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On the comparison of Shapley values for variance and standard deviation games

Part of: Linear inference, regression Game theory

Published online by Cambridge University Press:  16 September 2021

Marcello Galeotti  and
Giovanni Rabitti
Marcello Galeotti*
Affiliation:
University of Florence
Giovanni Rabitti*
Affiliation:
Heriot-Watt University
*
*Postal address: Department of Statistics, Informatics and Applications, University of Florence, Via delle Pandette 9, 50127 Florence, Italy. Email address: marcello.galeotti@unifi.it
**Postal address: Department of Actuarial Mathematics and Statistics, Heriot-Watt University and Maxwell Institute for Mathematical Sciences, EH14 4AS, Edinburgh, UK. Email address: g.rabitti@hw.ac.uk
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Abstract

Motivated by the problem of variance allocation for the sum of dependent random variables, Colini-Baldeschi, Scarsini and Vaccari (2018) recently introduced Shapley values for variance and standard deviation games. These Shapley values constitute a criterion satisfying nice properties useful for allocating the variance and the standard deviation of the sum of dependent random variables. However, since Shapley values are in general computationally demanding, Colini-Baldeschi, Scarsini and Vaccari also formulated a conjecture about the relation of the Shapley values of two games, which they proved for the case of two dependent random variables. In this work we prove that their conjecture holds true in the case of an arbitrary number of independent random variables but, at the same time, we provide counterexamples to the conjecture for the case of three dependent random variables.

Keywords

Covariance matrixsum of random variablesvector majorizationcooperative games

MSC classification

Primary: 91A12: Cooperative games
Secondary: 62J10: Analysis of variance and covariance
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 3 , September 2021 , pp. 609 - 620
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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