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Arrow's theorem in judgment aggregation
Social Choice and Welfare 29 (1):1933 (2007)
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This paper provides an introductory review of the theory of judgment aggregation. It introduces the paradoxes of majority voting that originally motivated the field, explains several key results on the impossibility of propositionwise judgment aggregation, presents a pedagogical proof of one of those results, discusses escape routes from the impossibility and relates judgment aggregation to some other salient aggregation problems, such as preference aggregation, abstract aggregation and probability aggregation. The present illustrative rather than exhaustive review is intended to give readers (...) 

How can the propositional attitudes of several individuals be aggregated into overall collective propositional attitudes? Although there are large bodies of work on the aggregation of various special kinds of propositional attitudes, such as preferences, judgments, probabilities and utilities, the aggregation of propositional attitudes is seldom studied in full generality. In this paper, we seek to contribute to filling this gap in the literature. We sketch the ingredients of a general theory of propositional attitude aggregation and prove two new theorems. (...) 

The aim of this article is to introduce the theory of judgment aggregation, a growing interdisciplinary research area. The theory addresses the following question: How can a group of individuals make consistent collective judgments on a given set of propositions on the basis of the group members' individual judgments on them? I begin by explaining the observation that initially sparked the interest in judgment aggregation, the socalled "doctinal" and "discursive paradoxes". I then introduce the basic formal model of judgment aggregation, (...) 

In solving judgment aggregation problems, groups often face constraints. Many decision problems can be modelled in terms the acceptance or rejection of certain propositions in a language, and constraints as propositions that the decisions should be consistent with. For example, court judgments in breachofcontract cases should be consistent with the constraint that action and obligation are necessary and sufficient for liability; judgments on how to rank several options in an order of preference with the constraint of transitivity; and judgments on (...) 

Judgment aggregation theory, or rather, as we conceive of it here, logical aggregation theory generalizes social choice theory by having the aggregation rule bear on judgments of all kinds instead of merely preference judgments. It derives from Kornhauser and Sager’s doctrinal paradox and List and Pettit’s discursive dilemma, two problems that we distinguish emphatically here. The current theory has developed from the discursive dilemma, rather than the doctrinal paradox, and the final objective of the paper is to give the latter (...) 

The rationality of individual agents is secured for the most part by their makeup or design. Some agents, however – in particular, human beings – rely on the intentional exercise of thinking or reasoning in order to promote their rationality further; this is the activity that is classically exempliﬁed in Rodin’s sculpture of Le Penseur. Do group agents have to rely on reasoning in order to maintain a rational proﬁle? Recent results in the theory of judgment aggregation show that under (...) 

While a large socialchoicetheoretic literature discusses the aggregation of individual judgments into collective ones, there is much less formal work on the transformation of judgments in group communication. I develop a model of judgment transformation and prove a baseline impossibility theorem: Any judgment transformation function satisfying some initially plausible conditions is the identity function, under which no opinion change occurs. I identify escape routes from this impossibility and argue that the kind of group communication envisaged by deliberative democats must be (...) 

This paper provides an introductory review of the theory of judgment aggregation. It introduces the paradoxes of majority voting that originally motivated the field, explains several key results on the impossibility of propositionwise judgment aggregation, presents a pedagogical proof of one of those results, discusses escape routes from the impossibility and relates judgment aggregation to some other salient aggregation problems, such as preference aggregation, abstract aggregation and probability aggregation. The present illustrative rather than exhaustive review is intended to give readers (...) 

Distributed cognition refers to processes which are (i) cognitive and (ii) distributed across multiple agents or devices rather than performed by a single agent. Distributed cognition has attracted interest in several fields ranging from sociology and law to computer science and the philosophy of science. In this paper, I discuss distributed cognition from a socialchoicetheoretic perspective. Drawing on models of judgment aggregation, I address two questions. First, how can we model a group of individuals as a distributed cognitive system? Second, (...) 

Axiomatic characterization results in social choice theory are usually compared either regarding the normative plausibility or regarding the logical strength of the axioms involved. Here, instead, we propose to compare axiomatizations according to the language used for expressing the axioms. In order to carry out such a comparison, we suggest a formalist approach to axiomatization results which uses a restricted formal logical language to express axioms. Axiomatic characterization results in social choice theory then turn into definability results of formal logic. (...) 

Suppose the members of a group (e.g., committee, jury, expert panel) each form a judgment on which worlds in a given set are possible, subject to the constraint that at least one world is possible but not all are. The group seeks to aggregate these individual judgments into a collective judgment, subject to the same constraint. I show that no judgment aggregation rule can solve this problem in accordance with three conditions: “unanimity,” “independence” and “nondictatorship,” Although the result is a (...) 

Our aim in this survey article is to provide an accessible overview of some key results and questions in the theory of judgment aggregation. We omit proofs and technical details, focusing instead on concepts and underlying ideas. 

ment on the same propositions, and is plagued by impossibility re 2. What is the role of independence in judgment aggregation sults. In this paper we study the central notion of independence in.. 

I present an original model in judgment aggregation theory that demonstrates the general impossibility of consistently describing decisionmaking purely at the group level. Only a type of unanimity rule can guarantee a group decision is consistent with supporting reasons, and even this possibility is limited to a small class of reasoning methods. The key innovation is that this result holds when individuals can reason in different ways, an allowance not previously considered in the literature. This generalizes judgment aggregation to subjective (...) 

In this paper we explore the relation between three areas: judgment aggregation, belief merging and social choice theory. Judgment aggregation studies how to aggregate individual judgments on logically interconnected propositions into a collective decision on the same propositions. When majority voting is applied to some propositions it may however give a different outcome than majority voting applied to another set of propositions. Starting from this socalled doctrinal paradox, the paper surveys the literature on judgment aggregation, and shows that the application (...) 

We analyse the computational complexity of three problems in judgment aggregation: (1) computing a collective judgment from a profile of individual judgments (the winner determination problem); (2) deciding whether a given agent can influence the outcome of a judgment aggregation procedure in her favour by reporting insincere judgments (the strategic manipulation problem); and (3) deciding whether a given judgment aggregation scenario is guaranteed to result in a logically consistent outcome, independently from what the judgments supplied by the individuals are (the (...) 

In judgment aggregation, unlike preference aggregation, not much is known about domain restrictions that guarantee consistent majority outcomes. We introduce several conditions on individual judgments su¢  cient for consistent majority judgments. Some are based on global orders of propositions or individuals, others on local orders, still others not on orders at all. Some generalize classic socialchoicetheoretic domain conditions, others have no counterpart. Our most general condition generalizes Sen’s triplewise valuerestriction, itself the most general classic condition. We also prove a (...) 

In the emerging literature on judgment aggregation over logically connected propositions, expert rights or liberal rights have not been investigated yet. A group making collective judgments may assign individual members or subgroups with expert knowledge on, or particularly affected by, certain propositions the right to determine the collective judgment on those propositions. We identify a problem that generalizes Sen’s ‘liberal paradox’. Under plausible conditions, the assignment of rights to two or more individuals or subgroups is inconsistent with the unanimity principle, (...) 

In solving judgment aggregation problems, groups often face constraints. Many decision problems can be modelled in terms the acceptance or rejection of certain propositions in a language, and constraints as propositions that the decisions should be consistent with. For example, court judgments in breachofcontract cases should be consistent with the constraint that action and obligation are necessary and su¢  cient for liability; judgments on how to rank several options in an order of preference with the constraint of transitivity; and (...) 

Standard impossibility theorems on judgment aggregation over logically connected propositions either use a controversial systematicity condition or apply only to agendas of propositions with rich logical connections. Are there any serious impossibilities without these restrictions? We prove an impossibility theorem without requiring systematicity that applies to most standard agendas: Every judgment aggregation function (with rational inputs and outputs) satisfying a condition called unbiasedness is dictatorial (or effectively dictatorial if we remove one of the agenda conditions). Our agenda conditions are tight. (...) 

Can groups be rational agents over and above their individual members? We argue that group agents are distinguished by their capacity to mimic the way in which individual agents act and that this capacity must “supervene” on the group members’ contributions. But what is the nature of this supervenience relation? Focusing on group judgments, we argue that, for a group to be rational, its judgment on a particular proposition cannot generally be a function of the members’ individual judgments on that (...) 

The widely discussed "discursive dilemma" shows that majority voting in a group of individuals on logically connected propositions may produce irrational collective judgments. We generalize majority voting by considering quota rules, which accept each proposition if and only if the number of individuals accepting it exceeds a given threshold, where different thresholds may be used for different propositions. After characterizing quota rules, we prove necessary and sufficient conditions on the required thresholds for various collective rationality requirements. We also consider sequential (...) 

In the theory of judgment aggregation, it is known for which agendas of propositions it is possible to aggregate individual judgments into collective ones in accordance with the Arrowinspired requirements of universal domain, collective rationality, unanimity preservation, nondictatorship and propositionwise independence. But it is only partially known (e.g., only in the monotonic case) for which agendas it is possible to respect additional requirements, notably nonoligarchy, anonymity, no individual veto power, or implication preservation. We fully characterize the agendas for which there (...) 

Several recent results on the aggregation of judgments over logically connected propositions show that, under certain conditions, dictatorships are the only propositionwise aggregation functions generating fully rational (i.e., complete and consistent) collective judgments. A frequently mentioned route to avoid dictatorships is to allow incomplete collective judgments. We show that this route does not lead very far: we obtain oligarchies rather than dictatorships if instead of full rationality we merely require that collective judgments be deductively closed, arguably a minimal condition of (...) 

How can different individuals' probability functions on a given sigmaalgebra of events be aggregated into a collective probability function? Classic approaches to this problem often require 'eventwise independence': the collective probability for each event should depend only on the individuals' probabilities for that event. In practice, however, some events may be 'basic' and others 'derivative', so that it makes sense first to aggregate the probabilities for the former and then to let these constrain the probabilities for the latter. We formalize (...) 

The aggregation of consistent individual judgments on logically interconnected propositions into a collective judgment on the same propositions has recently drawn much attention. Seemingly reasonable aggregation procedures, such as propositionwise majority voting, cannot ensure an equally consistent collective conclusion. The literature on judgment aggregation refers to such a problem as the \textit{discursive dilemma}. In this paper we assume that the decision which the group is trying to reach is factually right or wrong. Hence, we address the question of how good (...) 

Diversity of opinion both presents problems and aff ords opportunities. Diff erences of opinion can stand in the way of reaching an agreement within a group on what decisions to take. But at the same time, the fact that the differences in question could derive from access to different information or from the exercise of diff erent judgemental skills means that they present individuals with the opportunity to improve their own opinions. This paper explores the implications for solutions to the (...) 

The standard image of how consensus can be achieved is by pooling evidence and reducing if not eliminating disagreements. But rather than just pooling substantive evidence on a certain question, why not also take into account the formal, testimonial evidence provided by the fact that a majority of the group adopt a particular answer? Shouldn't we be reinforced by the discovery that we are on that majority side, and undermined by the discovery that we are not? Shouldn't this be so, (...) 

All existing impossibility theorems on judgment aggregation require individual and collective judgment sets to be consistent and complete, arguably a demanding rationality requirement. They do not carry over to aggregation functions mapping profiles of consistent individual judgment sets to consistent collective ones. We prove that, whenever the agenda of propositions under consideration exhibits mild interconnections, any such aggregation function that is "neutral" between the acceptance and rejection of each proposition is dictatorial. We relate this theorem to the literature. 

Which rules for aggregating judgments on logically connected propositions are manipulable and which not? In this paper, we introduce a preferencefree concept of nonmanipulability and contrast it with a preferencetheoretic concept of strategyproofness. We characterize all nonmanipulable and all strategyproof judgment aggregation rules and prove an impossibility theorem similar to the GibbardSatterthwaite theorem. We also discuss weaker forms of nonmanipulability and strategyproofness. Comparing two frequently discussed aggregation rules, we show that “conclusionbased voting” is less vulnerable to manipulation than “premisebased voting”, (...) 

Group decisions must often obey exogenous constraints. While in a preference aggregation problem constraints are modelled by restricting the set of feasible alternatives, this paper discusses the modelling of constraints when aggregating individual yes/no judgments on interconnected propositions. For example, court judgments in breachofcontract cases should respect the constraint that action and obligation are necessary and sufficient for liability, and judgments on budget items should respect budgetary constraints. In this paper, we make constraints in judgment aggregation explicit by relativizing the (...) 

The point of departure in my story is the contrast between two models of democratic voting process: popular democracy and what might be called committee democracy. On one interpretation, voting in popular democracy is a procedure whose function is to aggregate the individuals’ preferences to something like a collective preference, while in committee democracy what is being aggregated are committee members’ judgments. The relevant judgments on the agenda often address an evaluative question. It is such value judgments that this paper (...) 

How can different individuals' probability assignments to some events be aggregated into a collective probability assignment? Classic results on this problem assume that the set of relevant events  the agenda  is a sigmaalgebra and is thus closed under disjunction (union) and conjunction (intersection). We drop this demanding assumption and explore probabilistic opinion pooling on general agendas. One might be interested in the probability of rain and that of an interestrate increase, but not in the probability of rain or (...) 

Political theorists have offered many accounts of collective decisionmaking under pluralism. I discuss a key dimension on which such accounts differ: the importance assigned not only to the choices made but also to the reasons underlying those choices. On that dimension, different accounts lie in between two extremes. The ‘minimal liberal account’ holds that collective decisions should be made only on practical actions or policies and that underlying reasons should be kept private. The ‘comprehensive deliberative account’ stresses the importance of (...) 

We investigate judgment aggregation by assuming that some formulas of the agenda are singled out as premisses, and the Independence condition (formulawise aggregation) holds for them, though perhaps not for others. Whether premissbased aggregation thus de…ned is nondegenerate depends on how premisses are logically connected, both among themselves and with other formulas. We identify necessary and su¢ cient conditions for dictatorship or oligarchy on the premisses, and investigate when these results extend to the whole agenda. Our theorems recover or strengthen (...) 

The new …eld of judgment aggregation aims to …nd collective judgments on logically interconnected propositions. Recent impossibility results establish limitations on the possibility to vote independently on the propositions. I show that, fortunately, the impossibility results do not apply to a wide class of realistic agendas once propositions like “if a then b” are adequately modelled, namely as subjunctive implications rather than material implications. For these agendas, consistent and complete collective judgments can be reached through appropriate quota rules (which decide (...) 

This introduces the symposium on judgment aggregation. The theory of judgment aggregation asks how several individuals' judgments on some logically connected propositions can be aggregated into consistent collective judgments. The aim of this introduction is to show how ideas from the familiar theory of preference aggregation can be extended to this more general case. We first translate a proof of Arrow's impossibility theorem into the new setting, so as to motivate some of the central concepts and conditions leading to analogous (...) 

This paper provides an introductory review of the theory of judgment aggregation. It introduces the paradoxes of majority voting that originally motivated the field, explains several key results on the impossibility of propositionwise judgment aggregation, presents a pedagogical proof of one of those results, discusses escape routes from the impossibility and relates judgment aggregation to some other salient aggregation problems, such as preference aggregation, abstract aggregation and probability aggregation. The present illustrative rather than exhaustive review is intended to give readers (...) 

In this paper, I explore the possibility of applying the methods and results of Judgement Aggregation to the problem of logical disagreement. I develop and evaluate different ways in which individuals who logically disagree can generate a collective logic. I prove a version of the discursive paradox, where the majority voting of a group of structural logicians can give rise to a substructural logic; then I develop a more general impossibility result. After this, I analyze different ways to avoid the (...) 

The aggregation of individual judgments on logically interconnected propositions into a collective decision on the same propositions is called judgment aggregation. Literature in social choice and political theory has claimed that judgment aggregation raises serious concerns. For example, consider a set of premises and a conclusion where the latter is logically equivalent to the former. When majority voting is applied to some propositions (the premises) it may give a different outcome than majority voting applied to another set of propositions (the (...) 

The problem of the aggregation of inputs coming from different sources arises in several contexts. Examples are the combination of individual preferences (studied in social choice theory), opinions (judgment aggregation), and data (artificial intelligence). While a number of results are available in each of these disciplines, a question that has been addressed only recently is how similar these aggregation problems are, despite the different types of inputs they try to combine. 

Judgment aggregation theory, which concerns the translation of individual judgments on logical propositions into consistent group judgments, has shown that group consistency generally cannot be guaranteed if each proposition is treated independently from the others. Developing the right method of abandoning independence is thus a highpriority goal. However, little work has been done in this area outside of a few simple approaches. To ﬁll the gap, we compare four methods based on distance metrics between judgment sets. The methods generalize the (...) 

I propose a general collective decision problem consisting in many issues that are interconnected in two ways: by mutual constraints and by connections of relevance. Aggregate decisions should respect the mutual constraints, and be based on relevant information only. This general informational constraint has many special cases, including premisebasedness and Arrow’s independence condition; they result from special notions of relevance. The existence and nature of (nondegenerate) aggregation rules depends on both types of connections. One result, if applied to the preference (...) 

Agents are often assumed to have degrees of belief (“credences”) and also binary beliefs (“beliefs simpliciter”). How are these related to each other? A muchdiscussed answer asserts that it is rational to believe a proposition if and only if one has a high enough degree of belief in it. But this answer runs into the “lottery paradox”: the set of believed propositions may violate the key rationality conditions of consistency and deductive closure. In earlier work, we showed that this problem (...) 

Following Lauwers and Van Liedekerke (1995), this paper explores in a modeltheoretic framework the relation between Arrovian aggregation rules and ultraproducts, in order to investigate a source of impossibility results for the case of an infinite number of individuals and an aggregation rule based on a free ultrafilter of decisive coalitions. 



Arrow’s axiomatic foundation of social choice theory can be understood as an application of Tarski’s methodology of the deductive sciences—which is closely related to the latter’s foundational contribution to model theory. In this note we show in a modeltheoretic framework how Arrow’s use of von Neumann and Morgenstern’s concept of winning coalitions allows to exploit the algebraic structures involved in preference aggregation; this approach entails an alternative indirect ultrafilter proof for Arrow’s dictatorship result. This link also connects Arrow’s seminal result (...) 

In what follows, I appeal to Charles Babbage’s discussion of the division of mental labor to provide evidence that—at least with respect to the social acquisition, storage, retrieval, and transmission of knowledge—epistemologists have, for a broad range of phenomena of crucial importance to actual knowers in their epistemic practices in everyday life, failed adequately to appreciate the significance of socially distributed cognition. If the discussion here is successful, I will have demonstrated that a particular presumption widely held within the contemporary (...) 

In this paper, I investigate the relationship between preference and judgment aggregation, using the notion of ranking judgment introduced in List and Pettit. Ranking judgments were introduced in order to state the logical connections between the impossibility theorem of aggregating sets of judgments and Arrow’s theorem. I present a proof of the theorem concerning ranking judgments as a corollary of Arrow’s theorem, extending the translation between preferences and judgments defined in List and Pettit to the conditions on the aggregation procedure. 