this means that they cannot both hold. In the general case, there are several possible evaluations for a set of arguments and an attack relation on them. Several extensions of Dung’s theory have been proposed in the literature and recently researchers have started extending argumentation theory to multi-agent settings. Interestingly, some of these first attempts looked at the problem of judgment aggregation. Given an argumentation graph, Rahwan and Tohmé [61] addressed the question of how to aggregate the (possibly) different individual evaluations of the graph into a collective position. The similarity with the problem addressed in judgment aggregation is striking. Indeed, by drawing on a general impossibility theorem from judgment aggregation, they prove an impossibility result for collective argument evaluation. Welfare properties of collective argument evaluation have also been explored [60].

When dealing with judgment aggregation problems, aside from group inconsistency, it may happen that majority rule selects a consistent combination of reasons and conclusion that no individual voted for (a remembrance of another voting paradox, the multiple election paradox). Thus, in some contexts, a desirable property of group decisions is to ensure an outcome that does not conflict with some of the members’ positions. If we call such property ‘compatibility’, the question to address is whether we can define concrete aggregation operators that guarantee a consistent, unique and compatible group outcome. This question was tackled by Caminada and Pigozzi in [8], where three operators have been defined and investigated using the abstract argumentation framework. Welfare properties of these operators have subsequently been studied [9]. The reason for using abstract argumentation for judgment aggregation problems is twofold: on the one hand, the existence of different argumentation semantics allows to be flexible when defining which social outcomes are permissible. On the other hand, it allows to bring judgment aggregation from classical logic to nonmonotonic reasoning.

5 Conclusion

Judgment aggregation is a fairly recent discipline that addresses a simple problem that nevertheless turns out to be difficult to solve. In this short excursus, we have seen that, whereas the early phase of the literature was dominated by impossibility results, the current phase is exploring concrete judgment aggregation rules bringing the area closer to voting theory and computational social choice. Several exciting research questions are still open. In the last section we have outlined three of them. Our hope is that this survey will help to attract more people to work in this area and to stimulate further research.