Values and cores of fuzzy games with infinitely many players

The problem of the existence ofvalues (FA-valued, linear, positive, symmetric and efficient operators) on symmetric spaces of “fuzzy games” (that is, ideal set functions of bounded variation) arises naturally from 8], 18], 23] and 2], 3], 4] where it is implicitely approached for technical purposes. In our present work, this problem is approached in itself for the main reason that it is essentially related with the problem of the existence of significant countable additive measures lying in the cores of the “market games”. In fact, it is shown here that there exists a continuous value on the closed subspacebv′ICA ofIBV spanned by thebv′ functions of “fuzzy probability measures” (9]), this values is “diagonal” onpICA, the closed subspace ofbv′ICA spanned by the natural powers of the fuzzy measures and this is used to prove the main result stating that the cooperative markets contained inpICA have unique fuzzy measures in their cores which are exactly the corresponding diagonal values. This result is of interest because it is providing a tool of determiningCA measures lying in the cores of large classes of games which are not necessarily “non-atomic” and, specially, because it is opening a way toward a new approach of the “Value Equivalence Principle” for differentiable markets with a continuum of traders which are not “perfectly competitive”.