Subcalculus for set functions and cores of TU games.

This paper introduces a subcalculus for general set functions and uses this framework to study the core of TU games. After stating a linearity theorem, we establish several theorems that characterize mea- sure games having finite-dimensional cores. This is a very tractable class of games relevant in many economic applications. Finally, we show that exact games with Þnite dimensional cores are generalized linear production games.TU games; non-additive set functions; subcalculus; cores

Cluster-additive functions on stable translation quivers

Additive functions on translation quivers have played an important role in the representation theory of finite dimensional algebras, the most prominent ones are the hammock functions introduced by S. Brenner. When dealing with cluster categories (and cluster-tilted algebras), one should look at a corresponding class of functions defined on stable translation quivers, namely the cluster-additive ones. We conjecture that the cluster-additive functions on a stable translation quiver of Dynkin type A_n, D_n, E_6, E_7, E_8 are non-negative linear combinations of cluster-hammock functions (with index set a tilting set). The present paper provides a first study of cluster-additive functions and gives a proof of the conjecture in the case A_n.Comment: 24 page

On density properties of weakly absolutely continuous measures

It is shown that some set of all step functions (and the set of all uniform limits of ones) allows an embedding into some compact subset (with respect to weak-star topology) of the set of all finitely additive measures of bounded variation in the form of an everywhere dense subset. Precisely, we considered the set of all step functions (the set of all uniform limits of such functions) such that integral of absolute value of the functions with respect to non-negative finitely additive measure λ is equal to the unit. For these sets, the possibility of the embedding is proved for the cases of non-atomic and finite range measure λ; in the cases the compacts do not coincide. Namely, in the nonatomic measure case, it is shown that the mentioned sets of functions allow the embedding into the unit ball (in the strong norm-variation) of weakly absolutely continuous measures with respect to λ in the form of a everywhere dense subset. In the finite range measure case, it is shown that the mentioned sets of functions allow the embedding into the unit sphere of weakly absolutely continuous measures with respect to λ in the form of a everywhere dense subset. In the last case the sphere is closed in the weak-star topology. An interpretation of these results is given in terms of an approach connected with an extension of linear control problems in the class of finitely additive measures

Choquet integrals in potential theory

This is a survey of various applications of the notion of the Choquet integral to questions in Potential Theory, i.e. the integral of a function with respect to a non-additive set function on subsets of Euclidean n-space, capacity. The Choquet integral is, in a sense, a nonlinear extension of the standard Lebesgue integral with respect to the linear set function, measure. Applications include an integration principle for potentials, inequalities for maximal functions, stability for solutions to obstacle problems, and a refined notion of pointwise differentiation of Sobolev functions

A multicriteria analysis of stated preferences among freight transport alternatives

Stated preferences data can be of different types: choice data, rankings or ratings. In all cases, these data can be used in different ways as inputs of econometric discrete choice models. This allows to estimate the weights of the different attributes characterizing an alternative. For freight transport, an alternative's attributes would be, for example, reliability, safety, frequency, etc., besides time and cost. Depending on the data sample, number of alternatives and number of attributes, it is possible to proceed to an analysis of individual data or of aggregated data. In case one is interested to analyze individual behaviors in depth, the option exists to rely on some kind of multicriteria analysis for deriving individual utility functions (actually, decision functions) rather than on a classic discrete choice model. Such a procedure also can be useful for deriving individual utilities as input in a hybrid model combining individual utilities with group data. Such a multicriteria approach is envisaged in the context of a stated preference experiment that is currently applied to freight shippers in Belgium. The data in this case are rankings of alternatives, and there is multicritera method that is particularly well adapted for such data: the UTA models developed by Jacquet-Lagrèze and Siskos. It is based on the specification of an additive utility made of non-linear partial utility functions that are piecewise linear. This allows the convenient set-up of a linear goal programming problem which estimates all the functions and their weights. The paper intends to present the ranking experiment, and to use some of the preliminary interviews to illustrate this UTA methodology. Also, it will be shown how it can be used to derive equivalent money values for each attributes on the basis of the cost attribute, and how to distinguish valuations in terms of willingness to pay and willingness to accept a compensation.