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Refinement derivatives and values of games. (English) Zbl 1166.91307
Summary: A definition of setwise differentiability for set functions is given through refining the partitions of sets. Such a construction is closely related to the one proposed by J. Rosenmüller [Extreme games and their solutions. Lecture Notes in Economics and Mathematical Systems. 145. Berlin-Heidelberg-New York: Springer-Verlag (1977; Zbl 0353.90100)], L. Epstein [Rev. Econ. Stud. 66, No. 3, 579–608 (1999; Zbl 0953.91002)], and L. Epstein and M. Marinacci [J. Econ. Theory 100, No. 2, 235–273 (2001; Zbl 1005.91014)]. We present several classes of transferable utility (TU) games that are differentiable and study differentiation rules. The last part of this paper applies refinement derivatives to the computation of value of games. Following S. A. Hart and A. Mas-Colell [Econometrica 57, No. 3, 589–614 (1989; Zbl 0675.90103)], we define an operator through the refinement derivative of the potential of the game. We show that this operator is a value, when restricted to the spaces $$pM\infty$$ and $$POT_{2}$$. The latter space is closely related to Myerson’s balanced contributions axiom.

##### MSC:
 91A12 Cooperative games
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