Non-additive Set Research Articles

Certain Considerations on the Atomicity of Set Functions

Abstract In this paper, certain considerations concerning the atomicity of non-additive set functions are given. Several properties and examples are provided.

Atoms of Certain Types of Non-Additive Set Functions

Abstract In this paper, different considerations on atoms, pseudo-atoms and minimal atoms are given for several types of non-additive set functions. In this sense, their properties, different relationships, examples and counterexamples are given.

Nonadditive integration and some solutions of cooperative games

In the paper, we propose three schemes of nonadditive integration based on several extensions of nonadditive set function and integrand to the appropriate symmetric power of the original measurable space. A survey on the integral representation of some classic objects of the cooperative game theory, derived by nonadditive integration, is given. A universal approach for investigation of both finite and infinite games is developed. We pay a particular attention to the Shapley value, Owen multilinear extension, and support function of the core of a convex cooperative game.

Volterra-Choquet nonlinear operators

In this paper we study to what extent properties of the classical linear Volterra operators can be transferred to the nonlinear Volterra-Choquet operators, obtained by replacing the classical linear integral with respect to the Lebesgue measure, by the nonlinear Choquet integral with respect to a nonadditive set function. Compactness, Lipschitz and cyclicity properties are studied.

Convergence results in Birkhoff weak integrability

This article presents some properties of real Birkhoff weak integrable functions with respect to a non-additive set function, such as H??lder inequality, Minkowski inequality, Lebesgue type convergence theorems and Fatou type theorem.

Generalized pseudo-probability measure

A new approach to non-additive set functions is presented. The notion of non-additive measure is extended using non-commutative and non-associative operations with two parameters and the cumulative distribution function of a random variable. One new non-additive measure, generalized pseudo-probability measure, is defined and some of its basic properties are proven.

ON THE EXTENSION OF NON-ADDITIVE SET FUNCTIONS

In this paper we prove theorems on the extension of non-additive set functions domain of definition of which, generally speaking, is not a ring, on the sigma-ring of sets. It is shown that continuous from the top at zero, the exhaustive compositional submersion of the first or second kind can be continued from the multiplicative class of sets to the sigma-ring of sets to a complete quasitriangular submerse complete at zero. Conditions are found under which the composition sub-measure of the first (second) kind extends to the composition sub-measure of the same kind. The continuation of the composite submerses obtained in the work is, in general, not unique. Some particular types of submeasures are considered, for which uniqueness of continuation takes place.

Cases and Scenarios in Decisions Under Uncertainty

We offer a model that combines and generalizes case-based decision theory and expected utility maximization. It is based on the premise that an agent looks ahead and assesses possible future scenarios, but may not know how to evaluate their likelihood and may not be sure that the set of scenarios is exhaustive. Consequently, she also looks back at her memory for past cases, and makes decisions so as to maximize a combined function, taking into account both scenarios and cases. We allow for non-additive set functions, both over future scenarios and over past cases, to capture (i) incompletely specified or unforeseen scenarios, (ii) ambiguity, (iii) the absence of information about counterfactuals, and (iv) some forms of case-to-rule induction (abduction) and statistical inference. We axiomatize this model. Learning in this model takes several forms, and, in particular, changes the relative weights of the two forms of reasoning.

Compromising in bifocal distribution games: the average value

It is well known that, in distributions problems, fairness rarely leads to a single viewpoint (see, for instance, Young, Equity in theory and practice. Princeton University Press, Princeton, 1994). In this context, this paper provides interesting bases that support the simple and commonly observed behavior of reaching intermediate agreements when two prominent distribution proposals highlight a discrepancy in sharing resources. Specifically, we formalize such a conflicting situation by associating it with a ‘natural’ cooperative game, called bifocal distribution game, to show that both the Nucleolus (Schmeidler, SIAM J Appl Math 17:1163–1170, 1969) and the Shapley value (Shapley, Additive and non-additive set functions. Princeton University, Princeton, 1953a) agree on recommending the average of the two focal proposals. Furthermore, we analyze the interpretation of the previous result by means of axiomatic arguments.

Regularity aspects of non-additive set multifunctions

In this paper, continuity properties of set multifunctions, such as regularity and continuity from above and below (as well as others), are introduced by way of a Wijsman topology. Some of the relationships between them are established. Different examples, counterexamples and applications are provided.

Atomicity via regularity for non-additive set multifunctions

In this paper, an approach of atomicity problems is presented by means of regularity. Characterizations and physical interpretations of atoms and non-atomicity for set multifunctions taking values in the family of all nonempty subsets of a topological space are given.

Solving nonlinear programming problems based on the Choquet integral by a genetic algorithm

Nonadditive set functions represent contribution rate of individual feature attributes and combinations of feature attributes toward the target. Their nonadditivity describes the interaction among contributions. The generalized weighted Choquet integral with respect to a nonadditive set function serves as an aggregation tool, which may be used in the nonlinear objective function of optimization problems, to project n-dimensional feature space onto a real axis with a corresponding projection value. In this paper, we propose a special genetic algorithm model to identify the values of feature attributes that maximize the value of the objective function involving a generalized weighted Choquet integral. Thus, we may find the values of the feature attributes giving the greatest contribution toward the target. This is a generalization of linear programming to nonlinear cases.

Multi-Attribution Classification Based on Lower Integral with Genetic Algorithm

Fuzzy integral has been widely used in multi-attribution classification when the interactions exist between the attributions. Because the fuzzy measure defined on the attributions represents the weights of all the attributions and the interactions between them. The lower integral is a type of fuzzy integral with respect to fuzzy measures, which represents the minimum potential of efficiency for a group of attributions with interaction. The value of lower integrals can be evaluated through solving a linear programming problem. Considering the lower integral as a classifier, this paper investigates its implementation and performance. The difficult step in the implementation is how to learn the non-additive set function used in lower integrals. And Genetic algorithm is used to solve the problem. Finally, numerical simulations on some benchmark data sets are given.

Sensitivity analysis of biological Boolean networks using information fusion based on nonadditive set functions.

BackgroundAn algebraic method for information fusion based on nonadditive set functions is used to assess the joint contribution of Boolean network attributes to the sensitivity of the network to individual node mutations. The node attributes or characteristics under consideration are: in-degree, out-degree, minimum and average path lengths, bias, average sensitivity of Boolean functions, and canalizing degrees. The impact of node mutations is assessed using as target measure the average Hamming distance between a non-mutated/wild-type network and a mutated network.ResultsWe find that for a biochemical signal transduction network consisting of several main signaling pathways whose nodes represent signaling molecules (mainly proteins), the algebraic method provides a robust classification of attribute contributions. This method indicates that for the biochemical network, the most significant impact is generated mainly by the combined effects of two attributes: out-degree, and average sensitivity of nodes.ConclusionsThe results support the idea that both topological and dynamical properties of the nodes need to be under consideration. The algebraic method is robust against the choice of initial conditions and partition of data sets in training and testing sets for estimation of the nonadditive set functions of the information fusion procedure.

The Gould integral in Banach lattices

In this paper, we study a Gould type integral in a new frame of Banach lattices. We consider the Gould integral of real functions relative to a non-additive set function taking values in a Banach lattice. Some continuity properties of this integral and relationships between integrability and total measurability are presented.

Products of non-additive measures: a Fubini-like theorem

For non-additive set functions, the independent product, in general, is not unique and the Fubini theorem is restricted to slice-comonotonic functions. In this paper, we use the representation theorem of Gilboa and Schmeidler (Math Oper Res 20:197–212, 1995) to extend the Mobius product for non-additive set functions to non-finite spaces. We extend the uniqueness result of Ghirardato (J Econ Theory 73:261–291, 1997) for products of two belief functions and weaken the requirements on the marginals necessary to obtain the Fubini property in the product. More importantly, we show that for the Mobius product one side of the Fubini theorem holds for all integrable functions if one of the marginals either is a probability or a convex combination of a chain of unanimity games, i.e., we relax the requirement of slice-comonotonicity and enrich the set of possible applications.

On vector-valued Dobrakov submeasures

Ivan Dobrakov has initiated a theory of non-additive set functions defined on a ring of sets intended to be a non-additive generalization of the theory of finite non-negative countably additive measures. These set functions are now known as the Dobrakov submeasures. In this paper, we extend Dobrakov’s considerations to vector-valued submeasures defined on a ring of sets. The extension of such submeasures in the sense of Drewnowski is also given.

Inclusion and Chain Variations of Non-Additive Set Function on T-Tribe

Inclusion and Chain Variations of Non-Additive Set Function on T-Tribe

Dieudonné's theorem for non-additive set functions

A Cafiero type criterion and an extension of the classical Dieudonné's theorem for a class of non-additive set functions valued into a complete Hausdorff uniform space are established.

Credibility measure of fuzzy sets and applications

In order to measure a fuzzy event, credibility measure is proposed as a non-additive set function. In this paper, this concept is extended to fuzzy sets. First, a mean measure is defined by Lebesgue integral and some properties are investigated such as the monotonicity theorem, self-duality theorem and subadditivity theorem. Furthermore, by using Sugeno integral, an equilibrium measure is proposed and studied. Finally, these concepts are applied to optimisation problems, and the mean measure maximisation model and equilibrium measure maximisation model are proposed.