Begoña Subiza

A weak utility function for acyclic preferences

Abstract The purpose of this paper is to provide a continuous weak utility function for acyclic preferences which characterizes the maximal elements of the binary relation in terms of the maxima of this function.

Maximal elements of not necessarily acyclic binary relations

Abstract The existence of maximal elements for binary relations is analyzed without imposing transitivity or convexity conditions. An acyclic relation is constructed in such a way that some maximal elements of this new relation characterize maximal elements of the original one.

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.

A note on adjusting correlation matrices

A new algorithm for adjusting correlation matrices and for comparison with Fingers algorithm, which is used to compute Value-at-Risk in RiskMetrics for stress test scenarios. The solution proposed by the new methodology is always better than Fingers approach in the sense that it alters as little as possible those correlations that one would wish not to alter, but they change in order to obtain a consistent Finger correlation matrix.

A characterization of weak-monotone matrices

Abstract We provide some characterizations of weak-monotone matrices by using positive splittings. We prove that a matrix allowing a generalized B -splitting is weak-monotone iff the associated spectral radius is less than one. Next, we prove that weak- monotone matrices allow a generalized B -splitting. Weak-monotonicity is associated with strong positive solvability of a linear system Ax = b , b ⩾ 0.

CHOICE FUNCTIONS: RATIONALITY RE-EXAMINED

On analyzing the problem that arises whenever the set of maximal elements is large, and a selection is then required (see Peris & Subiza 1998), we realize that logical ways of selecting among maximals violate the classical notion and axioms of rationality. We arrive at the same conclusion if we analyze solutions to the problem of choosing from a tournament (where maximal elements do not necessarily exist). So, in our opinion the notion of rationality must be discussed, not only in the traditional sense of external conditions (Sen 1993), but in terms of the internal information provided by the binary relation.

A reformulation of von Neumann–Morgenstern stability: m-stability

The notion of a stable set (introduced by von Neumann and Morgenstern, 1944) is an important tool in the field of Decision Theory. However, stable sets may fail to exist. Other stability notions have been introduced in the literature in order to solve the non-existence problem. We propose a new notion, that we call m-stability, and compare it with previous proposals. Moreover, we analyze some properties (existence, uniqueness, unions and intersections, …) of the different notions of a stable set. Finally, we use the Shapley–Scarf market model with indivisible goods in order to show that the non-empty core is an m-stable set, and does not fulfill, in general, the other stability notions.

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.

Nontrivial pseudo-utility functions

Abstract A real valued function u: X → R is a pseudo-utility for a binary relation P on X if for all x, y ϵ X such that u(x) > u(y) then xPy. This kind of numerical representation is trivial in the sense that it always exists (choose u(x) as a constant function). In this paper we consider and analyze the existence of a nontrivial type of pseudo-utility function that gives valuable information on the binary relation.

Numerical representation of choice functions

In the social sciences, the most common description of individual choice consistsof assuming that the agent has an a priori ordering, or a ranking over the differentalternatives, that is, the agent knows his preference relation. Then, rational behaviorrequires choosing the best elements, according to this criterion, in every feasibleset presented for choice (i.e., to choose the maximal elements).A different approach is given by removing the assumption that the agent knowsa priori his preference relation. In this case, the way of analyzing the rationality ofthe choice function consists of observing the different choices individuals make whendifferent subsets of alternatives are presented for choice, and comparing them. Thus,rationality is based on the analysis of some coherent properties between the differentchoices individuals make when the feasible set changes.