Abstract
In the paper, we consider conditions providing the coincidence of the cores and superdifferentials of fuzzy cooperative games with side payments. It turned out that weak homogeneity is one of the simplest sufficient conditions. Moreover, by applying the so-called \(S^* \)-representation of a fuzzy game introduced by the author, we show that for any \(v\) with nonempty core \(C(v) \) there exists some game \(u \) such that \(C(v) \) coincides with the superdifferential of \(u \). By applying the subdifferential calculus, we describe a structure of the core for classical fuzzy extensions of the ordinary cooperative game (e.g., the Aubin and Owen extensions) as well as for some new continuations, like the generalized Airport game.
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Notes
Recall [3] that \(C(v^*)=\{x\in \mathbf {R}^N\big |e^*_N\cdot x=v^*(e^*_N),\;\tau ^*\cdot x\geq v^*(\tau ^*),\quad \tau ^*\in \sigma _F^*\} \).
A function \(v \) on \(\sigma _F^* \) is said to be concave with respect to the center of gravity \(e_N^* \) [3] if for any representation of \(e_N^* \) in the form of a convex combination \(e_N^*=\sum \limits _{k\in K}\lambda _k\tau _k,\tau _k\in \sigma _F^* \), \(k\in K \), one has the inequality \(v(e_N^*)\geq \sum \limits _{k\in K}\lambda _kv(\tau _k)\) .
For example, the well-known multilinear Owen extension [10] is inhomogeneous.
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This work was supported by the Russian Foundation for Basic Research, project no. 19-010-00910.
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Translated by V. Potapchouck
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Vasil’ev, V.A. Core and Superdifferential of a Fuzzy \(TU\)-Cooperative Game. Autom Remote Control 82, 926–934 (2021). https://doi.org/10.1134/S0005117921050155
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DOI: https://doi.org/10.1134/S0005117921050155