Katsushige Fujimoto

Axiomatic characterizations of probabilistic and cardinal-probabilistic interaction indices

Abstract In the framework of cooperative game theory, the concept of interaction index, which can be regarded as an extension of that of value, has been recently proposed to measure the interaction phenomena among players. Axiomatizations of two classes of interaction indices, namely probabilistic interaction indices and cardinal-probabilistic interaction indices, generalizing probabilistic values and semivalues, respectively, are first proposed. The axioms we utilize are based on natural generalizations of axioms involved in the axiomatizations of values. In the second half of the paper, existing instances of cardinal-probabilistic interaction indices encountered thus far in the literature are also axiomatized.

A HIERARCHICAL DECOMPOSITION OF CHOQUET INTEGRAL MODEL

In this paper we give a nesessary and sufficient condition for a Choquet integral model to be decomposable into an equivalent hierarchical Choquet integral model constructed by hierarchical combinations of some ordinary Choquet integral models. The condition is obtained by Inclution-Exclusion Covering (IEC). Moreover we show some properties on the set of IECs.

Separated hierarchical decomposition of the Choquet integral

The paper gives a necessary and sufficient condition for a Choquet integral to be decomposable into an equivalent separated hierarchical Choquet-integral system, which is a hierarchical combination of ordinary Choquet integrals with mutually disjoint domains.

Some characterizations of the systems represented by Choquet and multi-linear functionals through the use of Mo¨bius inversion

The systems represented by the Choquet or the multi-linear fuzzy integral with respect to fuzzy measure is equivalently decomposable into hierarchically sub-systems through the use of Inclusion-Exclusion Covering (IEC) (Theorem 4.1,4.2). Hence, IEC is one of very useful concepts/indexes for structural analysis of the fuzzy integral systems (short for: the systems represented by the Choquet or the multi-linear fuzzy integral) However, it is quite difficult to identify all IECs. This paper shows a method for identifying it easily, through the use of Mobius inversion (Theorem 5.1).

Axiomatic characterizations of generalized values

In the framework of cooperative game theory, the concept of generalized value, which is an extension of that of value, has been recently proposed to measure the overall influence of coalitions in games. Axiomatizations of two classes of generalized values, namely probabilistic generalized values and generalized semivalues, which extend probabilistic values and semivalues, respectively, are first proposed. The axioms we utilize are based on natural extensions of axioms involved in the axiomatizations of values. In the second half of the paper, special instances of generalized semivalues are also axiomatized.

Hierarchical decomposition theorems for Choquet integral models

In this paper, we give two necessary and sufficient conditions for a Choquet integral model to be decomposable into two equivalent hierarchical Choquet integral models constructed by hierarchical combinations of some ordinary Choquet integral models. These conditions are obtained by inclusion-exclusion covering (IEC). Moreover, we show some properties on the set of IECs.<<ETX>>

Canonical hierarchical decomposition of Choquet integral over finite set with respect to null additive fuzzy measure

In this paper, we provide the necessary and sufficient condition for a Choquet integral model to be decomposable into a canonical hierarchical Choquet integral model constructed by hierarchical combinations of some ordinary Choquet integral models. This condition is characterized by the pre-Znclusion-Exclusion Covering (pre-IEC). Moreover, we show that the pre-IEC is the subdivision of an IEC and that the additive hierarchical structure is the most fundamental one on considering a hierarchical decomposition of the Choquet integral model.

k-Additivity and C-decomposability of bi-capacities and its integral

k-Additivity is a convenient way to have less complex (bi-)capacities. This paper gives a new characterization of k-additivity, introduced by Grabisch and Labreuche, of bi-capacities and contrasts between the existing characterization and the new one, that differs from the one of Saminger and Mesiar. Besides, in the same way for capacities, a concept of C-decomposability, distinct from the proposal of Saminger and Mesiar, but closely linked to k-additivity, is introduced for bi-capacities. Moreover, the concept of C-decomposability applies to the Choquet integral with respect to bi-capacities.

Canonical separated hierarchical decomposition of the Choquet integral over a finite set

The paper shows the existence of a canonical separated hierarchical decomposition of the Choquet integral over a finite set. The decomposed system is a hierarchical combination of Choquet integrals...

Cooperative Game as Non-Additive Measure

This chapter surveys cooperative game theory as an important application based on non-additive measures. In ordinary cooperative game theory, it is implicitly assumed that all coalitions of N can be formed; however, this is in general not the case. Let us elaborate on this, and distinguish several cases: 1) Some coalitions may not be meaningful. 2) Coalitions may not be “black and white”. In order to deal with such situations, various generalizations/extensions of the theory have been proposed, e.g., bi-cooperative games, games on networks, games on combinatorial structures. We give a survey on values and interaction indices for these extended cooperative game theory.