Modeling interaction phenomena using fuzzy measures: On the notions of interaction and independence.

*(English)*Zbl 1017.28011Fuzzy measures characterize the strength of coalitions of elements in several domains such as game theory or multicriteria decision making. The information contained in a fuzzy measure can be exploited to quantify several properties and relationships of elements and of sets. Recall, e.g., the importance index of Shapley (1953), the interaction index of Murofushi and Soneda (1993) or the marginal interaction index of Grabisch et al. (2000). In this paper, several new quantitative characteristics are introduced and studied. First of all, the marginal interaction index is generalized from couples of elements to couples of disjoint subsets (\(p\)-tuples of subsets). Several properties of this generalization are studied, especially for some distinguished classes of fuzzy measures (sub- or super-additive, sub- or super-modular, \(k\)-monotone, \(k\)-additive). Next, interaction indices among pairwise disjoint subsets are defined and investigated. Finally, measures of marginal amount of interaction, marginal mutual independence, measures of marginal complementarity, etc., are discussed. All introduced quantities bring new lights into description of information concentrated in fuzzy measures. Their applications, especially in the game theory and multicriteria decision making can be expected in the near future.

Reviewer: Radko Mesiar (Bratislava)

##### MSC:

28E10 | Fuzzy measure theory |

##### Keywords:

non-additive measure; discrete fuzzy measures; marginal interaction; marginal mutual independence; measures of marginal amount of interaction; marginal complementarity
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\textit{I. Kojadinovic}, Fuzzy Sets Syst. 135, No. 3, 317--340 (2003; Zbl 1017.28011)

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