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Judgment aggregation: (Im)possibility theorems
Journal of Economic Theory 1 (126):286298 (2006)
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Many think that sentences about what metaphysically explains what are true iff there exist grounding relations. This suggests that sceptics about grounding should be error theorists about metaphysical explanation. We think there is a better option: a theory of metaphysical explanation which offers truth conditions for claims about what metaphysically explains what that are not couched in terms of grounding relations, but are instead couched in terms of, inter alia, psychological facts. We do not argue that our account is superior (...) 

Can we design a perfect democratic decision procedure? Condorcet famously observed that majority rule, our paradigmatic democratic procedure, has some desirable properties, but sometimes produces inconsistent outcomes. Revisiting Condorcet’s insights in light of recent work on the aggregation of judgments, I show that there is a conflict between three initially plausible requirements of democracy: “robustness to pluralism”, “basic majoritarianism”, and “collective rationality”. For all but the simplest collective decision problems, no decision procedure meets these three requirements at once; at most (...) 

Social choice theory is the study of collective decision processes and procedures. It is not a single theory, but a cluster of models and results concerning the aggregation of individual inputs (e.g., votes, preferences, judgments, welfare) into collective outputs (e.g., collective decisions, preferences, judgments, welfare). Central questions are: How can a group of individuals choose a winning outcome (e.g., policy, electoral candidate) from a given set of options? What are the properties of different voting systems? When is a voting system (...) 

To investigate the relationship between logical reasoning and majority voting, we introduce logic with groups Lg in the style of Gentzen’s sequent calculus, where every sequent is indexed by a group of individuals. We also introduce the settheoretical semantics of Lg, where every formula is interpreted as a certain closed set of groups whose members accept that formula. We present the cutelimination theorem, and the soundness and semantic completeness theorems of Lg. Then, introducing an inference rule representing majority voting to (...) 

Judgmentaggregation theory has always focused on the attainment of rational collective judgments. But so far, rationality has been understood in static terms: as coherence of judgments at a given time, defined as consistency, completeness, and/or deductive closure. This paper asks whether collective judgments can be dynamically rational, so that they change rationally in response to new information. Formally, a judgment aggregation rule is dynamically rational with respect to a given revision operator if, whenever all individuals revise their judgments in light (...) 

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 (...) 

In a recent paper on ‘The Many as One’, Lewis A. Kornhauser and Lawrence G. Sager look at an issue that we take to be of great importance in political theory. How far should groups in public life try to speak with one voice, and act with one mind? How far should public groups try to display what Ronald Dworkin calls integrity? We do not expect the many on the market to be integrated in this sense. But should we expect (...) 

Decisionmaking typically requires judgments about causal relations: we need to know the causal effects of our actions and the causal relevance of various environmental factors. We investigate how several individuals' causal judgments can be aggregated into collective causal judgments. First, we consider the aggregation of causal judgments via the aggregation of probabilistic judgments, and identify the limitations of this approach. We then explore the possibility of aggregating causal judgments independently of probabilistic ones. Formally, we introduce the problem of causalnetwork aggregation. (...) 

The constitution of a collective judgment is analyzed from a contractarian point of view. The optimal collective judgment is defined as the one that maximizes the sum of the utility each member gets from the collective adoption of that judgment. It is argued that judgment aggregation is a different process from the aggregation of information and public deliberation. This entails that the adoption of a collective judgment should not make any rational member of the group change her individual opinion, and (...) 



We introduce a conceptual model for reaching group decisions. Our model extends a wellknown, singleagent cognitive model, the recognitionprimed decision (RPD) model. The RPD model includes a recognition phase and an evaluation phase. Group extensions of the RPD model, applicable to a group of RPD agents, have been considered in the literature, however the proposed models do not formalize how distributed and possibly inconsistent information can be combined in either phase. We show how such information can be utilized by aggregating (...) 

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 …lling this gap in the literature. We sketch the ingredients of a general theory of propositional attitude aggregation and prove two new theorems. (...) 

In the emerging literature on judgment aggregation over logically connected proposi tions, expert rights or liberal rights have not been investigated yet. A group making collective judgments may assign individual members or subgroups with expert know ledge 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 (...) 

This essay raises new objections to the two dominant approaches to understanding the justification of group beliefs—_inflationary_ views, where groups are treated as entities that can float freely from the epistemic status of their members’ beliefs, and _deflationary_ views, where justified group belief is understood as nothing more than the aggregation of the justified beliefs of the group's members. If this essay is right, we need to look in an altogether different place for an adequate account of justified group belief. (...) 

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. 

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 (...) 

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 (...) 

In this paper, I introduce the emerging theory of judgment aggregation as a framework for studying institutional design in social epistemology. When a group or collective organization is given an epistemic task, its performance may depend on its ‘aggregation procedure’, i.e. its mechanism for aggregating the group members’ individual beliefs or judgments into corresponding collective beliefs or judgments endorsed by the group as a whole. I argue that a group’s aggregation procedure plays an important role in determining whether the group (...) 

The combination 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 in which 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 (...) 

Can collectives be wise? The thesis that they can has recently received a lot of attention. It has been argued that, in many judgmental or decisionmaking tasks, suitably organized groups can outperform their individual members. In this paper, I discuss the lessons we can learn about collective wisdom from the emerging theory of judgment aggregation, as distinct from the literature on Condorcet’s jury theorem. 

A necessary condition for a group being taken as a rational agent is that its choices and judgements are ‘logically contestable’, but this can lead to problems of aggregation, as Arrow impossibility theorem or the discursive dilemma. This paper proposes a contractarian or constitutional approach: the relevant thing is what aggregation mechanisms would be preferred by the members of the group. Two distinctions need to be made: first, judgement aggregation is not aggregation of decisions, and judgement aggregation needs be distinguished (...) 

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 (...) 

I model sequential decisions over multiple interconnected propositions and investigate pathdependence in such decisions. The propositions and their interconnections are represented in propositional logic. A sequential decision process is pathdependent if its outcome depends on the order in which the propositions are considered. Assuming that earlier decisions constrain later ones, I prove three main results: First, certain rationality violations by the decisionmaking agent—individual or group—are necessary and sufficient for pathdependence. Second, under some conditions, pathdependence is unavoidable in decisions made by (...) 

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 (...) 

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, (...) 



What is the relationship between degrees of belief and binary beliefs? Can the latter be expressed as a function of the former—a socalled “beliefbinarization rule”—without running into difficulties such as the lottery paradox? We show that this problem can be usefully analyzed from the perspective of judgmentaggregation theory. Although some formal similarities between belief binarization and judgment aggregation have been noted before, the connection between the two problems has not yet been studied in full generality. In this paper, we seek (...) 

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 (...) 

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 (...) 

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 (...) 

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. (...) 

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, (...) 

Judgment aggregation problems are language dependent in that they may be framed in different yet equivalent ways. We formalize this dependence via the notion of translation invariance, adopted from the philosophy of science, and we argue for the normative desirability of translation invariance. We characterize the class of translation invariant aggregation functions in the canonical judgment aggregation model, which requires collective judgments to be complete. Since there are reasonable translation invariant aggregation functions, our result can be viewed as a possibility (...) 

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 (...) 

In response to recent work on the aggregation of individual judgments on logically connected propositions into collective judgments, it is often asked whether judgment aggregation is a special case of Arrowian preference aggregation. We argue for the converse claim. After proving two impossibility theorems on judgment aggregation (using "systematicity" and "independence" conditions, respectively), we construct an embedding of preference aggregation into judgment aggregation and prove Arrow’s theorem (stated for strict preferences) as a corollary of our second result. Although we thereby (...) 

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 (...) 

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 (...) 

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 (...) 

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. (...) 

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 (...) 

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 new field of judgment aggregation aims to merge many individual sets of judgments on logically interconnected propositions into a single collective set of judgments on these propositions. Judgment aggregation has commonly been studied using classical propositional logic, with a limited expressive power and a problematic representation of conditional statements ("if P then Q") as material conditionals. In this methodological paper, I present a simple unified model of judgment aggregation in general logics. I show how many realistic decision problems can (...) 

I propose a relevancebased independence axiom on how to aggregate individual yes/no judgments on given propositions into collective judgments: the collective judgment on a proposition depends only on people’s judgments on propositions which are relevant to that proposition. This axiom contrasts with the classical independence axiom: the collective judgment on a proposition depends only on people’s judgments on the same proposition. I generalize the premisebased rule and the sequentialpriority rule to an arbitrary priority order of the propositions, instead of a (...) 

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. 

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 it may give a different outcome than majority voting applied to another set of propositions. This problem is (...) 

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 (...) 

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”, (...) 

Majority cycling and related social choice paradoxes are often thought to threaten the meaningfulness of democracy. But deliberation can prevent majority cycles – not by inducing unanimity, which is unrealistic, but by bringing preferences closer to singlepeakedness. We present the first empirical test of this hypothesis, using data from Deliberative Polls. Comparing preferences before and after deliberation, we find increases in proximity to singlepeakedness. The increases are greater for lower versus higher salience issues and for individuals who seem to have (...) 

Groups and other sorts of collective entities are frequently said to believe things. Sarah Huckabee Sanders, for instance, was asked by reporters at White House press conferences whether the Trump administration ‘believes in climate change’ or ‘believes that slavery is wrong’. Similarly, it is said on the website of the Aclu of Illinois that the organization ‘firmly believes that rights should not be limited based on a person’s sexual orientation or gender identity’. A widespread philosophical view is that belief on (...) 