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

Motivated by the problem of variance allocation for the sum of dependent random variables, Colini-Baldeschi, Scarsini and Vaccari [2018, Methodol Comput Appl Probab, 20:19–933] have recently introduced the Shapley values for variance and standard deviation games. These Shapley values constitute a criterion satisfying nice properties useful to allocate 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 have also formulated a conjecture about the relation of the Shapley values of the two games, which they have 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.