Topological methods of preference and judgment aggregation

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.

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.

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,

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.

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 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.

This thesis aims at understanding and improving topology optimization techniques focusing on density-based level-set methods and geometrical nonlinearities. Central in this work are the numerical modeling of the mechanical response of a design and the consistency of the optimization process itself. Concerning the first topic, we investigate different means to improve the robustness of density-based numerical models including geometrical nonlinearities. The conventional approach (scaling the local material properties) can result in convergence problems due to excessive deformation in low-stiffness finite elements. To avoid excessive deformation, we combine the element connectivity parameterization method (adapting the connectivity between finite elements) with level-set-based topology optimization. Furthermore, we achieve greater robustness of analysis method using a second and improved approach called element deformation scaling. This approach eliminates the need for solving internal equilibrium equations (needed for the element connectivity parameterization method) via an explicit relation between the local internal and global external displacement field. The second focus of this thesis is on the optimization process of level-set-based topology optimization, and in particular its numerical consistency. We observe that signed-distance reinitialization of the level-set function affects the shape of a design in the optimization process. To minimize this effect, we propose a discrete level-set method that is based on an approximate Heaviside function and focusing on the implementation. Furthermore, we also propose a level-set-based topology optimization method using an exact Heaviside function and mathematical programming that effectively eliminates the need for reinitialization. We demonstrate that our density-based level-set method is closely related to conventional density-based topology optimization methods, while offering the advantage of more control over the geometrical complexity. On the other hand, we confirm that the dependence of the final result on the initial design which remains one of the big challenges for level-set-based topology optimization. The potential of the proposed level-set method is shown by applying it on problems with stress constraints and geometrical nonlinearities and performing manufacturing tolerant topology optimization. Finally, this thesis offers a review of level-set methods for structural topology optimization to identify and discuss the different approaches that are available in literature. We can distinguish between level-set methods by examination of their design parameterization, sensitivities, update procedures and regularization techniques. A level-set-based design parameterization offers the advantage of a crisp distinction between subdomains. For this reason, X-FEM approaches and conforming discretizations are an interesting option to retain the crisp nature of the level-set-based description of the design. Many level-set methods are combined with density-based numerical models and are, therefore, closely related to conventional density-based topology optimization methods. In particular, recently proposed projection methods have much in common with a level-set-based design description. The results of level-set-based TO methods often rely heavily on regularization techniques that introduce inconsistencies in the optimization process. Numerical consistency does not necessarily lead to the best search direction, but is essential to find a Karush-Kuhn-Tucker point of the discretized optimization problem.

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.

Many decision making problems of business and management are formulated in terms of Multiple Attribute Decision Making (MADM): given a set of alternatives evaluated with multiple criteria, find the alternative which according to the Decision Maker (DM), has the most preferred combination of criteria values (attributes), or rank alternatives from the most preferred one to the least preferred one. The MADM methods incorporate mechanisms of building preference models based on information obtained from the DM. In a wide variety of such methods, the DM is supposed to provide information in terms of weights of criteria, usually understood as criteria’s priorities. These weights serve as parameters of the method-pecific preference models. The DM can define weights directly, or by using special weight elicitation techniques such as AHP, MAVT and others. Our concerns are that when using weight-based methods, the DM cannot ensure the correctness of the preference model. First, different weight-based methods use different kinds of preference models, which prioritize criteria based on weights in different manners. Second, interpretation of weights in some MADM methods is far from intuitive. Thus, a situation may occur when an inexperienced DM thinks of weights differently than they actually work in the method, and expresses the preference information incorrectly. In this paper we demonstrate the differences between how weights are interpreted in several methods: simple additive weighting, TOPSIS, VIKOR and PROMETHEE. We do it by comparing rankings produced with methods based on randomly generated data. We demonstrate that differences of interpreting weights significantly contribute to differences in produced rankings. A solution to this problem could be twofold: first, increasing awareness of differences between method-specific weight-based prioritizing mechanisms, and second, providing interpretations of weights for popular methods in the language understandable by the DMs.

We develop a general, formal model of social choice in which voters have continuous preferences over a one-dimensional space. Our model is parameterized by different restrictions that we introduce regarding the way voter preferences change in time as well as the optimization criteria (that correspond to a normative continuum of fairness definitions) desired from an aggregation method---that outputs a continuous, one-dimensional curve---given such inputs. We discuss the applicability of the model to different real-world situations and, as a first step towards an analysis of the different model realizations, we concentrate on identifying those cases that are computationally feasible to compute.

Asthma is a complex respiratory disease characterised by spatial ventilation heterogeneity (VH) in the lung. Multiple breath washout (MBW) and forced oscillation technique (FOT) testing has been shown to quantify VH at the mouth. Methods to provide fresh insight into the mechanisms driving VH are urgently needed to support treatment development. Computed tomography (CT) with parametric response map (PRM) analysis provides spatial and functional lung data. Integrated data sets of imaging features and VH markers present the problem of extracting meaningful information from high dimensional objects; topological data analysis (TDA) has demonstrated recent success in solving this problem. In this thesis topological methods are applied to extract information about lung anatomy and function from imaging data to study association with VH in innately high dimensional integrated data sets. It was sought to describe the spatial distributions of voxel based metrics, visualise the lungs using TDA methods, and combine PRM with simulated airway tree models to improve computational modelling of FOT. Data was obtained from a clinical trial involving 52 subjects, 41 asthmatic (11 healthy). An algorithm was developed to quantify directional Hounsfield unit change (ΔHU) gradients. TDA was applied to integrated data sets and voxel sets with measures of airway tree depth. FOT modelling was augmented with ΔHU about simulated acinar airways.Inferior-to-superior ΔHU gradient best differentiated VH markers. Visualisations of the lungs illustrated gradient reversal and increased heterogeneity in ΔHU in asthmatics. FOT modelling using ΔHU parameterisation showed increased correlation between clinically measured and simulated FOT small airway resistance (R5-R20). In this thesis additional insight is presented into mechanisms affecting VH markers derived from MBW and FOT. Topological methods to analyse imaging data were developed and implemented, including workflows to visualise the lungs and lung function. Linking simulated airway tree geometry with ΔHU showed improvement in FOT modelling.

Introduction to Judgment Aggregation