Reasoning about judgment and preference aggregation

Economics and Philosophy, 2007

Which rules for aggregating judgments on logically connected propositions are manipulable and which not? In this paper, we introduce a preference-free concept of non-manipulability and contrast it with a preference-theoretic concept of strategy-proofness. We characterize all non-manipulable and all strategy-proof judgment aggregation rules and prove an impossibility theorem similar to the Gibbard--Satterthwaite theorem. We also discuss weaker forms of non-manipulability and strategy-proofness. Comparing two frequently discussed aggregation rules, we show that “conclusion-based voting” is less vulnerable to manipulation than “premise-based voting”, which is strategy-proof only for “reason-oriented” individuals. Surprisingly, for “outcome-oriented” individuals, the two rules are strategically equivalent, generating identical judgments in equilibrium. Our results introduce game-theoretic considerations into judgment aggregation and have implications for debates on deliberative democracy.

Social Choice and Welfare, 2008

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 rationality, compatible even with empty judgment sets. We derive several characterizations of oligarchies and provide illustrative applications to Arrowian preference aggregation and Kasher and Rubinstein's group identi…cation problem.

This paper presents a modal logic for modelling individual and collective choices over a set of feasible alternatives. The logic extends propositional logic with a binary modality so that a formula can express not only properties of alternatives but also priorities of individuals over the properties. More importantly, each formula of this logic determines a preference ordering over alternatives based on the priorities over properties that the formula expresses. In such a way, preferences of multiple agents can be represented by a set of formulas in the same logic. This allows us to treat the problem of collective choice in a multi-agent system as aggregation of logical formulas. We further use this language to express a few plausible collective choice rules. Similar to preference aggregation, we specify collective choice rules by Arrow’s conditions. Interestingly, all Arrowian conditions are plausible under the new setting except Independence of Irrelevant Alternatives. This gives us a natural way to avoid Arrow’s impossibility result. Finally, we develop a model checking algorithm to automatically generate individual and collective choices in the logic.

In this paper, we discuss the approach based on Social Choice Theory and Judgment Aggregation to the definition of collective reasoning. We shall make explicit the aggregative nature of the notion of collective reasoning that is defined in the Judgment Aggregation account and we shall stress that the notion of logical coherence plays a fundamental role in defining collective attitudes. Unfortunately, as several results in Judgment Aggregation show, coherence is not compatible with fair aggregation procedures. On closer inspection, the notion of coherence that is jeopardized by Judgment Aggregation is based on classical logic. In this work, we propose to revise the standard view of rationality of Judgment Aggregation by exploring the realm of non-classical logics. In particular, we will present possibility results for substructural logics. Those logics, we argue, provide a viable notion of collective reasoning.

1988

This paper presents a modal logic for modelling individual and collective choices over a set of feasible alternatives. The logic extends propositional logic with a binary modality so that a formula can express not only properties of alternatives but also priorities of individuals over the properties. More importantly, each formula of this logic determines a preference ordering over alternatives based on the priorities over properties that the formula expresses. In such a way, preferences of multiple agents can be represented by a set of formulas in the same logic. This allows us to treat the problem of collective choice in a multi-agent system as aggregation of logical formulas. We further use this language to express a few plausible collective choice rules. Similar to preference aggregation, we specify collective choice rules by Arrow's conditions. Interestingly, all Arrowian conditions are plausible under the new setting except Independence of Irrelevant Alternatives. This gives us a natural way to avoid Arrow's impossibility result. Finally, we develop a model checking algorithm to automatically generate individual and collective choices in the logic.

Jounral of Applied Non-Classical Logics, 2017

This work contributes to the theory of judgment aggregation by discussing a number of significant non-classical logics. After adapting the standard framework of judgment aggregation to cope with non-classical logics, we discuss in particular results for the case of Intuitionistic Logic, the Lambek calculus, Linear Logic and Relevant Logics. The motivation for studying judgment aggregation in non-classical logics is that they offer a number of modelling choices to represent agents' reasoning in aggregation problems. By studying judgment aggregation in logics that are weaker than classical logic, we investigate whether some well-known impossibility results, that were tailored for classical logic, still apply to those weak systems.

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 breach-of-contract 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 rationality conditions of consistency and deductive closure to a constraint set, whose variation yields more or less strong notions of rationality. This approach of modelling constraints explicitly contrasts with that of building constraints as axioms into the logic, which turns compliance with constraints into a matter of logical consistency and thereby conflates requirements of or...

Lecture Notes in Computer Science, 2014

Similar to Arrow's impossibility theorem for preference aggregation, judgment aggregation has also an intrinsic impossibility for generating consistent group judgment from individual judgments. Removing some of the pre-assumed conditions would mitigate the problem but may still lead to too restrictive solutions. It was proved that if completeness is removed but other plausible conditions are kept, the only possible aggregation functions are oligarchic, which means that the group judgment is purely determined by a certain subset of participating judges. Instead of further challenging the other conditions, this paper investigates how the judgment from each individual judge affects the group judgment in an oligarchic environment. We explore a set of intuitively demanded conditions under abstentions and design a feasible judgment aggregation rule based on the agents' hierarchy. We show this proposed aggregation rule satisfies the desirable conditions. More importantly, this rule is oligarchic with respect to a subset of agenda instead of the whole agenda due to its literal-based characteristics.

Proceedings of the 13th …, 2011

Many voting rules are based on some minimization principle. Likewise, in the field of logic-based knowledge representation and reasoning, many belief change or inconsistency handling operators also make use of minimization. Surprisingly, minimization has not played a major role in the field of judgment aggregation, in spite of its proximity to voting theory and logic-based knowledge representation and reasoning. Here we make a step in this direction and study six judgment aggregation rules; two of them, based on distances, have been previously defined; the other four are new, and all inspired both by voting theory and knowledge representation and reasoning. We study the inclusion relationships between these rules and address some of their social choice theoretic properties.

Social Choice and Welfare, 2007

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. JEL Classi…cation: D70, D71

Computational Intelligence, 2017

Judgment aggregation deals with the problem of how collective judgments on logically connected propositions can be formed based on individual judgments on the same propositions. The existing literature on judgment aggregation mainly focuses on the anonymity condition requiring that individual judgments be treated equally. However, in many real-world situations, a group making collective judgments may assign individual members or subgroups different priorities to determine the collective judgment. Based on this consideration, this paper relaxes the anonymity condition by giving a hierarchy over individuals so as to investigate how the judgment from each individual affects the group judgment in such a hierarchical environment. Moreover, we assume that an individual can abstain from voting on a proposition and the collective judgment on a proposition can be undetermined, which means that we do not require completeness at both individual and collective levels. In this new setting, we first identify an impossibility result and explore a set of plausible conditions in terms of abstentions. Secondly, we develop an aggregation rule based on the hierarchy of individuals and show that the aggregation rule satisfies those plausible conditions. The computational complexity of this rule is also investigated. Finally, we show that the proposed rule is (weakly) oligarchic over a subset of agenda. This is by no means a negative result. In fact, our result reveals that with abstentions, oligarchic aggregation is not necessary to be a single-level determination but can be a multiple-level collective decision-making, which partially explains its ubiquity in the real world.

Dagstuhl Reports, 2014

This report documents the programme and the outcomes of Dagstuhl Seminar 14202 on "Judgment Aggregation for Artificial Intelligence". Judgment aggregation is a new group decision-making theory that lies in the intersection of logic and social choice; it studies how to reach group decisions on several logically interconnected issues by aggregation of individual judgments. Until recently research in judgment aggregation was dominated by its originating context of philosophy, political science and law. Presently, however we are witnessing increasing work in judgment aggregation from researchers in computer science. Since researchers from such diverse disciplinary backgrounds working on judgment aggregation each publish within their own discipline with virtually no cross-discipline cooperation on concrete projects, it is essential that they are given an opportunity to connect to each other and become aware of the workings of the other side. This seminar has provided such an op...

Synthese, 2004

The "doctrinal paradox" or "discursive dilemma" shows that propositionwise majority voting over the judgments held by multiple individuals on some interconnected propositions can lead to inconsistent collective judgments on these propositions. List and Pettit (2002) have proved that this paradox illustrates a more general impossibility theorem showing that there exists no aggregation procedure that generally produces consistent collective judgments and satisfies certain minimal conditions. Although the paradox and the theorem concern the aggregation of judgments rather than preferences, they invite comparison with two established results on the aggregation of preferences: the Condorcet paradox and Arrow's impossibility theorem. We may ask whether the new impossibility theorem is a special case of Arrow's theorem, or whether there are interesting disanalogies between the two results. In this paper, we compare the two theorems, and show that they are not straightforward corollaries of each other. We further suggest that, while the framework of preference aggregation can be mapped into the framework of judgment aggregation, there exists no obvious reverse mapping. Finally, we address one particular minimal condition that is used in both theorems-an independence condition-and suggest that this condition points towards a unifying property underlying both impossibility results.

SYNTHESE-DORDRECHT-, 2006