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

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

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

- 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.

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.

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

Sharing the cost of maximum quality optimal spanning trees

Minimum cost spanning tree problems have been widely studied in operation research and economic literature. Multi-objective optimal spanning trees provide a more realistic representation of different actual problems. Once an optimal tree is obtained, how to allocate its cost among the agents defines a situation quite different from what we have in the minimum cost spanning tree problems. In this paper, we analyze a multi-objective problem where the goal is to connect a group of agents to a source with the highest possible quality at the cheapest cost. We compute optimal networks and propose cost allocations for the total cost of the project. We analyze properties of the proposed solution; in particular, we focus on coalitional stability (core selection), a central concern in the literature on minimum cost spanning tree problems.This work is supported by the Spanish Ministerio de Economía y Competitividad, under project ECO2016-77200-P. Financial support from the Generalitat Valenciana (BEST/2019 Grants) to visit the UNSW is also acknowledged

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.

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.