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.

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.

There is a widespread interest among institutions and economic agents for a reduction of the environmental impact of the production system. An important role seems to be played by the ability of public institutions to push the transition toward a green economy also through the application of fiscal policies that envisage a system of rewards and penalties, respectively, for those companies which adopt green strategies and those which do not. It is clear that readjusting older production systems to new pollution regulations can lead in the short term to profitability reductions for the companies implementing them, even though it is possible to assume increases in profitability over medium-long time horizons. One possible approach to this issue is the classical econometric one, which analyzes the effect of different parameters of multivariate models, that influence the level of pollution due to production systems with different propensity for environmental protection. Optimal control models have been also considered with control variables relating to the technologies of production systems and public incentive policies for the green economy: see for example (Tan et al. in J Syst Sci Inf 9(1):61–73, 2021). In recent years, many scholars have studied the relationship between environmental regulation and enterprise technological innovation using evolutionary games, involving mainly economic incentives and fiscal strategies (see see Suyong et al. in Appl Math Comput 355(15):343–355, 2019; Zhang and Li in Appl Math Model 63:577–590, 2018). In our article, we propose a dynamical model where the public administration uses pollution penalties as a control variable in order to push a production sector toward better performances concerning two targets, pollution level and profitability. To this end, we consider the effects of competitiveness among firms and technology innovation.

This paper analyzes an intertemporal optimization problem in which agents derive utility from three goods: leisure, a public environmental good and the consumption of a produced good. The global analysis of the dynamic system generated by the optimization problem shows that global indeterminacy may arise: given the initial values of the state variables, the economy may converge to different steady states, by choosing different initial values of the control variable.