Introduction to Judgment Aggregation

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 provide a new proof of Arrow’s theorem, our main aim is to identify the analogue of Arrow’s theorem in judgment aggregation, to clarify the relation between judgment and preference aggregation, and to illustrate the generality of the judgment aggregation model.

Agents that must reach agreements with other agents need to reason about how their preferences, judgments, and beliefs might be aggregated with those of others by the social choice mechanisms that govern their interactions. The emerging field of judgment aggregation studies aggregation from a logical perspective, and considers how multiple sets of logical formulae can be aggregated to a single consistent set. As a special case, judgment aggregation can be seen to subsume classical preference aggregation. We present a modal logic that is intended to support reasoning about judgment aggregation scenarios (and hence, as a special case, about preference aggregation): the logical language is interpreted directly in judgment aggregation rules. We present a sound and complete axiomatisation. We show that the logic can express aggregation rules such as majority voting; rule properties such as independence; and results such as the discursive paradox, Arrow's theorem and Condorcet's paradox--which are derivable as formal theorems of the logic. The logic is parameterised in such a way that it can be used as a general framework for comparing the logical properties of different types of aggregation--including classical preference aggregation. As a case study we present a logical study of, including a formal proof of, the neutrality lemma, the main ingredient in a well-known proof of Arrow's theorem.

Judgment aggregation is a field in which individuals are required to vote for or against a certain decision (the conclusion) while providing reasons for their choice. The reasons and the conclusion are logically connected propositions. The problem is how a collective judgment on logically interconnected propositions can be defined from individual judgments on the same propositions. It turns out that, despite the fact that the individuals are logically consistent, the aggregation of their judgments may lead to an inconsistent group outcome, where the reasons do not support the conclusion. However, in this paper we claim that collective irrationality should not be the only worry of judgment aggregation. For example, judgment aggregation would not reject a consistent combination of reasons and conclusion that no member voted for. In our view this may not be a desirable solution. This motivates our research about when a social outcome is `compatible' with the individuals' judgments. The key notion that we want to capture is that any individual member has to be able to defend the collective decision. This is guaranteed when the group outcome is compatible with its members views. Judgment aggregation problems are usually studied using classical propositional logic. However, for our analysis we use an argumentation approach to judgment aggregation problems. Indeed the question of how individual evaluations can be combined into a collective one can also be addressed in abstract argumentation. We introduce three aggregation operators that satisfy the condition above, and we offer two definitions of compatibility. Not only does our proposal satisfy a good number of standard judgment aggregation postulates, but it also avoids the problem of individual members of a group having to become committed to a group judgment that is in conflict with their own individual positions.

Preference aggregation and judgment aggregation are two basic research models of group decision making. And preference aggregation has been deeply studied in social choice theory. However, researches of social choice theory gradually focus on judgment aggregation which appears recently. Judgment aggregation focuses on how to aggregate many consistent logical formulas into one, from the perspective of logic. We try to start with judgment aggregation model based on logic and then explore different solutions to problem of judgment aggregation.

In recent years, judgement aggregation has emerged as an important area of social choice theory. Judgement aggregation is concerned with aggregating sets of individual judgements over logically connected propositions into a set of collective judgements. It has been shown that even seemingly weak conditions on the aggregation function make it impossible to find functions that produce rational collective judgements from all possible rational individual judgements. This implies that the step from individual judgements to collective judgements requires trade-offs between different desiderata, such as universal domain, rationality, epistemological quality, and unbiasedness. The typical application for judgement aggregation is the problem of group decision making. Juries and expert committees are the stock examples. However, the relevance of judgement aggregation goes beyond these cases. In this survey, I review some core results in the field of judgement aggregation and social epistemology and discuss their implications for the analysis of distributed thinking.

Judgment aggregation studies how to aggregate individual judgments on logically correlated propositions into collective judgments. Different logics can be used in judgment aggregation, for which Dietrich and Mongin have proposed a generalized model based on general logics. Despite its generality, however, all nonmonotonic logics are excluded from this model. This paper argues for using nonmonotonic logic in judgment aggregation. Then it generalizes Dietrich and Mongin’s model to incorporate a large class of nonmonotonic logics. This generalization broadens the theoretical boundaries of judgment aggregation by proving that, even if these nonmonotonic logics are employed, certain typical impossibility results still hold.

Agents that must reach agreements with other agents need to reason about how their preferences, judgments, and beliefs might be aggregated with those of others by the social choice mechanisms that govern their interactions. The recently emerging field of judgment aggregation studies aggregation from a logical perspective, and considers how multiple sets of logical formulae can be aggregated to a single consistent set. As a special case, judgment aggregation can be seen to subsume classical preference aggregation. We present a modal logic that is intended to support reasoning about judgment aggregation scenarios (and hence, as a special case, about preference aggregation): the logical language is interpreted directly in judgment aggregation rules. We present a sound and complete axiomatisation of such rules. We show that the logic can express aggregation rules such as majority voting; rule properties such as independence; and results such as the discursive paradox, Arrow's theorem and Condorcet's paradox - which are derivable as formal theorems of the logic. The logic is parameterised in such a way that it can be used as a general framework for comparing the logical properties of different types of aggregation - including classical preference aggregation.

Judgment aggregation is a recent theory that combines aggregation problems previously studied by social choice theory with logic. Social choice theory is a vast subject, including not only the study of preference aggregation and voting theory but also topics like social welfare and justice. Given the tight links between judgment aggregation and preference aggregation, in this first chapter we give a concise survey on some historical aspects of preference aggregation and then introduce and motivate the more recent field of judgment aggregation. The chapter builds on [Sen99, Sen86, Bla58] for the historical overview on social choice theory, and on [KS93, Kor92] for the informal introduction to judgment aggregation.

Computational social choice is a new interdisciplinary research area at the intersection of social choice theory and computer science that studies computational aspects of social choice mechanisms as well as applications of social choice ideas in the design and analysis of computing systems. Research on computational social choice now encompasses a variety of topics, including the algorithmic aspects of voting rules, computational barriers to strategic behaviour, collective decision-making in multi-agent systems, preference elicitation and communication issues in voting, fair division, computational aspects of weighted voting games, collective decision-making in combinatorial domains, logic-based formalisms for social choice problems, belief and judgement aggregation, and social software. The aim of this special issue is to build on the spectacular success of two workshops on computational social choice (COMSOC’06, Amsterdam and COMSOC’08, Liverpool), and provide a forum for reseachers in the area to present their ideas on a variety of topics. This special issue is composed of nine research articles and a short note that span a fairly wide range of issues, including voting, mechanism design, judgment aggregation, and fairness, and a similarly wide range of techniques, such as logic, experiments, and computational complexity. The paper by Agotnes, van der Hoek and Wooldridge defines a new modal logic for representing and reasoning about preference and judgment aggregation. The authors show that their logic can be used to express a variety of concepts and results from the social choice domain, including majority voting, the discursive paradox, Arrow’s theorem and Condorcet’s paradox. Baharad, Goldberger, Koppel and Nitzan propose a maximum likelihood estimation-based approach for aggregating voters’ opinions over several issues. They argue that their method is vastly superior to simple majority voting, and use both analytical and empirical evaluation to support their argument. Balan, Richards and Luke deal with the problem of selecting a sequence of actions when the actions have multiple beneficiaries. Their goal is to select the actions according to a lexicographic fairness criterion. The explore the complexity of this problem for both finite and infinite sequences of actions and present both algorithms and intractability results for this problem. The paper by Caminada and Pigozzi applies argumentation theory to judgment aggregation, and proposes aggregation operators satisfying not only the usual property of collective rationality, but another highly desirable property,

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 Arrow-inspired requirements of universal domain, collective rationality, unanimity preservation, nondictatorship and propositionwise independence. But it is only partially known for which agendas it is possible to respect additional requirements, notably non-oligarchy, anonymity, no individual veto power, or implication preservation. We fully characterize the agendas for which there are such possibilities, thereby answering the most salient open questions about propositionwise judgment aggregation. Our results build on earlier results by Nehring and Puppe (2002), Nehring (2006) and Dietrich and List (2007a).

There are two different fundamental conceptions of the task of social aggregation: interest aggregation and judgment aggregation. While the former aims at the fair and efficient trade-off of personal interests, the latter aims the best resolution of disagreement. Both are possible interpretations of the standard preference aggregation and preference-based voting problems. This note makes two broad points. First, the judgment aggregation conception is not yet well understood and remains undertheorized. Second, the difference between the two conceptions matters; it is a difference that makes a difference. To substantiate the first point, we argue that judgment aggregation as resolution of disagreement cannot be subsumed under statistical aggregation in the manner of Condorcet’s jury theorem, and sketch a more specific articulation as “reflexive consensus under arguable disagreement”. To substantiate the second point, we illustrate the difference in the context of two important voting rules, the Kemeny rule and Majority Judgment.

In the framework of judgment aggregation, we assume that some formulas of the agenda are singled out as premisses, and that both Independence (formula-wise aggregation) and Unanimity Preservation hold for them. Whether premiss-based aggregation thus defined is compatible with conclusion-based aggregation, as defined by Unanimity Preservation on the non-premisses, depends on how the premisses are logically connected, both among themselves and with other formulas. We state necessary and sufficient conditions under which the combination of both approaches leads to dictatorship (resp. oligarchy), either just on the premisses or on the whole agenda. Our analysis is inspired by the doctrinal paradox of legal theory and is arguably relevant to this field as well as political science and political economy. When the set of premisses coincides with the whole agenda, a limiting case of our assumptions, we obtain several existing results in judgment aggregation theory.

My focus is on aggregation of individual value rankings of alternatives to a collective value ranking. This is compared with aggregation o individual prefrences to a collective preference. While in an individual preference ranking the alternatives are ordered in accordance with one’s preferences, the order in a value ranking expresses one’s comparative evaluation of the alternatives, from the best to the worst. I suggest that, despite their formal similarity as rankings, this difference in the nature of individual inputs in two aggregation scenarios has important implications: The kind of procedure that looks fine for aggregation of judgments is inappropriate for aggregation of preferences. The procedure I have in mind consists in similarity maximization, or – more precisely – in minimization of the average distance from individual inputs. When applied to judgment aggregation, this procedure can also be approached from the epistemic perspective: the questions are posed concerning its advantages as a truth-tracker. From that perspective, what matters is not only the probability of the outcome of the procedure being true, but also the expected verisimilitude of the outcome: its expected distance from truth.

Judgment aggregation and agenda manipulation

Arrow’s Impossibility Theorem is a classical result in social choice theory (a branch of economic theory), which states that any system of rules for combining (“aggregating”) individual preference relations into a single representative relation results in a “dictatorship” where the combined preference only reflects the wishes of a single individual (provided that the aggregation rule satisfies two basic criteria). Since the 1980s, this result has been reformulated and understood using algebraic topology. The topological approach offers some geometric intuition as to why Arrow’s theorem holds, and can also be used to find alternative hypotheses which may escape the dictatorship outcome. A thorough examination of such topological models constitutes the main body of this thesis. Recently, social choice theory has been generalized (resulting in a field called “judgment aggregation”), and results analogous to Arrow's theorem have been established. The second part of this thesis introduces this field of study, and studies how some of the techniques from topological social choice theory can be extended to understand dictatorship outcomes in the theory of judgment aggregation. Although the analysis is restricted to a rather simple case, it nonetheless highlights the potential for a more general topological model of judgment aggregation, and exposes the main challenges that must be overcome in constructing such a theory.