Refinement Derivatives and Values of Games

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 Rosenmuller [Rosenmuller, J. 1977. Extreme Games and Their Solutions. Springer-Verlag, Berlin], Epstein [Epstein, L. 1999. A definition of uncertainty aversion. Rev. Econom. Stud. 66 579–608], and Epstein and Marinacci [Epstein, L., M. Marinacci. 2001. The core of large differentiable TU games. J. Econom. Theory100 235–273]. 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 Hart and Mas-Colell [Hart, S., A. Mas-Colell. 1989. Potential, value and consistency. Econometrica57 589–614], 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∞ and POT₂. The latter space is closely related to Myerson's balanced contributions axiom.