NUMERICAL REPRESENTATION OF ACYCLIC PREFERENCES

In this paper, it is shown that, under certain conditions on a preference relation defined on a set X, there exists a numerical representation by means of set-valued real functions. This kind of representation extends the usual utility function as well as the representation by means of two real functions. The continuity of this representation is also discussed.

CONDORCET CHOICE FUNCTIONS AND MAXIMAL ELEMENTS

Choice functions on tournaments always select the maximal element (Condorcet winner), provided they exist, but this property does not hold in the more general case of weak tournaments. In this paper we analyze the relationship between the usual choice functions and the set of maximal elements in weak tournaments. We introduce choice functions selecting maximal elements, whenever they exist. Moreover, we compare these choice functions with those that already exist in the literature.choice functions, tournaments, maximal elements.

A characterization of acyclic preferences on countable sets

In this paper a new numerical representation of preferences (by means of set-valued real functions) is proposed. Our representation extends the usual utility function (in case preferences are preorder-type) as well as the pairwise representation (in case preferences are interval-order type). Then, we provide a characterization of acyclic preference relations on countable sets as those admitting a set-valued numerical representation.

Choosing among maximals

In a choice situation, it is usually assumed that the agents select the maximal elements inaccordance with their preference relation. Nevertheless, there are situations in which a selectioninside this maximal set is needed. In such a situation we can select randomly some of thesemaximal elements, or we can choose among them according to the behaviour of these maximalelements. In order to illustrate this, let´s imagine a preference relation >=, defined on a finite setA={x1,x2,...,xn}, such that x1 is indifferent to each alternative and x2 is strictly preferred to everyxi,i >=3. Both x1 and x2 are maximal elements, but we can say that x2 is a better maximalthan x1. In this paper we define selections of the set of maximal elements of a preference relationby choosing the better ones among them.Binary relation, maximal elements

Numerical representation for lower quasi-continuous preferences

A weaker than usual continuity condition for acyclic preferences is introduced. For preorders this condition turns out to be equivalent to lower continuity, but in general this is not true. By using this condition, a numerical representation which is upper semicontinuous is obtained. This fact guarantees the existence of maxima of such a function, and therefore the existence of maximal elements of the binary relation.Numerical representation, maximal elements, lower quasi-continuous preferences

Numerical representation of partial orderings

In this paper a numerical representation of preferences by means of subsets of the real line is proposed. This representation turns out to be natural for partial orderings. Some results on this representation, extending those for utility functions and pairwise representations, are provided for this case.

Maximal elements of non necessarily acyclic binary relations

The existence of maximal elements for binary preference relations is analyzed without imposing transitivity or convexity conditions. From each preference relation a new acyclic relation is defined in such a way that some maximal elements of this new relation characterize maximal elements of the original one. The result covers the case whereby the relation is acyclic.

A demand function for pseudotransitive preferences

This paper deals with the existence and properties of the demand correspondence when agents' preferences are pseudotransitive. It is shown that a consumption plan belongs to the demand mapping if and only if it is a maximizer of a real-valued weak utility function. Further properties, as hemicontinuity and convex-valuedness of the demand mapping, are also analyzed.Pseudotransitive preferences, demand function, weak utility function

Equal-loss solution for monotonic coalitional games

A new solution concept to monotonic cooperative games with nontransferable utility is introduced. This proposal, called the coalitional equal-loss solution, is based on the idea that players withing a coalition should have equal losses from a point of maximum expectations. The proposal generalizes the rational equal-loss solution defined on the subclass of bargaining problems as well as the Shapley value defined on the subclass of superadditive cooperative games with transferable utility.Coalitional Games, Rational Equal-Loss Solution, Shapley Value

Revealed preference axioms for rational choice on nonfinite sets

Following the work of Bandyopadhyay and Sengupta, we analyze the rationalization of a choice function in terms of the revealed preference but in a more general context: choice sets with a continuum of alternatives. Firstly it is proved that some results which are verified in the finite case are not true in this context and new conditions to characterize the different kinds of rationality are stated. Furthermore, we analyze the continuity of the revealed preference and a characterization of open revealed preferences in terms of the hemicontinuity of the choice function is obtained.Revealed preference on nonfinite sets, rational choice function