Bernard Monjardet

The median procedure in cluster analysis and social choice theory

Abstract Classical approachs for fitting and aggregation problems, specially in cluster analysis, social choice theory and paired comparisons methods, consist in the minimization of a remoteness function between relational data and a relational model. The notion of median, with its algebraic, metric, geometrical and statistical aspects, allow a unified treatment of many of base problems. Properties of median procedures are organized according to four directions: stabilities and axiomatic characterizations; Arrow-like properties; combinatorial properties; effective computational possibilities. Finally, interesting mathematical problems, related to the notion of median are surveyed.

Biorder families, valued relations and preference modelling

Abstract Several compatibility conditions are studied for families of interval orders or semiorders, involving for instance step-type matrices and functional representations. Our approach uses the basic notion of biorder or Guttman scale. The results answer a question raised by Roberts, who in fact treated the particular case of nested families. They provide in a very general setting various formalizations for the notion of probabilistic consistency of a subject in a binary choice process. Other possible applications are also mentioned.

Utility Maximization, Choice and Preference

The utility maximization paradigm forms the basis of many economic, psychological, cognitive and behavioral models. Since it was first devised in the eighteenth century, numerous examples have revealed the deficiencies of the concept.This book makes a contribution to overcome those deficiencies by taking into account insensitivity of measurement threshold and context of choice. It covers classic theory as a special, context-free case and gives a systematic overview of new models of utility maximization within a context-dependent threshold as well as related preference and choice models.The second edition has been updated to include the most recent developments and a new chapter on classic and new results for infinite sets. The presented models will be helpful to scientists in economics, decision making theory, social choice theory, behavioral and cognitive sciences, and related fields.

Metrics on partially ordered sets-A survey

We review more or less recent results on metrics associated with valuations or (and) minimum weighted paths in posets, metric characterizations of posets and we point out some applications of these results to problems involving uses of distances in social sciences.

Arrowian characterizations of latticial federation consensus functions

Abstract In this paper we begin the construction of an abstract axiomatic theory of consensus functions in order to account for several classes of similar concrete results in the domain of the axiomatic approach to consensus. Such results, which belong to various fields, are mentioned in our introduction and conclusion. Among the consensus functions defined on a semilattice L , i.e. the functions L n → L , we study the federation consensus functions associated with cohereditary families of subsets of {1,…, n } called ‘federations’ (or simple games). We obtain several axiomatic characterizations of all federation consensus functions, or of significant subclasses of such functions, and especially of the meet projection (‘oligarchic’) consensus functions. These characterizations depend on the structural properties of the considered semilattice, which are the key to understanding the concrete results mentioned above and allow us to obtain new results. We end by pointing out some further lines of research.

An Axiomatic Theory of Tournament Aggregation

An axiomatic theory for aggregation of individual preferences is developed. Many authors, since K. J. Arrow, have studied the case where every individual preference is an ordering. We study here the case where every individual preference is a tournament for instance, in “paired comparisons”. The original results obtained can be compared to those of the classic theory. For example, we prove, in this context, a generalisation of Arrows theorem and we emphasize duality between Arrows results and Black-Inada-Sens results technically by means of a Galois connection between two lattices. We used social functions defined by means of “families of majorities” simple games,

On the comparison of the Spearman and Kendall metrics between linear orders

Abstract This paper bears on the comparison of two well-known metrics between linear orders called the Kendall and Spearman metrics or/and of their normalized versions, respectively, known as the Kendall tau and the Spearman rho . Using a combinatorial approach based on the partial order intersection of the two compared linear orders, one first proves a relation between these two metrics and a semi-metric, equivalent to the classical Daniels inequality (1948) and to a Guilbaud formula (1980). Then this approach allows to express the difference tau-rho as a simple function of parameters of this same partial order, to compute the maximum value of this difference and to characterize the corresponding pairs of linear orders. Finally, it also leads to discover an ordinal monotonicity property of the Spearman metric.

Norbert Wiener on the theory of measurement (1914, 1915, 1921)

Abstract These words, written by Norbert Wiener in 1919 when he was 24 years old, appear near the end of the third of three extraordinary papers on the theory of relations and measurement that he began before his twentieth birthday. The papers use the notation of Principia Mathematica, which is fairly inaccessible to modern readers. However, Wieners contributions to measurement theory deserve to be remembered because they include important concepts that were rediscovered by others and now have a central place in the representational theory of measurement and in graph theory. Our purpose is to recount in modern terms Wieners work in these areas.

The presence of lattice theory in discrete problems of mathematical social sciences. Why

Abstract In this paper we point out many fields of mathematical social sciences where lattices are present, for instance the discrete resources allocation problems. However, we do not intend to survey the use of lattice theory in such discrete problems. Rather, we present an overview on the ubiquity and polymorphism of finite lattice theory, which accounts for its presence in these problems (like in many others). Indeed, on one hand lattices are on the border of two fundamental mathematical structures, namely algebra and order, and on the other hand they are naturally induced by several other significant mathematical objects like set systems, operators, binary relations or implications. We set out these different aspects and we illustrate them in the case of two significant classes of lattices, the lower locally distributive lattices and the distributive lattices.

A Generalisation of Probabilistic Consistency: Linearity Conditions for Valued Preference Relations

We show how the theory of “probabilistic consistency” developped in the framework of pair comparison methods with forced choice con be extended to arbitrary valued preference relations. Especially we generalize classical “stochastic transitivity” conditions securing the “linearity” of valued relations and we study the implications between such conditions.