1,300 research outputs found

    Social choice of convex risk measures through Arrovian aggregation of variational preferences

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    It is known that a combination of the Maccheroni-Marinacci-Rustichini (2006) axiomatisation of variational preferences with the Fƶllmer-Schied (2002,2004) representation theorem for concave monetary utility functionals provides an (individual) decision-theoretic foundation for convex risk measures. The present paper is devoted to collective decision making with regard to convex risk measures and addresses the existence problem for non-dictatorial aggregation functions of convex risk measures - in the guise of variational preferences - satisfying Arrow-type rationality axioms (weak universality, systematicity, Pareto principle). We prove an impossibility result for finite electorates, viz. a variational analogue of Arrow's impossibility theorem. For infinite electorates, the possibility of rational aggregation of variational preferences (i.e. convex risk measures) depends on a uniform continuity condition for the variational preference profiles: We shall prove variational analogues of both Campbell's impossibility theorem and Fishburn's possibility theorem. Methodologically, we adopt the model-theoretic approach to aggregation theory inspired by Lauwers-Van Liedekerke (1995). In an appendix, we apply the Dietrich-List (2010) analysis of logical aggregation based on majority voting to the problem of variational preference aggregation. The fruit is a possibility theorem, but at the cost of considerable and - at least at first sight - rather unnatural restrictions on the domain of the variational preference aggregator.variational preference representation, convex risk measure, multiple priors preferences, Arrow-type preference aggregation, judgment aggregation, abstract aggregation theory, model theory, first-order predicate logic, ultrafilter, ultraproduct

    Social Preference Under Twofold Uncertainty

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    We investigate the conflict between the ex ante and ex post criteria of social welfare in a new framework of individual and social decisions, which distinguishes between two sources of uncertainty, here interpreted as an objective and a subjective source respectively. This framework makes it possible to endow the individuals and society not only with ex ante and ex post preferences, as is usually done, but also with interim preferences of two kinds, and correspondingly, to introduce interim forms of the Pareto principle. After characterizing the ex ante and ex post criteria, we present a first solution to their conflict that extends the former as much possible in the direction of the latter. Then, we present a second solution, which goes in the opposite direction, and is also maximally assertive. Both solutions translate the assumed Pareto conditions into weighted additive utility representations, and both attribute to the individuals common probability values on the objective source of uncertainty, and different probability values on the subjective source. We discuss these solutions in terms of two conceptual arguments, i.e., the by now classic spurious unanimity argument and a novel informational argument labelled complementary ignorance. The paper complies with the standard economic methodology of basing probability and utility representations on preference axioms, but for the sake of completeness, also considers a construal of objective uncertainty based on the assumption of an exogeneously given probability measure. JEL classification: D70; D81

    Measuring voting power in convex policy spaces

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    Classical power index analysis considers the individual's ability to influence the aggregated group decision by changing its own vote, where all decisions and votes are assumed to be binary. In many practical applications we have more options than either "yes" or "no". Here we generalize three important power indices to continuous convex policy spaces. This allows the analysis of a collection of economic problems like e.g. tax rates or spending that otherwise would not be covered in binary models.Comment: 31 pages, 9 table

    Methods for Ordinal Peer Grading

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    MOOCs have the potential to revolutionize higher education with their wide outreach and accessibility, but they require instructors to come up with scalable alternates to traditional student evaluation. Peer grading -- having students assess each other -- is a promising approach to tackling the problem of evaluation at scale, since the number of "graders" naturally scales with the number of students. However, students are not trained in grading, which means that one cannot expect the same level of grading skills as in traditional settings. Drawing on broad evidence that ordinal feedback is easier to provide and more reliable than cardinal feedback, it is therefore desirable to allow peer graders to make ordinal statements (e.g. "project X is better than project Y") and not require them to make cardinal statements (e.g. "project X is a B-"). Thus, in this paper we study the problem of automatically inferring student grades from ordinal peer feedback, as opposed to existing methods that require cardinal peer feedback. We formulate the ordinal peer grading problem as a type of rank aggregation problem, and explore several probabilistic models under which to estimate student grades and grader reliability. We study the applicability of these methods using peer grading data collected from a real class -- with instructor and TA grades as a baseline -- and demonstrate the efficacy of ordinal feedback techniques in comparison to existing cardinal peer grading methods. Finally, we compare these peer-grading techniques to traditional evaluation techniques.Comment: Submitted to KDD 201

    Condorcet admissibility: Indeterminacy and path-dependence under majority voting on interconnected decisions

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    Judgement aggregation is a model of social choice where the space of social alternatives is the set of consistent evaluations (`views') on a family of logically interconnected propositions, or yes/no-issues. Unfortunately, simply complying with the majority opinion in each issue often yields a logically inconsistent collection of judgements. Thus, we consider the Condorcet set: the set of logically consistent views which agree with the majority in as many issues as possible. Any element of this set can be obtained through a process of diachronic judgement aggregation, where the evaluations of the individual issues are decided through a sequence of majority votes unfolding over time, with earlier decisions possibly imposing logical constraints on later decisions. Thus, for a fixed profile of votes, the ultimate social choice can depend on the order in which the issues are decided; this is called path dependence. We investigate the size and structure of the Condorcet set ---and hence the scope and severity of path-dependence ---for several important classes of judgement aggregation problems.judgement aggregation; diachronic; path-dependence; indeterminacy; Condorcet; median rule; majoritarian

    Universal Social Orderings

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    We propose the concept of a universal social ordering, defined on the set of pairs of an allocation and a preference profile of any finite population. It is meant to unify evaluations and comparisons of social states with populations of possibly different sizes with various characteristics. The universal social ordering not only evaluates policy options for a given population but also compares social welfare across populations, as in international or intertemporal comparisons of living standards. It also makes it possible to evaluate policy options which affect the size of the population or the preferences of its members. We study how to extend the theory of social choice in order to select such orderings on a rigorous axiomatic basis. Key ingredients in this analysis are attitudes with respect to population size and the bases of interpersonal comparisons.social choice, universal social orderings, maximin principle, interpersonal comparisons

    Decision-Making in the Context of Imprecise Probabilistic Beliefs

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    Coherent imprecise probabilistic beliefs are modelled as incomplete comparative likelihood relations admitting a multiple-prior representation. Under a structural assumption of Equidivisibility, we provide an axiomatization of such relations and show uniqueness of the representation. In the second part of the paper, we formulate a behaviorally general axiom relating preferences and probabilistic beliefs which implies that preferences over unambiguous acts are probabilistically sophisticated and which entails representability of preferences over Savage acts in an Anscombe-Aumann-style framework. The motivation for an explicit and separate axiomatization of beliefs for the study of decision-making under ambiguity is discussed in some detail.
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