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A mesh-based notion of differential for TU games. (English) Zbl 1246.91010
Nonadditive set functions have been used to model transferable utility (TU) games. Given a measurable space $$(\Omega,\Sigma)$$, a set function $$\nu:\Sigma\rightarrow \mathbb{R}$$ such that $$\nu(\emptyset)=0$$ is called a TU game. The set $$\Omega$$ and the $$\sigma$$-algebra $$\Sigma$$ are interpreted respectively as a set of players and a set of possible coalitions $$S$$, while $$\nu(S)$$ represents the payoff among the members of $$S$$. The necessity of infinite dimensional analysis arises as games with a continuum of players appear naturally whenever there is a “large” number of negligible individuals, like for example consumers in a perfect competition economy. Since in the finite case, the Shapley value [cf. L. S. Shapley, Contrib. Theory of Games, II, Ann. Math. Stud. No. 28, 307–317 (1953; Zbl 0050.14404)] of a player represents an average of the so-called marginal contribution to the worth of every coalition he can join, in an infinite dimensional context the importance of developing suitable derivative notions seems to be clear in order to generalize this solution concept and treat various classes of games.
Differentiation of nonadditive set functions in cooperative game theory dates back to [R. J. Aumann and L. S. Shapley, Values of non-atomic games. Princeton, N. J.: Princeton University Press (1974; Zbl 0311.90084)]. Since then, other notions have been developed and applied mainly to the study of the core of TU games. All these concepts are very general and based on the use of refinements of finite partitions and on the limit made with respect to the partial order induced by them, to capture the idea of increasing smallness of the increment. The common and fundamental idea underlying these notions is that of approximating a generally nonadditive set function (the game) with an additive and hence more treatable one (its derivative). Hence, a measure coincides with its derivative, while nonadditive games in the limit behave almost as additive set functions.
In the current paper, the authors introduce the notion of Burkill-Cesari (BC) differentiability, where refinements substitute for the classical notion of mesh in order to measure the size of a partition. The new formulation is inspired by the Burkill-Cesari integrability introduced in the sixties in [L. Cesari, Trans. Am. Math. Soc. 102, 94–113 (1962; Zbl 0115.26902)]. This notion allows to manage chaotic families of decompositions of sets. With respect to the choice of a suitable mesh, a natural one is induced if one considers monotonic strongly non-atomic games. Moreover, they compare the new notion with the Epstein-Marinacci refinement differential (cf. [L. G. Epstein and M. Marinacci, J. Econ. Theory 100, No. 2, 235–273 (2001; Zbl 1005.91014)]) as well as with the Epstein $$\mu$$-differentiability (cf. [L. G. Epstein, Rev. Econ. Stud. 66, No. 3, 579–608 (1999; Zbl 0953.91002)]). Various kinds of absolute continuity for games are known in the literature (see for example [L. Montrucchio and P. Semeraro, Math. Oper. Res. 33, No. 1, 97–118 (2008; Zbl 1166.91307)]), and they need them throughout the paper in order to prove several results, such as for example the calculus rules. They also provide a study and some representation results on the core of Burkill integrable games. The paper includes a large list of suitable and ad hoc examples and counterexamples.

##### MSC:
 91A12 Cooperative games
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##### References:
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