p-Symmetric fuzzy measures

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 measures are studied, as well as their Choquet integral. Finally, a degree of interaction between the subsets of indifference is defined.

💡 Research Summary

The paper introduces a novel class of fuzzy measures called p‑symmetric fuzzy measures, which generalize the traditional notion of symmetric fuzzy measures by allowing the universal set to be partitioned into p “subsets of indifference”. Within each indifference subset, the measure treats all elements as indistinguishable: for any two elements x_i and x_j belonging to the same subset, and for any auxiliary set C that does not contain them, the equality µ(C ∪ {x_i}) = µ(C ∪ {x_j}) holds. This concept extends the idea of symmetry (the case p = 1) and bridges the gap between fully symmetric measures and the much richer family of k‑additive measures.

The authors first recall basic definitions: a fuzzy measure µ on a finite set X satisfies boundary conditions (µ(∅)=0, µ(X)=1) and monotonicity; the Möbius transform m(A) and Shapley interaction index I(A) provide alternative representations. They then formalize the notion of indifference for single elements and for subsets, proving that indifference is hereditary (any subset of an indifference set is also an indifference set) and that null sets (sets that never change the measure when added) are particular indifference sets.

A p‑symmetric measure is defined as a fuzzy measure whose coarsest partition of X into indifference subsets consists of exactly p non‑empty blocks {A₁,…,A_p}. The coarseness requirement guarantees uniqueness of the partition: a p‑symmetric measure cannot simultaneously be q‑symmetric for q > p. The paper shows that the entire measure is completely determined by the cardinalities of the intersections of any subset C with each block A_i. Consequently, the number of independent parameters needed to specify a p‑symmetric measure is
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