We define Bel coalitional games, that generalize classical coalitional games by introducing uncertainty in the framework. Unlike Bayesian coalitional games, uncertainty is modelled through the Dempster-Shafer theory and every agent can have different knowledge. We propose the notion of contract in our framework, that specifies how agents divide the values of the coalitions and we use the Choquet integral to model the agents' preferences between contracts. In a second step, we define the ex-ante core and the ex- t -interim core, where, in the latter, we need the Dempster conditional rule to update the mass functions of agents. In particular, in the last step of the ex- t -interim case and when the set of states reduces to a singleton, i.e., when there is no uncertainty, we recover the classical definition of the core. We give some conditions for the nonemptiness of the ex-ante and the ex- t -interim core of Bel coalitional games, considering different types of agents' knowledge and different kind of games. Finally, we present Bel market games as an economical application.

In this paper, we study a frequentist approach to belief updating in the framework of Dempster-Shafer Theory (DST). We propose several mechanisms that allow the gathering of possibly ambiguous pieces of evidence over time to obtain a belief mass assignment. We then use our approach to study the impact of ambiguous evidence on the belief distribution of agents in social networks. We illustrate our approach by taking three representative situations. In the first one, we suppose that there is an unknown state of nature, and agents form belief in the set of possible states. Nature constantly sends a signal which reflects the true state with some probability but which can also be ambiguous. In the second situation, there is no ground truth, and agents are against or in favor of some ethical or societal issues. In the third situation, there is no ground state either, but agents have opinions on left, center, and right political parties. We show that our approach can model various phenomena often observed in social networks, such as polarization or bounded confidence effects.

Several practical applications can be formulated as a problem of estimating a source of interest from a set of mixed data collected by different sensors. Although a lot of effort has been done to address the optimization task in signal extraction, there is a lack in the literature on how to evaluate the contribution of each sensor in the extraction process. In this letter, we propose a model-agnostic approach that can be used to interpret both the contribution of each sensor in the estimated source and the interaction effects between them. Our proposal is based on a solution concept from game theory, called Shapley value. Numerical experiments on synthetic and real data attest the use of our proposal in blind source extraction problems.

We extend the notion of belief function to the case where the underlying structure is no more the Boolean lattice of subsets of some universal set, but any lattice, which we will endow with a minimal set of properties according to our needs. We show that all classical constructions and definitions (e.g., mass allocation, commonality function, plausibility functions, necessity measures with nested focal elements, possibility distributions, Dempster rule of combination, decomposition with regard to simple support functions, etc.) remain valid in this general setting. Moreover, our proof of decomposition of belief functions into simple support functions is much simpler and general than the original one by Shafer. © 2008 Wiley Periodicals, Inc.

Multicriteria decision analysis studies decision problems in which the alternatives are evaluated on several dimensions or viewpoints. In the problems we consider in this article, the scales used for assessing the alternatives with respect to a viewpoint are bipolar and univariate or unipolar and bivariate. In the former case, the scale is divided in two zones by a neutral point; a positive feeling is associated to the zone above the neutral point and a negative feeling to the zone below this point. On unipolar bivariate scales, an alternative can receive both a positive evaluation and a negative evaluation, reflecting contradictory feelings or stimuli. The article discusses procedures and models that have been proposed to aggregate multicriteria evaluations when the scale of each criterion is of one of these two types. We present both a constructive view and a descriptive view on this question; the descriptive approach is concerned with characterizations of models of preference, whereas the constructive approach aims at building preferences by questioning the decision maker. We show that these views are complementary. © 2008 Wiley Periodicals, Inc.

This paper investigates the statistical properties of the Choquet and Sugeno integrals, used as multiattribute models. The investigation is done on an empirical basis, and focuses on two topics: the distribution of the output of these integrals when the input is corrupted with noise, and the robustness of these models when they are identified using some set of learning data through some learning procedure.

We provide a survey of recent developments about capacities (or fuzzy measures) and cooperative games in characteristic form, when they are defined on more general structures than the usual power set of the universal set, namely lattices. In a first part, we give various possible interpretations and applications of these general concepts, and then we elaborate about the possible definitions of usual tools in these theories, such as the Choquet integral, the Möbius transform, and the Shapley value.

We consider the interval ]-1, 1[ and intend to endow it with an algebraic structure like a ring. The motivation lies in decision making, where scales that are symmetric w.r.t. 0 are needed in order to represent a kind of symmetry in the behaviour of the decision maker. A former proposal due to Grabisch was based on maximum and minimum. In this paper, we propose to build our structure on t-conorms and t-norms, and we relate this construction to uninorms. We show that the only way to build a group is to use strict t-norms, and that there is no way to build a ring. Lastly, we show that the main result of this paper is connected to the theory of ordered Abelian groups.

In this paper we propose a generalization of the concept of symmetric fuzzy measure based in a decomposition of the universal set in what we have called subsets of indifference. Some properties of these are studied, as well as their Choquet integral. Finally, a degree of interaction between the subsets of indifference is defined.

The problem of combining pieces of information issued from several sources can be encountered in various fields of application. This paper aims at presenting the different aspects of information fusion in different domains, such as databases, regulations, preferences, sensor fusion, etc., at a quite general level. We first present different types of information encountered in fusion problems, and different aims of the fusion process. Then we focus on representation issues which are relevant when discussing fusion problems. An important issue is then addressed, the handling of conflicting information. We briefly review different domains where fusion is involved, and describe how the fusion problems are stated in each domain. Since the term fusion can have different, more or less broad, meanings, we specify later some terminology with respect to related problems, that might be included in a broad meaning of fusion. Finally we briefly discuss the difficult aspects of validation and evaluation. © 2001 John Wiley & Sons, Inc.

The problem of assessing the value of a candidate is viewed here as a multiple combination problem. On the one hand, a candidate can be evaluated according to different criteria, and on the other hand, several experts are supposed to assess the value of candidates according to each criterion. Criteria are not equally important, experts are not equally competent or reliable. Moreover, levels of satisfaction of criteria, or levels of confidence are only assumed to take their values in linearly ordered scales, whose nature is rather qualitative. The problem is discussed within two frameworks, the transferable belief model (TBM) and the qualitative possibility theory (QPT). They respectively offer a quantitative and a qualitative setting for handling the problem, thus providing a way to emphasize what are the underlying assumptions in each approach. © 2001 John Wiley & Sons, Inc.

This paper addresses the correlation problem which is central in sensor fusion, from the viewpoint of possibility theory. This problem aims at separating pieces of information pertaining to different objects and to gather those which are likely to pertain to the same object. We present two different views of the problem, one based on similarity relations, while the other discusses the problem in a logical framework. © 2001 John Wiley & Sons, Inc.

The intepretation of aggregation functions in multicriteria decision making is often based on indices such as importance indices that measure the importance of one criterion. For instance, for the Choquet integral, the importance index is the so-called Shapley value. The use of an index must always be limited to the specific context it has been designed for. Here, we are interested in determining on which criteria acts should be improved if we want their global evaluation to increase as much as possible. The Shapley value is not suited for describing this. So, we introduce a new index of importance which represents the mean worth for acts to reach higher scores in a set of criteria. This index is defined for general aggregation functions with the help of several axioms. This importance index is then applied to the Choquet integral. In particular, we computer the worth to reach higher levels in one attribute, and in a couple of attributes. Interestingly, this leads to quantities that are closely related to the Shapley and interaction indices.

This short overview paper points out the striking similarity between decision under uncertainty and multicriteria decision making problems, two areas which have been developed in an almost completely independent way until now. This pertains both to additive and non-additive (including qualitative) approaches existing for the two decision paradigms. This leads to an emphasis on the remarkable formal equivalence between postulates underlying these approaches (like between the “sure-thing principle” and mutual preferential independence of criteria). This analogy is exploited by surveying classical results as well as very recent advances. This unified view should be fruitful for a better understanding of the postulates underlying the approaches, for cross-fertilization, and for adapting artificial intelligence uncertainty representation frameworks to preference modelling. © 2000 John Wiley & Sons, Inc.

The Choquet integral (1953) is a fuzzy measure used as a powerful aggregation operator over a finite set of elements. We present a graphical interpretation of the Choquet integral, viewed as an aggregation operator in the case of two elements. The interpretation relies on the interaction representation introduced by the author.

In this paper, we address the problem of identification of fuzzy measures through different representations, namely the Möbius, the Shapley and the Banzhaf interaction representations. In the first part of the paper, we recall the main results concerning these representations, and give a simple algorithm to compute them. Then we determine the bounds of the Möbius and the interaction representations for fuzzy measures. Lastly, the identification of fuzzy measures by minimizing a quadratic error criterion is addressed. We give expressions of the quadratic program for all the considered representations, and study the uniqueness of the solution.

A real-time track classifier based on fuzzy logic is presented here. The application deals with false alarm rejection in an airborne Infra-Red Search and Track (IRST) system. The approach is based on the use of a set of rules representing heterogeneous knowledge on behavior and appearance of physical objects. Uncertainty rules are considered to model knowledge on classes, while gradual rules are used to compute some auxiliary variables. Examples of rules are given, as well as results on simulated data. © 1997 John Wiley & Sons, Inc.

This special issue gathers eight articles which illustrate various aspects of a new trend of application-oriented researches called “Fuzzy Information Engineering.” Seven of these articles are revised and expanded versions of articles presented in a series of two invited sessions, organized by the guest editors of this special issue at the Fourth European Congress on Intelligent Techniques and Soft Computing (EUFIT'96) in Aachen, Germany on September 4, 1996. We first briefly restate what Information Engineering covers and what the contribution of fuzzy set-based methods is to this research trend, before providing a short presentation of the articles in the issue. © 1997 John Wiley & Sons, Inc.

This paper introduces three different representations of fuzzy measures, through the Möbius transformation, and the expression of importance and interaction. This leads naturally to the concept of k-order additive measures. It is shown how these concepts can be used in decision making, especially multicriteria evaluation.

An equivalence relation between connectives (or operators) in the framework of multiple criteria decision-making is introduced. Two operators are said to be equivalent if they lead to the same ranking of alternatives. Some general results to find equivalence classes of operators are given through the concept of level surfaces. In a second part, these results are applied to the case of discrete fuzzy integrals, which are considered here as n-place operators. First, some general results on the situation of fuzzy integrals among fuzzy operators are given and then the equivalence classes of fuzzy integrals.>