Statistical Comparison of Aggregation Rules for Votes

If individual voters observe the true ranking on a set of alternatives with error, then the social choice problem, that is, the problem of aggregating their observations, is one of statistical inference. This study develops a statistical methodology that can be used to evaluate the properties of a given or aggregation rule. These techniques are then applied to some well-known rules.Vote aggregation, ranking rules, figure skating, maximum likelihood, optimal inference, Monte Carlo, Kemeny, Borda

Perspectives on Preference Aggregation

For centuries, the mathematical aggregation of preferences by groups, organizations or society has received keen interdisciplinary attention. Extensive 20th century theoretical work in Economics and Political Science highlighted that competing notions of “rational social choice” intrinsically contradict each other. This led some researchers to consider coherent “democratic decision making” a mathematical impossibility. Recent empirical work in Psychology qualifies that view. This nontechnical review sketches a quantitative research paradigm for the behavioral investigation of mathematical social choice rules on real ballot, experimental choice, or attitudinal survey data. The paper poses a series of open questions. Some classical work sometimes makes assumptions about voter preferences that are descriptively invalid. Do such technical assumptions lead the theory astray? How can empirical work inform the formulation of meaningful theoretical primitives? Classical “impossibility results” leverage the fact that certain desirable mathematical properties logically cannot hold universally in all conceivable electorates. Do these properties nonetheless hold in empirical distributions of preferences? Will future behavioral analyses continue to contradict the expectations of established theory? Under what conditions and why do competing consensus methods yield identical outcomes?

Aggregation of Rankings: a Brief Review of Distance-Based Rules

Some researchers have addressed the problem of aggregating individual preferences or rankings by seeking a ranking that is closest to the individual rankings. Their methods differ according to the notion of distance that they use. The best known method of this sort is due to Kemeny. The first part of this paper offers a brief survey of some of these methods. Another way of approaching the aggregation of rankings is as a problem of optimal statistical inference, in which an expected loss is minimised. This approach requires a loss function, a concept closely related the notion of distance between rankings. The second part of this paper examines two classes of parametric functions and proposes one class for the optimal statistical inference problem.Vote aggregation, ranking rules, distance, loss function, maximum likelihood, optimal inference, Kemeny

A comprehensive study of implicator-conjunctor based and noise-tolerant fuzzy rough sets: definitions, properties and robustness analysis

© 2014 Elsevier B.V. Both rough and fuzzy set theories offer interesting tools for dealing with imperfect data: while the former allows us to work with uncertain and incomplete information, the latter provides a formal setting for vague concepts. The two theories are highly compatible, and since the late 1980s many researchers have studied their hybridization. In this paper, we critically evaluate most relevant fuzzy rough set models proposed in the literature. To this end, we establish a formally correct and unified mathematical framework for them. Both implicator-conjunctor-based definitions and noise-tolerant models are studied. We evaluate these models on two different fronts: firstly, we discuss which properties of the original rough set model can be maintained and secondly, we examine how robust they are against both class and attribute noise. By highlighting the benefits and drawbacks of the different fuzzy rough set models, this study appears a necessary first step to propose and develop new models in future research.Lynn D’eer has been supported by the Ghent University Special Research Fund, Chris Cornelis was partially supported by the Spanish Ministry of Science and Technology under the project TIN2011-28488 and the Andalusian Research Plans P11-TIC-7765 and P10-TIC-6858, and by project PYR-2014-8 of the Genil Program of CEI BioTic GRANADA and Lluis Godo has been partially supported by the Spanish MINECO project EdeTRI TIN2012-39348-C02-01Peer Reviewe

Statistical Comparison of Aggregation Rules for Votes

If individual voters observe the true ranking on a set of alternatives with error, then the social choice problem, that is, the problem of aggregating their observations, is one of statistical inference. This study develops a statistical methodology that can be used to evaluate the properties of a given voting or aggregation rule. These techniques are then applied to some well-known rules

Perspectives on preference aggregation

For centuries, the mathematical aggregation of preferences by groups, organizations or society has received keen interdisciplinary attention. Extensive 20th century theoretical work in Economics and Political Science highlighted that competing notions of “rational social choice” intrinsically contradict each other. This led some researchers to consider coherent “democratic decision making” a mathematical impossibility. Recent empirical work in Psychology qualifies that view. This nontechnical review sketches a quantitative research paradigm for the behavioral investigation of mathematical social choice rules on real ballot, experimental choice, or attitudinal survey data. The paper poses a series of open questions. Some classical work sometimes makes assumptions about voter preferences that are descriptively invalid. Do such technical assumptions lead the theory astray? How can empirical work inform the formulation of meaningful theoretical primitives? Classical “impossibility results” leverage the fact that certain desirable mathematical properties logically cannot hold universally in all conceivable electorates. Do these properties nonetheless hold in empirical distributions of preferences? Will future behavioral analyses continue to contradict the expectations of established theory? Under what conditions and why do competing consensus methods yield identical outcomes

Defining Bonferroni means over lattices

In the face of mass amounts of information and the need for transparent and fair decision processes, aggregation functions are essential for summarizing data and providing overall evaluations. Although families such as weighted means and medians have been well studied, there are still applications for which no existing aggregation functions can capture the decision makers\u27 preferences. Furthermore, extensions of aggregation functions to lattices are often needed to model operations on L-fuzzy sets, interval-valued and intuitionistic fuzzy sets. In such cases, the aggregation properties need to be considered in light of the lattice structure, as otherwise counterintuitive or unreliable behavior may result. The Bonferroni mean has recently received attention in the fuzzy sets and decision making community as it is able to model useful notions such as mandatory requirements. Here, we consider its associated penalty function to extend the generalized Bonferroni mean to lattices. We show that different notions of dissimilarity on lattices can lead to alternative expressions.<br /