Judgment aggregation in nonmonotonic logic

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 and agenda manipulation

**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.

Introduction to Judgment Aggregation

Logical puzzles like the doctrinal paradox raise the problem of how to aggregate individual judgements into a collective judgement, or alternatively, how to merge collectively inconsistent knowledge bases. In this paper, we view judgement aggregation as a function on propositional logic valuations, and we investigate how logic constrains judgement aggregation. In particular, we show that there is no non-dictatorial decision method for aggregating sets of judgements in a logically consistent way if the decision method is local, i.e., only depends on the individual judgements on the proposition under consideration.

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).

Complexity of manipulation and bribery in premise-based judgment aggregation with simple formulas

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.

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.

We deal the problem of aggregation of individual judgments for a global evaluation of a candidate or a product. In our theoretically oriented approach, aggregation operators are compared with each other based on their mathematical properties. We show that any monotone and strategy-proof operators is characterized by a particular collection of decision makers.

In this paper, we investigate group decision-making in a multi-criteria complex environment using an analytical hierarchy process (AHP) seeking to solve the problem of how to effectively aggregate individual preferences to reach a group consensus. Currently, the two methods considered most useful for aggregating individual preferences are the aggregation of individual judgments (AIJ) and the aggregation of individual priorities (AIP). These two aggregation methods involve two independent stages, which disconnect the processes from each other, and the relationship between the two has not yet been researched. Besides, the authorities of different experts can vary, as they are restricted by the individuals’ expertise and experiences as well as the complexity of real-world applications. In this article, a new consensus facilitation model for AHP group decision-making that takes into account experts’ weights, called EWAHP-GDM, is established to address these aspects. In this model, two aggregation methods are unified and integrated in group decision-making with AHP. We present the quantification steps for the experts’ weights and conduct an empirical analysis to verify our model by comparing it with four classic models/methods. The numerical results demonstrate that our model can provide effective investment strategy selection in investment decision-making.

The present thesis focuses on the study of the mathematical structures underlying input/output logic. Building on insights from Abstract Algebraic Logic and on the recent introduction of subordination algebras and related structures as a semantic environment for input/output logic, the present thesis establishes a strong and systematic bridge between areas of logic which have been investigated for decades independently of each other, and by doing so, it opens new possibilities of interpretations and areas of applications for these formal frameworks. Motivated by a research program investigating ‘logic at work’ rather than ‘logic in isolation’, input/output logic has been introduced for modelling the interaction between logical inferences and other agency-related notions such as conditional obligations, goals, ideals, preferences, actions, and beliefs, in the context of the formalization of normative systems in philosophical logic and AI, but also in connection and combination with a very diverse range of issues, also relevant to present-day AI, spanning from formal argumentation, to causal reasoning and non-monotonic reasoning. The first contribution of this thesis is a generalization of the framework of input/output logic from classical propositional logic to selfextensional logics, a large class of logics which prominently includes those logical systems the consequence relation of which corresponds to a partial order on the algebras canonically associated with it. The various notions of permission systems, namely, negative permission, positive static permission and dynamic permission, are generalized and studied uniformly in the context of selfextensional logics, both in themselves, and in connection with normative systems. In the same context, the notion of dual negative permission system is introduced and studied. The second contribution of this thesis is a systematic study of the algebraic counterparts of normative, permission, and dual permission systems based on the algebras of the classes canonically associated with (fully) selfextensional logics. In particular, the proof of the main characterization result of the previous chapter is refined and streamlined, and further extended to characterize properties of various notions of permission systems also in relation to normative systems, by means of modal axioms in an expanded slanted signature which includes negative modal operators (intuitively representing prohibitions). Moreover, these characterization results are then applied to resolve different issues: from the logical characterizations of output operators for both normative and permission systems, to the dual characterizations of conditions on subordination algebra (resp. precontact algebras) and on their associated spaces; the algebraization of positive static permissions; the modal characterization of the notion of cross-coherence; the clarification of the relation between certain conditions on positive bi-subordination lattices and Dunn’s axioms for positive modal logic. The third contribution of this thesis generalizes the modal characterizations, achieved in the previous chapter, of finitely many conditions on normative and permission systems, as well as on subordination algebras and related structures, to those conditions corresponding to the infinite class of clopen-analytic inequalities. Conversely, Kracht (first-order) formulas are introduced as the class of those first order conditions on algebras with subordination, precontact, and dual precontact relations which correspond to clopen-analytic axioms; finally, as an application of these results, Celani’s dual characterization results between subordination lattices and subordination spaces is generalized to the environment of algebras with relations based on distributive lattices and the infinite class of inductive Kracht formulas.