A KKM-RESULT AND AN APPLICATION FOR BINARY AND NON-BINARY CHOICE FUNCTIONS

By generalizing the classical Knaster-Kuratowski-Mazurkiewicz Theorem, we obtain a result that provides sufficient conditions to ensure the non-emptiness of several kinds of choice functions. This result generalizes well-known results on the existence of maximal elements for binary relations (Bergstrom, 1975; Walker, 1977; Tian, 1993), on the non-emptiness of non-binary choice functions (Nehring, 1996; Llinares and Sánchez, 1999) and on the non-emptiness of some classical solutions for tournaments (top cycle and uncovered set) on non-finite sets.Binary Choice Function; Non-Binary Choice Function

Affirmative Action and School Choice

This paper proposes a reform for school allocation procedures in order to help integration policies reach their objective. For this purpose, we suggest the use of a natural two-step mechanism. The (equitable) first step is introduced as an adaptation of the deferred-acceptance algorithm designed by Gale and Shapley (1962), when students are divided into two groups. The (efficient) second step captures the idea of exchanging places inherent to Gale’s Top Trading Cycle. This latter step could be useful for Municipal School Boards when implementing some integration policies.Integration Policy; School Allocation; Affirmative Action

On Integration Policies and Schooling

This paper proposes a reform for school allocation procedures in order to help integration policies reach their objective. For this purpose, we suggest the use of a natural two-step mechanism. The (stable) first step is introduced as an adaptation of the deferred-acceptance algorithm designed by Gale and Shapley (1962), when students are divided into two groups. The (efficient) second step captures the idea of exchanging places inherent to Gale's Top Trading Cycle. This latter step could be useful for Municipal School Boards when implementing some integration policies.Integration Policy; School Allocation; Affirmative Action

- CHOICE FUNCTIONS: RATIONALITY RE-EXAMINED.

On analyzing the problem that arises whenever the set of maximal elements is large, and aselection is then required (see Peris and Subiza, 1998), we realize that logical ways of selectingamong maximals violate the classical notion and axioms of rationality. We arrive at the sameconclusion if we analyze solutions to the problem of choosing from a tournament (where maximalelements 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 internalinformation provided by the binary relation.Rationality; Choice Functions; Maximal Elements.

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 Solution for General Exchange Markets with Indivisible Goods when Indifferences are Allowed

It is well known that the core of an exchange market with indivisible goods is always non empty, although it may contain Pareto inefficient allocations. The strict core solves this shortcoming when indifferences are not allowed, but when agents' preferences are weak orders the strict core may be empty. On the other hand, when indifferences are allowed, the core or the strict core may fail to be stable sets, in the von Neumann and Morgenstern sense. We introduce a new solution concept that improves the behaviour of the strict core, in the sense that it solves the emptiness problem of the strict core when indifferences are allowed in the individuals' preferences and whenever the strict core is non-empty, our solution is included in it. We define our proposal, the MS-set, by using a stability property (m-stability) that the strict core fulfills. Finally, we provide a min-max interpretation for this new solution

ADJUSTING CORRELATION MATRICES

The article proposes a new algorithm for adjusting correlation matrices and for comparison with Finger's 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 Finger's approach in the sense that it alters as little as possible those correlations that we do not wish to alter but they change in order to obtain a consistent Finger correlation matrix.Stochastic, Volatility, Skewness, Kurtosis, Pricing.

Folk solution for simple minimum cost spanning tree problems

A minimum cost spanning tree problem analyzes how to efficiently connect a group of individuals to a source. Once the efficient tree is obtained, the addressed question is how to allocate the total cost among the involved agents. One prominent solution in allocating this minimum cost is the so-called Folk solution. Unfortunately, in general, the Folk solution is not easy to compute. We identify a class of mcst problems in which the Folk solution is obtained in an easy way. This class includes elementary cost mcst problems.Financial support from Generalitat de Catalunya (2014SGR325 and 2014SGR631) and Ministerio de Economía y Competitividad (ECO2013-43119-P) is acknowledged

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

Cost sharing solutions defined by non-negative eigenvectors

The problem of sharing a cost M among n individuals, identified by some characteristic ci∈R+,ci∈R+, appears in many real situations. Two important proposals on how to share the cost are the egalitarian and the proportional solutions. In different situations a combination of both distributions provides an interesting approach to the cost sharing problem. In this paper we obtain a family of (compromise) solutions associated to the Perron’s eigenvectors of Levinger’s transformations of a characteristics matrix A. This family includes both the egalitarian and proportional solutions, as well as a set of suitable intermediate proposals, which we analyze in some specific contexts, as claims problems and inventory cost games.Financial support from Spanish Ministry of Economy and Competitiveness under Project ECO2013-43119 is gratefully acknowledged. Silva-Reus also acknowledges financial support from the Generalitat Valenciana, Spain under Project PROMETEO/2009/068