\(p\)-symmetric bi-capacities. (English) Zbl 1249.28021

Summary: Bi-capacities have been recently introduced as a natural generalization of capacities (or fuzzy measures) when the underlying scale is bipolar. They allow to build more flexible models in decision making, although their complexity is of order \(3^n\), instead of \(2^n\) for fuzzy measures. In order to reduce the complexity, the paper proposes the notion of \(p\)-symmetric bi-capacities, in the same spirit as for \(p\)-symmetric fuzzy measures. The main idea is to partition the set of criteria (or states of nature, individuals,\(\ldots \)) into subsets whose elements are all indifferent for the decision maker.


28E05 Nonstandard measure theory
28E10 Fuzzy measure theory
03H05 Nonstandard models in mathematics
28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
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