Logical Constraints on Judgement 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 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.

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.

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.

Judgment aggregation and agenda manipulation

International audience

We explore how judgment aggregation and belief merging in the framework of fuzzy logic can help resolve the “Doctrinal Paradox.” We also illustrate the use of fuzzy aggregation functions in social choice theory.

Drawing on dual process theory, we suggest that the benefits that arise from combining different individual judgments will be heightened when these judgments are based on different cognitive processes. We test this hypothesis in two experimental studies in which participants were prompted to make judgments relying on an analytical process, on their intuition, or in a control condition in which no particular instructions were given. Our results show that an aggregation of intuitive and analytical judgments provides more accurate estimates than any other aggregation procedure, and that this advantage increases with the number of judgments that are aggregated. Moreover, we find evidence that this result is driven by a lower correlation between errors of intuitive and analytical judgments compared to errors of judgments that are based on the same cognitive process or judgment errors in the control condition.

**Read paper on the following link:** https://ifaamas.org/Proceedings/aamas2022/pdfs/p263.pdf **Abstract:** We simulate voting rules for multiwinner elections in a model of judgment aggregation that distinguishes between rationality and feasibility constraints. These constraints restrict the structure of the individual judgments and of the collective outcome computed by the rule, respectively. We extend known results regarding the simulation of single-winner voting rules to the multiwinner setting, both for elections with ordinal preferences and for elections with approval-based preferences. This not only provides us with a new tool to analyse multiwinner elections, but it also suggests the definition of new judgment aggregation rules, by generalising some of the principles at the core of well-known multiwinner voting rules to this richer setting. We explore this opportunity with regards to the principle of proportionality. Finally, in view of the computational difficulty associated with many judgment aggregation rules, we investigate the computational complexity of our embeddings and of the new judgment aggregation rules we put forward.

In the literature on judgment aggregation, an important open question is how to measure the distance between any two judgment sets. This is relevant for issues of social choice: if two individuals hold different beliefs then we might want to find a compromise that lies somewhere between them. We propose a set of axioms that determine a measure of distance uniquely. This measure differs from the widely used Hamming metric. The difference between Hamming’s metric and ours boils down to one axiom. Given judgment sets A and B, this axiom says that if the propositions in $${A \\cap B}$$ jointly imply that the propositions in A−B share the same truth value, then the disagreement between A and B over those propositions in A−B should be counted as a single disagreement. We consider the application of our metric to judgment aggregation, and also use the metric to measure the distance between preference rankings. Copyright Springer-Verlag 2012

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.

Convex combinations in judgment aggregation

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).

An enterprise resource planning (ERP) system is a critical investment that can significantly affect future competitiveness and performance of a company. The analytic hierarchy process (AHP) method is often applied to select an ERP system since it is well suited to multiple criteria decision-making problems. This study presents a simulation-based AHP (SiAHP) method for group decision making and is applied to the real-world problem of selecting a suitable ERP system for a Korean homeshopping company. To enhance the fitness of a group AHP method and to facilitate the ERP system selection process, this paper proposes a simulation-based approach for building a group consensus instead of forming point estimates that are aggregated from individual preference judgments. To be specific, the proposed method is based on observations from empirically observed frequency distributions and does not use aggregation procedures, compared to typical group AHP for obtaining a group solution. This approach, reflecting the diversification of group members' opinions as they are, is conceived to be useful as a tool for obtaining insights into agreements and disagreements with respect to the alternatives among the individuals of a group. The real-world example demonstrates the feasibility of our proposed approach.

Drawing on dual-process theory, we suggest that the benefits that arise from combining several quantitative individual judgments will be heightened when these judgments are based on different cognitive processes. We tested this hypothesis in three experimental studies in which participants provided estimates for the dates of different historical events (Study 1, N = 152), made probabilistic forecasts for the outcomes of soccer games (Study 2, N = 98), and estimated the weight of individuals on the basis of a photograph (Study 3, N = 3,695). For each of these tasks, participants were prompted to make judgments relying on an analytical process, on their intuition, or (in a control condition) on no specific instructions. Across all three studies, our results show that an aggregation of intuitive and analytical judgments provides more accurate estimates than any other aggregation procedure and that this advantage increases with the number of aggregated judgments.