Acyclic domains of linear orders: a survey

Among the many significant contributions that Fishburn made to social choice theory some have focused on what he has called "acyclic sets", i.e. the sets of linear orders where majority rule applies without the "Condorcet effect" (majority relation never has cycles). The search for large domains of this type is a fascinating topic. I review the works in this field and in particular consider a recent one that allows to show the connections between some of them that have been unrelated up to now.acyclic set;alternating scheme;distributive lattice;effet Condorcet;linear order,maximal chain,permutoèdre lattice, single-peaked domain,weak Bruhat order,value restriction.

Some order dualities in logic, games and choices

We first present the concept of duality appearing in order theory, i.e. the notions of dual isomorphism and of Galois connection. Then, we describe two fundamental dualities, the duality extension/intention associated with a binary relation between two sets, and the duality between implicational systems and closure systems. Finally, we present two "concrete" dualities occuring in social choice and in choice functions theories.antiexchange closure operator, Galois connection, implicational system, path-independent choice function, simple game.

Condorcet domains and distributive lattices

Condorcet domains are sets of linear orders where Condorcet's effect can never occur. Works of Abello, Chameni-Nembua, Fishburn and Galambos and Reiner have allowed a strong understanding of a significant class of Condorcet domains which are distributive lattices -in fact covering distributive sublattices of the permutoèdre lattice- and which can be obtained from a maximal chain of this lattice. We describe this class and we study three particular types of such Condorcet domains.Acyclic set, alternating scheme, Condorcet effect, distributive lattice, maximal chain of permutations, permutoèdre lattice.

Guilbaud's 1952 theorem on the logical problem of aggregation

In a paper published in 1952, shortly after publication of Arrow's celebrated impossibility result, the French mathematicien Georges-Théodule Guilbaud has obtained a dictatorship result for the logical problem of aggregation, thus anticipating the literature on abstract aggregation theory and judgment aggregation. We reconstruct the proof of Guilbaud's theorem, which is also of technical interest, because it can be seen as the first use of ultrafilters in social choice theory.Aggregation ; judgment aggregation ; logical connectives ; simple game ; ultrafilter

Consensus theories: an oriented survey

This article surveys seven directions of consensus theories: Arrowian results, federation consensus rules, metric consensus rules, tournament solutions, restricted domains, abstract consensus theories, algorithmic and complexity issues. This survey is oriented in the sense that it is mainly – but not exclusively – concentrated on the most significant results obtained, sometimes with other searchers, by a team of French searchers who are or were full or associate members of the Centre d'Analyse et de Mathématique Sociale (CAMS).Consensus theories ; Arrowian results ; aggregation rules ; metric consensus rules ; median ; tournament solutions ; restricted domains ; lower valuations ; median semilattice ; complexity

The duality between the anti-exchange closure operators and the path independent choice operators on a finite set,

In this paper, we show that the correspondence discovered by Koshevoy ([18]) and Johnson and Dean ([15],[16]) between anti-exchange closure operators and path independent choice operators is a duality between two semilattices of such operators. Then we use this duality to obtain results concerning the "ordinal" representations of path independent choice functions from the theory of anti-exchange closure operators.Anti-exchange closure operator, choice function, convex geometry, path independence,partial order, semilattice.

Lattices of choice functions and consensus problems

. In this paper we consider the three classes of choice functionssatisfying the three significant axioms called heredity (H), concordance (C) and outcast (O). We show that the set of choice functions satisfying any one of these axioms is a lattice, and we study the properties of these lattices. The lattice of choice functions satisfying (H) is distributive, whereas the lattice of choice functions verifying (C) is atomistic and lower bounded, and so has many properties. On the contrary, the lattice of choice functions satisfying(O) is not even ranked. Then using results of the axiomatic and metric latticial theories of consensus as well as the properties of our three lattices of choice functions, we get results to aggregate profiles of such choice functions into one (or several) collective choice function(s).Aggregation, choice function, concordance, consensus, distance, distributive, heredity, lattice, outcast

Guilbaud's Theorem : An early contribution to judgment aggregation

In a paper published in 1952, the French mathematician Georges-Théodule Guilbaud has generalized Arrow's impossibility result to the "logical problem of aggregation", thus anticipating the literature on abstract aggregation theory and judgment aggregation. We reconstruct the proof of Guilbaud's theorem, which is also of technical interest, because it can be seen as the first use of ultrafilters in social choice theory.Arrow's theorem, aggregation rule, judgment aggregation, logical connexions, simple game, ultrafilter.

Aggregation and residuation

In this paper, we give a characterization of meet-projections in simple atomistic lattices that generalizes results on the aggregation of partitions in cluster analysis.Aggregation theory ; dependence relation ; meet projection ; partition ; residual map ; simple lattice

Modèles ordinaux de préférences

URL des Cahiers : https://halshs.archives-ouvertes.fr/CAHIERS-MSECahiers de la Maison des Sciences Economiques 2005.97 - ISSN : 1624-0340Dans ce texte à visée didactique on présente les principaux modèles d'ordres utilisés pour représenter les préférences d'un sujet sur un ensemble fini de biens de nature variée. On part du modèle d'ordre fort correspondant au cas où la préférence est représentée par une fonction d'utilité numérique, modèle qui implique que la relation d'indifférence du sujet est transitive. Les modèles d'ordre quasi-fort et d'ordre d'intervalles permettent de ne plus supposer l'indifférence transitive, tout en conservant des propriétés de représentation numériques avec l'introduction de seuils. Des résultats sur la caractérisation et la représentation numérique des relations de Ferrers, relations qui généralisent les relations d'ordres précédentes, permettent d'obtenir simplement les résultats sur ces relations d'ordres. Des compléments d'ordre historique ou mathématique sont proposés au lecteur