On the Aumann-Shapley value
This paper generalizes Theorem A in [\textit{R. J. Aumann} and \textit{L. S. Shapley}, Values of non-atomic games. Princeton, N. J.: Princeton University Press (1974; Zbl 0311.90084), p. 20], where the underlying space is an algebra of subsets isomorphic to \(([0,1], \mathcal{B})\), \(\mathcal{B}\) the Borel sets. In studying games with fuzzy coalitions, \textit{D. Butnariu} and \textit{E. P. Klement} [Triangular norm-based measures and games with fuzzy coalitions. Dordrecht: Kluwer Academic Publishers (1993; Zbl 0804.90145)] extended some of the results in Aumann and Shapley to a triangular norm-based \(T_\infty\)-tribe of fuzzy sets. Continuing their previous work such as in [\textit{A. Avallone, A. Basile} and \textit{P. Vitolo}, Positivity 10, No. 4, 701--719 (2006; Zbl 1113.28003)], the authors use a \(\sigma-\)complete difference lattice \(L\) (or, a lattice-ordered effect algebra). The space \(BL\) is the Banach space of all real-valued set functions vanishing at 0 (i.e., games) of bounded variation on \(L\) with the variation norm. Taking a measure \(\mu\) on \(L\) to an abelian group, they introduce the properties that \(\mu\) is \textit{non-atomic} (using the notion of an orthogonal sequence) and \textit{modular} (i.e., \(\mu (a \vee b) + \mu(a \wedge b) =\mu (a) + \mu (b)\) for all \(a, b\in L\)). The subspace \(\mathfrak{b}\mathfrak{v}'\)NA of \(BV\) is spanned by functions of the form \(f\circ\mu\) where \(\mu : L \to [0,1]\) is a non-atomic \(\sigma\)-additive measure and \(f : [0,1] \to \mathbb{R}\) is of bounded variation and continuous at 0 and at 1. In this setting, the paper proves the existence of a continuous Aumann-Shapley value \(\varphi\) (a mapping from set functions to additive set functions with certain conditions) on \(\mathfrak{b}\mathfrak{v}'\)NA.
scientific article; zbMATH DE number 5534415
scientific article