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

Marcello Galeotti, Giovanni Rabitti

Research output: Contribution to journalArticlepeer-review

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.
Original languageEnglish
Pages (from-to)609-620
Number of pages12
JournalJournal of Applied Probability
Volume58
Issue number3
DOIs
Publication statusPublished - 16 Sep 2021

Keywords

  • Keywords: Covariance matrix
  • cooperative games
  • sum of random variables
  • vector majorization

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Statistics, Probability and Uncertainty

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