With a focus on the risk contribution in a portofolio of dependent risks, Colini-Baldeschi et al. (2018) introduced Shapley values for variance and standard deviation games. In this note we extend their results, introducing tail variance as well as tail standard deviation games. We derive closed-form expressions for the Shapley values for the tail variance game and we analyze the vector majorization problem for the two games. In particular, we construct two examples showing that the risk contribution rankings for the two games may be inverted depending on the conditioning threshold and the tail fatness. Motivated by these examples, we formulate a conjecture for general portfolios. Lastly, we discuss risk management implications, including the characterization of tail covariance premiums and reinsurance pricing for peer-to-peer insurance policies.

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.

Two games are inseparable by semivalues if both gamesobtain the same allocation whatever semivalue is considered. The problem ofseparability by semivalues reduces to separability from the nullgame. For four or more players, the vector subspace of gamesinseparable from the null game by semivalues contains gamesdifferent to zero-game. Now, for five or more players, theconsideration of a priori coalition blocks in the player set allowsus to reduce in a significant way the dimension of the vectorsubspace of games inseparable from the null game. For thesesubspaces we provide basis formed by games of a particular type.

The computation of leastcore and prenucleolus is an efficient way ofallocating a common resource among n players. It has, however,the drawback being a linear programming problem with2n - 2 constraints. In this paper we showhow, in the case of convex production games, generate constraints by solving small sizelinear programming problems,with both continuous and integer variables. The approach is extended to games with symmetries (identical players), and to games with partially continuous coalitions. We also study thecomputation of prenucleolus, and display encouraging numerical results.

We present here a pricing model which is an extension of the cooperative game concept and which includes a notion of elastic demand. We present some existence results as well as an algorithm, and we conclude by discussing a specific problem related to network pricing.