Aggregation of Value Judgments Differs from Aggregation of Preferences

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.

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 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.

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.

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,

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.

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.

How can several 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 sigma -algebra 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 interest-rate increase, but not in the probability of rain or an interest-rate increase. We characterize linear pooling and neutral pooling for general agendas, with classic results as special cases for agendas that are sigma -algebras. As an illustrative application, we also consider probabilistic preference aggregation. Finally, we unify our results with existing results on binary judgment aggregation and Arrovian preference aggregation. We show that the same kinds of axioms (independence and consensus preservation) have radically different implications for different aggregation problems: linearity for probability aggregation and dictatorship for binary judgment or preference aggregation.

Introduction to Judgment Aggregation

An allocation problem is a type of aggregation problem in which the values of individuals' opinions on some set of variables (canonically a set of mutually exclusive and exhaustive possibilities) sum to a constant. This paper shows that for realistic allocation problems, namely ones in which the set of possible opinion values is finite, the only universal aggregation methods that satisfy two commonly invoked conditions are the dictatorial ones. The two conditions are, first, that the aggregate opinion on any variable depends only on the individuals' opinions on that variable and not on their opinions on any other variable and, secondly, that the aggregate opinion minimally preserves unanimous individual opinion. Connections between this result and similar ones in the literature on judgment aggregation are explored and their significance assessed.

A semi-linguistic fuzzy approach to multi-actor decision-making: Application to aggregation of experts' judgments

This paper studies the problem of majority-rule-based collective decision-making where the agents' preferences are represented by CP-nets (Conditional Preference Networks). As there are exponentially many alternatives, it is impractical to reason about the individual full rankings over the alternative space and apply majority rule directly. Most existing works either do not consider computational requirements, or depend on a strong assumption that the agents have acyclic CP-nets that are compatible with a common order on the variables. To this end, this paper proposes an efficient SAT-based approach, called MajCP (Majority-rule-based collective decision-making with CP-nets), to compute the majority winning alternatives. Our proposed approach only requires that each agent submit a CP-net; the CP-net can be cyclic, and it does not need to be any common structures among the agents' CP-nets. The experimental results presented in this paper demonstrate that the proposed approach is computationally efficient. It offers several orders of magnitude improvement in performance over a Brute-force algorithm for large numbers of variables.

Aggregating multiple rankings in a database is an important task well studied by the database community. High-stakes application domains include hiring, lending, and education where multiple decision makers rank candidates and their input is then combined into an overall consensus ranking. While state-of-art aggregation methods succeed in giving equal say to each decision maker, to date no methods ensure fair representation for groups of candidates being ranked, who risk being impacted by discriminatory bias. We present the first solution to this open problem of guaranteeing fairness for disadvantaged groups of candidates, while still producing a good consensus of the base rankings. We design a family of exact fair algorithms achieving optimality for fair rank aggregation. We also develop approximate methods achieving fairness with guaranteed minimal approximation error scaling to millions of candidates in the rankings. A comparative study evaluates our proposed methods, revealing trade-offs between aggregation accuracy and different degrees of unfair bias in a rich variety of rank aggregation scenarios. Our real-world case study demonstrates that our solutions mitigate unfair bias using real-world data.

This paper provides a “liberal paradox” that applies to the framework of abstract argumentation and complements the liberal paradox in preference aggregation. In abstract argumentation, arguments are viewed as abstract entities whose validities are determined according to a binary attack relation. When forming a collective attack relation, parts of it may be reserved to members of the society who hold expert knowledge. I prove that when only a binary evaluation of each argument is permitted, even under a minimal condition of rationality, the assignment of expert rights to two or more agents may be inconsistent with the condition of strong unanimity. Since argumentation aggregation is a particular case of judgement aggregation, this result might be a corollary of Dietrich and List (Soc Choice Welf 31(1):59–78, 2008), if the agenda I consider turns out to be connected in their sense, an issue that this paper has not been able to settle.