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

CentER Discussion Paper Series No. 2012-091

35 Pages Posted: 28 Nov 2012 Last revised: 9 Dec 2018

See all articles by Tim J. Boonen

Tim J. Boonen

University of Amsterdam

Anja De Waegenaere

Tilburg University - Center for Economic Research (CentER)

Henk Norde

Tilburg University - Center For Economic Research; Tilburg University - Department of Econometrics & Operations Research

Date Written: December 9, 2018

Abstract

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.

Keywords: Aumann-Shapley value, non-differentiability, piecewise linear fuzzy games, Mertens value, capital allocation

JEL Classification: C71, G32

Suggested Citation

Boonen, Tim J. and De Waegenaere, Anja M.B. and Norde, Henk W., The Family of Weighted Aumann-shapley Values With an Application in Risk Capital Allocations (December 9, 2018). CentER Discussion Paper Series No. 2012-091. Available at SSRN: https://ssrn.com/abstract=2181938 or http://dx.doi.org/10.2139/ssrn.2181938

Tim J. Boonen (Contact Author)

University of Amsterdam ( email )

Roetersstraat 11
Amsterdam, 1018 WB
Netherlands

HOME PAGE: http://www.uva.nl/profiel/b/o/t.j.boonen/t.j.boonen.html

Anja M.B. De Waegenaere

Tilburg University - Center for Economic Research (CentER) ( email )

P.O. Box 90153
Tilburg, 5000 LE
Netherlands

Henk W. Norde

Tilburg University - Center For Economic Research ( email )

P.O. Box 90153
Tilburg, 5000 LE
Netherlands
+31 13 466 3245 (Phone)

Tilburg University - Department of Econometrics & Operations Research ( email )

Tilburg, 5000 LE
Netherlands

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