The Family of Weighted Aumann-shapley Values With an Application in Risk Capital Allocations

This paper introduces the family of Weighted Aumann-Shapley values for piecewise linear fuzzy games. The regular Aumann-Shapley value is not well-defined in case some differentiability condition is not satisfied. As an alternative, we introduce a family of allocation rules inspired by the Shapley value in a fuzzy setting. We take a grid on the fuzzy participation set, define paths on this grid and construct an allocation rule based on such a path. Then, we define a rule as the limit of the average of these allocation rules, when the grid size converges to zero. This procedure is a special case of the limiting approach of Aumann and Shapley (1974). Their approach is not working on the class of piecewise linear fuzzy games. We show that if the Aumann-Shapley value is well-defined, all members of the family of Weighted Aumann-Shapley values coincide with it. We show moreover that the Mertens value (Mertens, 1988) does not belong to the family of Weighted Aumann-Shapley values. Finally, we apply the Weighted Aumann-Shapley values in the context of risk capital allocation problems