Oligarchy and soft incompleteness

The assumption that the social preference relation is complete is demanding. We distinguish between “hard” and “soft” incompleteness, and explore the social choice implications of the latter. Under soft incompleteness, social preferences can take values in the unit interval. We motivate interest in soft incompleteness by presenting a version of the strong Pareto rule that is suited to the context of a [0, 1]-valued social preference relation. Using a novel approach to the quasi-transitivity of this relation we prove a general oligarchy theorem. Our framework allows us to make a distinction between a “strong” and a “weak” oligarchy, and our theorem identifies when the oligarchy must be strong and when it can be weak. Weak oligarchy need not be undesirable

Oligarchy and soft incompleteness

The assumption that the social preference relation is complete is demanding. We distinguish between “hard” and “soft” incompleteness, and explore the social choice implications of the latter. Under soft incompleteness, social preferences can take values in the unit interval. We motivate interest in soft incompleteness by presenting a version of the strong Pareto rule that is suited to the context of a [0, 1]-valued social preference relation. Using a novel approach to the quasi-transitivity of this relation we prove a general oligarchy theorem. Our framework allows us to make a distinction between a “strong” and a “weak” oligarchy, and our theorem identifies when the oligarchy must be strong and when it can be weak. Weak oligarchy need not be undesirable

Quantitative Concept Analysis

Formal Concept Analysis (FCA) begins from a context, given as a binary relation between some objects and some attributes, and derives a lattice of concepts, where each concept is given as a set of objects and a set of attributes, such that the first set consists of all objects that satisfy all attributes in the second, and vice versa. Many applications, though, provide contexts with quantitative information, telling not just whether an object satisfies an attribute, but also quantifying this satisfaction. Contexts in this form arise as rating matrices in recommender systems, as occurrence matrices in text analysis, as pixel intensity matrices in digital image processing, etc. Such applications have attracted a lot of attention, and several numeric extensions of FCA have been proposed. We propose the framework of proximity sets (proxets), which subsume partially ordered sets (posets) as well as metric spaces. One feature of this approach is that it extracts from quantified contexts quantified concepts, and thus allows full use of the available information. Another feature is that the categorical approach allows analyzing any universal properties that the classical FCA and the new versions may have, and thus provides structural guidance for aligning and combining the approaches.Comment: 16 pages, 3 figures, ICFCA 201

Judgment aggregation on restricted domains

We show that, when a group takes independent majority votes on interconnected propositions, the outcome is consistent once the profile of individual judgment sets respects appropriate structural conditions. We introduce several such conditions on profiles, based on ordering the propositions or ordering the individuals, and we clarify the relations between these conditions. By restricting the conditions to appropriate subagendas, we obtain local conditions that are less demanding but still guarantee consistent majority judgments. By applying the conditions to agendas representing preference aggregation problems, we show parallels of some conditions to existing social-choice-theoretic conditions, specifically to order restriction and intermediateness, restricted to triples of alternatives in the case of our local conditions.mathematical economics;