2. Shapley value

Consider the situation in which a team of players generate a value. The Shapley value originates from game theory [Reference Shapley34] as a method to attribute to every player of the team the fair part of the value they contributed. Let v(J) be the value generated by the coalition of players, $J \subseteq N =\{1 ,2 ,\ldots ,n\}$ , with the assumption $v(\varnothing ) =0$ . The total value of the game produced by the team is then v(N).

Consider the following desirable properties for an attribution method:

• Efficiency: $\sum _{i =1}^{n}\phi _{i} (v) =v (N)$ .

• Symmetry: If $v (J \cup \{i\}) = v (J \cup \{j\})$ for all $J \subseteq N \setminus \{i ,j\}$ , then $\phi_{i}(v) = \phi_{j}(v)$ .

• Dummy player: If $v (J \cup \{i\}) =v (J)$ for all $J \subseteq N$ , then $\phi _{i} (v) =0$ .

• Linearity: If two value functions v and w have Shapley values, respectively, $\phi_{i}(v)$ and $\phi_{i}(w)$ , then the game with value $\alpha v + \beta w$ has Shapley values $\alpha\phi_{i}(v) + \beta\phi_{i}(w)$ for all $\alpha ,\beta \in \mathbb{R}$ .

The efficiency property states that the sum of the Shapley values of all players must equal the total value of the game to be shared. The symmetry property requires that players making the same contributions to any coalition must be paid equal shares. If the marginal contribution $v (J \cup\{i\}) - v (J)$ is null whatever coalition the ith player joins, this player receives a null share and is called a dummy player. The linearity property states that if two games are combined, then the received share of each player must be the combination of their shares from the two games.

Shapley [Reference Shapley34] proved that the value given by

\begin{equation*} \phi_{i}(v) = \sum_{J \subseteq N \setminus \{i\}}\frac{(n- \vert J\vert - 1)! \vert J\vert!}{n!}[v (J \cup \{i\}) -v (J)] \end{equation*}

is the unique attribution method satisfying these four properties, where $\vert J\vert $ denotes the cardinality of J. This is called the Shapley value. We note that the Shapley value for the ith player is based on the marginal increase in the value $v (J \cup \{i\}) - v(J)$ when they join coalition J, and such marginal increase is then averaged over all possible coalitions J.

Equivalently, the Shapley value can be expressed in terms of permutations as [Reference Colini-Baldeschi, Scarsini and Vaccari8]

\begin{equation*} \phi_{i}(v) = \frac{1}{n!}\sum_{\psi \in \mathcal{P} (N)}(v(P^{\psi}(i)\cup i) - v(P^{\psi}(i))),\end{equation*}

where $\mathcal{P}(N)$ is the set of all permutations of N, and $P^{\psi}(i)$ is the set of all players who precede i in the order directed by the permutation $\psi $ .

One important feature of the Shapley value is that it is considered a fair attribution method. To formalize the concept of fairness, consider the following property of balanced contributions.

• Balanced contributions: Denote by $v_{ -k}$ the game obtained by restricting the set of players to $N \setminus \{ k\}$ . Then $\phi$ satisfies the axiom of balanced contributions if $\phi_{i}(v) - \phi_{i}(v_{-j}) = \phi_{j}(v) - \phi_{j}(v_{-i})$ for all $i ,j \in N$ .

The Shapley value was characterized in [Reference Hart and Mas-Colell19] using the efficiency and the balance contribution properties. This property can be interpreted in terms of fairness since, for two cooperating players, it states that the value produced by them is the same with respect to what they would produce without cooperation. It was stated in [Reference Algaba, Fragnelli and Sanchez-Soriano3] that the Shapley value is the unique efficient attribution method satisfying this equal contribution property, making it ‘a benchmark for fairness’ (p. 23). This fairness feature becomes a relevant justification for the adoption of the Shapley value as allocation method.

3. Variance and standard deviation games of [Reference Colini-Baldeschi, Scarsini and Vaccari8]

Given a set of risks $(X_{1}, X_{2}, \ldots, X_{n})$ , pose $S = \sum _{i =1}^{n}X_{i}$ and $S_{J} = \sum _{i \in J}X_{i}$ for every $J\subseteq N =\{1 ,2 ,\ldots ,n\}$ . The value function for the variance game was defined in [Reference Colini-Baldeschi, Scarsini and Vaccari8] as $\nu(J) = \mathrm{Var}[S_{J}]$ . The intuition is to regard risks as players producing a total value of the game $\nu (N) = \mathrm{Var}[S]$ , i.e. the variance of the portfolio. Thus, the Shapley value for the variance game quantifies the contribution of every risk to the porfolio variance. It was proved that the Shapley value for the variance game is

(1) \begin{equation} \phi_{i}(\nu) = \mathrm{Cov}[X_{i},S], \end{equation}

where $\mathrm{Cov}[\cdot ,\cdot ]$ denotes the covariance. The representation in (1) is very convenient, since in general it is not possible to find an explicit representation of the Shapley values and expensive numerical techniques must be adopted [Reference Plischke, Rabitti and Borgonovo31].

We note that this expression of the Shapley values for the variance game in (1) is well studied in the actuarial literature. In particular, the normalized Shapley value for the variance game,

(2) \begin{equation} \widetilde{\phi}_{i}(\nu) = \frac{\phi_{i}(\nu)}{\nu(N)} = \frac{\mathrm{Cov}[X_{i},S]}{\mathrm{Var}[S]}, \end{equation}

is known as the covariance allocation method [Reference Dhaene, Tsanakas, Valdez and Vanduffel13, Reference Overbeck28]:

A second game was introduced in [Reference Colini-Baldeschi, Scarsini and Vaccari8]. It defined the standard deviation game considering the value function $\mu(J) = \sqrt{\mathrm{Var}[S_{J}]}$ for all $J\subseteq N$ . However, in such a case it is not possible to express the resulting Shapley value $\phi_{i}(\mu)$ in closed form. Then, [Reference Colini-Baldeschi, Scarsini and Vaccari8] compared the resulting Shapley values for the variance game with those for the standard deviation game in terms of vector majorization. In particular, a conjecture was formulated on the majorization of the Shapley values. Given two vectors $\textbf{x},\textbf{y}\in \mathbb{R}^{n}$ , $\textbf{x}$ is said to be majorized by $\textbf{y}$ ( $\textbf{x}\leq \textbf{y}$ ) if

\begin{equation*} \sum_{i=k}^{n}x_{(i)}\leq \sum_{i=k}^{n}y_{(i)}\quad \text{for all } k \in \{2,\ldots,n\}, \qquad \sum_{i=1}^{n}x_{i}=\sum_{i=1}^{n}y_{i},\end{equation*}

where $x_{(1)}\leq x_{(2)}\leq \cdots \leq x_{(n)}$ is the increasing rearrangement of $\textbf{x}$ . Using this notation, the conjecture can be formulated as follows.

Conjecture 1. (Colini-Baldeschi et al. [Reference Colini-Baldeschi, Scarsini and Vaccari8].) For any $n\times n$ covariance matrix $\Sigma$ , if $\nu$ is the corresponding variance game and $\mu$ the corresponding standard deviation game, then

(3) \begin{equation} \frac{1}{\mu (N)}\Phi (\mu )\leq \frac{1}{\nu (N)}\Phi (\nu ), \end{equation}

where $\Phi$ denotes the vector of the Shapley values.

This conjecture was proved to hold true in the independent case in [Reference Chen, Hu and Tang5, Reference Galeotti and Rabitti17]. However, [Reference Galeotti and Rabitti17] provided two counterexamples to the conjecture considering three dependent random variables. In general, such counterexamples can be seen to derive from Theorem 1.

3.1. Inverted majorization

Denote by y and x the normalized vectors of Shapley values relative, respectively, to the variance and the standard deviation game, i.e.

\begin{equation*} y_{i} = \frac{\phi_{i}(\nu)}{\sum_{i=1}^{n}\phi_{i}(\nu)} = \frac{\phi_{i}(\nu)}{\nu(N)}, \qquad x_{i} = \frac{\phi_{i}(\mu)}{\mu(N)}.\end{equation*}

Then the following theorem holds.

Theorem 1. Consider three non-negative valued risks $(X_{1}, X_{2}, X_{3})$ such that $\mathrm{Var}[X_{1}] < \mathrm{Var}[X_{2}] < \mathrm{Var}[X_{3}]$ , whereas the Shapley values relative to the variance game are equal, i.e. $\mathrm{Cov}[X_{i}, X_{1} + X_{2} + X_{3}] = \frac{1}{3}\mathrm{Var}[X_{1} + X_{2} + X_{3}]$ , $i = 1, 2, 3$ . Then, denoting by $\textbf{y}$ and $\textbf{x}$ the vectors of normalized Shapley values relative, respectively, to the variance game $\nu$ and the standard deviation game $\mu$ , $\textbf{y}$ is majorized by $\textbf{x}$ .

Remark 1. It is easily checked that the theorem also holds in the cases $\mathrm{Var}[X_{1}] = \mathrm{Var}[X_{2}] < \mathrm{Var}[X_{3}]$ and $\mathrm{Var}[X_{1}] < \mathrm{Var}[X_{2}] = \mathrm{Var}[X_{3}]$ .

The proof can be found in [Reference Galeotti and Rabitti17]. Theorem 1 provides counterexamples to the conjecture in [Reference Colini-Baldeschi, Scarsini and Vaccari8], as under its conditions the majorization in (3) is inverted. Conversely, if the three risks were independent, the conjectured majorization would hold, as proved in [Reference Chen, Hu and Tang5, Reference Galeotti and Rabitti17]. In such a case the Shapley values relative to the variance game are equal to the variances of the risks.

As we saw in Section 2 the Shapley value is a fair method to share the total value of the game among individual risks. The Shapley values of the two games are fair allocations of the two total game values, namely the portfolio variance and standard deviation. However, coming to the relationship between the two Shapley values, Theorem 1 shows that, in the dependent case, the Shapley value for the variance allocation may produce a risk ranking which is different from the one produced by the Shapley value for the standard deviation game. In fact, under the assumptions of the theorem, considering the variance and standard deviation Shapley values as risk metrics, it occurs that, by a small modification of the random variables’ distributions, the riskiest random variable relative to the variance game is not the riskiest one in the standard deviation game. This implies that, depending on the choice of the value of the game, one random variable $X_{i}$ can be considered proportionally less or more risky for the portfolio. Then, simply changing the game (i.e. the way we measure risk in the portfolio), an effect of risk transfer takes place among the random variables, since the risk ranking is changed.

4. Tail variance and standard deviation games

In this paper we want to extend the results of [Reference Colini-Baldeschi, Scarsini and Vaccari8] to the tail variance and the tail standard deviation. This allows the risk analyst to identify and quantify the most important risks driving the tail variability; it can be done introducing a tail threshold s and conditioning the loss as follows.

In parallel to the results of [Reference Colini-Baldeschi, Scarsini and Vaccari8], we can characterize the tail variance game.

Proof. First of all, we define new variables $(\widetilde{X}_{1},\widetilde{X}_{2},\ldots,\widetilde{X}_{n})$ as follows:

(4) \begin{equation} \widetilde{X}_{i} = \left\{ \begin{array}{c@{\quad}l} x & \text{if}\ X_{i} = x > s, \\[5pt] x & \text{if}\ X_{i} = x \leq s \text{ and } \sum_{j \neq i}X_{j} > s - x, \\[5pt] 0 & \text{if}\ \sum_{j=1}^{n}X_{j} \leq s - x. \end{array} \right. \end{equation}

Then, it is easily checked that, for any subset J of $\{1,2,\ldots,n\}$ ,

\begin{equation*} \mathrm{Var}\Bigg[\sum_{i\in J}\widetilde{X}_{i}\Bigg] = \mathrm{Var}\Bigg[\sum_{i \in J}X_{i} \mid S > s\Bigg]. \end{equation*}

It follows that the tail variance game $\nu_{s}$ , relative to the variables $(X_{1}, X_{2}, \ldots,X_{n})$ , is equivalent to the variance game relative to $(\widetilde{X}_{1}, \widetilde{X}_{2}, \ldots, \widetilde{X}_{n})$ . Hence, we can apply the result of [Reference Colini-Baldeschi, Scarsini and Vaccari8], implying $\phi_{i}(\nu_{s}) = \mathrm{Cov}\big[\widetilde{X}_{i},\sum_{j=1}^{n}\widetilde{X}_{j}\big]$ . On the other hand, we get $\mathrm{Cov}\big[\widetilde{X}_{i},\sum_{j=1}^{n}\widetilde{X}_{j}\big] = \mathrm{Cov}[X_{i},S \mid S > s]$ , which proves the theorem.

This closed-form expression generates the Shapley value for the tail variance game, avoiding the problem of estimating the value for all the $2^{n}$ possible coalitions, which becomes computationally demanding as n increases. The Shapley value may be negative: if $X_{i}$ contributes to hedging the total tail risk, it is rewarded with a negative Shapley value. In the actuarial literature we can find different works concerning this allocation principle. For instance, [Reference Valdez37] derived an analytical expression for $\mathrm{Cov}[X_{i},S \mid S > s]$ in the case of sums of multivariate normal random variables. This result was extended in [Reference Valdez38] to the case of elliptical distributions. Nonetheless, in our contribution for the first time this allocation principle is interpreted as a Shapley value. If we normalize the Shapley values for the tail variance game, we obtain

\begin{equation*} \widetilde{\phi}_{i}(\nu_{s}) = \frac{\phi_{i}(\nu_{s})}{\nu_{s}(N)} = \frac{\mathrm{Cov}[X_{i},S \mid S > s]}{\mathrm{Var}[S \mid S > s]},\end{equation*}

which are again Shapley values by linearity. In analogy to the covariance allocation principle in (2), we call the Shapley value $\widetilde{\phi}_{i}(\nu_{s})$ the tail covariance allocation principle. Note that this allocation is a pure number independent of the currency unit. It represents the fractional contribution of the random variable $X_{i}$ to the tail covariance.

In parallel to [Reference Colini-Baldeschi, Scarsini and Vaccari8], we can introduce the tail standard deviation game.

The related Shapley value allocates the tail standard deviation as the total value of the game. However, as in the unconditioned case, no closed-form expression is available. In order to compare the rankings of the two Shapley values arising from Definitions 1 and 2, we consider the majorization problem in the next section.

5. The majorization problem and the dependence on the threshold

It is interesting to investigate whether the analogue of Conjecture 1 could hold true in the tail conditional case. The answer is negative for suitable heavy-tail variables, as our following example will show.

To illustrate our scenario, we consider the situation where non-negative random variables refer to losses of an insurance portfolio. The insurance premiums are paid only if the total loss of the portfolio overtakes a given threshold $s\geq 0$ . Thus, when $s>0$ , we have observed how that is equivalent to considering a new portfolio $(\widetilde{X}_{1},\widetilde{X}_{2},\ldots,\widetilde{X}_{n})$ defined by (4). Therefore, assuming $X_{1},X_{2},\ldots,X_{n}$ to be independent, $\widetilde{X}_{1},\widetilde{X}_{2},\ldots,\widetilde{X}_{n}$ are no longer so. Hence, we wonder whether the majorization $\widetilde{\textbf{x}}(s)\leq \widetilde{\textbf{y}}(s)$ , holding when $s=0$ , may not hold or even be inverted for higher values of s.

To this end we consider two examples, where three independent non-negative variables $X_{1} ,X_{2} ,X_{3}$ have, respectively, light-tail (exponential) and heavy-tail (slightly modified Pareto ones) distributions. By applying Theorem 1 we prove that in the latter case the inverted majorization occurs for suitable high values of the tail threshold, while the majorization is preserved in the light-tail case. In the following, we provide outlines of the proofs of Propositions 1 and 2, omitting the details that appear in the full proofs in the Appendix.

5.1. A light-tail example without inversion

Consider three independent non-negative variables $X_{1} ,X_{2} ,X_{3}$ with distributions given by $F_{1}(x) = 1 - {\mathrm{e}}^{-(1 + \varepsilon)x}$ , $F_{2}(x) = F_{3}(x) = 1 - {\mathrm{e}}^{-x}$ . We have $\mathrm{Var}[X_{1}] = (1 + \varepsilon)^{-2} < \mathrm{Var}[X_{2}] = \mathrm{Var}[X_{3}] = 1$ , while $\mathbb{E}[X_{1}] = (1 + \varepsilon)^{-1} < \mathbb{E}[X_{2}] = \mathbb{E}[X_{3}] = 1$ .

Proposition 1. Consider the tail-conditioned variables $\widetilde{X}_{i}$ . Then, no matter how high the tail threshold s is and how small the parameter $\varepsilon $ is, $\phi_{1}(\nu_{s}) < \phi_{2}(\nu_{s}) = \phi_{3}(\nu_{s})$ , so that the inverted majorization caused by Theorem 1 does not occur.

Proof (outline). Taking $s>0$ sufficiently high and $\varepsilon >0$ sufficiently small, we want to estimate the sign of the difference

\begin{equation*} G(\varepsilon,s) = \phi_{1}(\nu_{s}) - \phi_{2}(\nu_{s}) = \mathrm{Cov}[\widetilde{X}_{1},\widetilde{X}_{1}+\widetilde{X}_{2}+\widetilde{X}_{3}] - \mathrm{Cov}[\widetilde{X}_{2},\widetilde{X}_{1}+\widetilde{X}_{2}+\widetilde{X}_{3}]. \end{equation*}

Explicitly, recalling that $\widetilde{X}_{2}$ and $\widetilde{X}_{3}$ are identically distributed,

(5) \begin{equation} G(\varepsilon, s) = \mathbb{E}[\widetilde{X}_{1}^{2}] - \mathbb{E}[\widetilde{X}_{2}^{2}] + \mathbb{E}[\widetilde{X}_{1}\widetilde{X}_{3}] - \mathbb{E}[\widetilde{X}_{2}\widetilde{X}_{3}] - \mathbb{E}^{2}[\widetilde{X}_{1}] - \mathbb{E}[\widetilde{X}_{1}]\mathbb{E}[\widetilde{X}_{3}] + 2\mathbb{E}^{2}[\widetilde{X}_{2}]. \end{equation}

First of all, we observe that, for $i\neq j\neq k$ ,

\begin{equation*} \mathrm{Prob}(\widetilde{X}_{i} > x) = \mathrm{Prob}(X_{i} > x \text{ and } X_{j} + X_{k} > s - x) = \overline{F}_{i}(x)\int_{s-x}^{+\infty}(f_{j} \ast f_{k})(z) \,{\mathrm{d}} z, \end{equation*}

where $\overline{F_{i}}(z) = 1 - F_{i}(z)$ , $f_{i}(z) = F_{i}^{\prime}(z)$ , and $f_{j}\ast f_{k}$ denotes the convolution product, so that $\int_{s-x}^{+\infty}(f_{j} \ast f_{k})(z) \,{\mathrm{d}} z = (\overline{F_{j} \ast F_{k}})(s - x)$ . Moreover,

(6) \begin{equation} \mathrm{Prob}(\widetilde{X}_{i} \leq x \text{ and } \widetilde{X}_{j} \leq y) = 1 - \mathrm{Prob}(\widetilde{X}_{i} > x) - \mathrm{Prob}(\widetilde{X}_{j} > y) + \mathrm{Prob}(\widetilde{X}_{i} > x, \widetilde{X}_{j} > y), \end{equation}

where

\begin{align*}\mathrm{Prob}(\widetilde{X}_{i} > x, \widetilde{X}_{j} > y) = \mathrm{Prob}(X_{i} > x, X_{j} > y, X_{k} > s - x - y) = \overline{F}_{i}(x)\overline{F}_{j}(y)\overline{F}_{k}(s-x-y).\end{align*}

We can now proceed to compute the terms in (5) by expanding them as formal power series of $\varepsilon$ . Observing that $G(0,s) = 0$ for all $s > 0$ and $G(\varepsilon,0) < 0$ for all $\varepsilon > 0$ , we are led to the formal power series $G(\varepsilon, s) = {\mathrm{e}}^{-s}\sum\varepsilon^{m}p_{m}(s)$ . Since the functions $p_{m}(s)$ grow at most polynomially in s, taking $\varepsilon \leq {\mathrm{e}}^{-s}$ means the series $\sum\varepsilon^{m}p_{m}(s)$ converges for any high value of s. In this case, the sign of $G(\varepsilon, s)$ when s is sufficiently high is given by the sign of $p_{1}(s)$ . Therefore, if it happens that, for sufficiently high values of s, $p_{1}(s) > 0$ , there exists some pair $(\overline{\varepsilon},\overline{s})$ satisfying $G(\overline{\varepsilon},\overline{s}) = 0$ and therefore meeting the conditions of Theorem 1. The first observation concerns the computation of $\mathbb{E}[\widetilde{X}_{1}]$ and $\mathbb{E}[\widetilde{X}_{2}] = \mathbb{E}[\widetilde{X}_{3}]$ . It is checked that, in any case, $\mathbb{E}[\widetilde{X}_{i}] = {\mathrm{e}}^{-s}a_{i}(s) + {\mathrm{e}}^{-s}(\varepsilon b_{i}(s) + \mathrm{h.o.t.}(\varepsilon))$ , where $a_{i}(s)$ and $b_{i}(s)$ grow polynomially in s, and $\mathrm{h.o.t.}(\varepsilon)$ means higher-order terms in $\varepsilon$ . Therefore, the coefficients of $\varepsilon$ in $\mathbb{E}^{2}[\widetilde{X}_{i}]$ and $\mathbb{E}[\widetilde{X}_{i}]\mathbb{E}[\widetilde{X}_{j}]$ all tend to zero when $s \rightarrow +\infty$ , as ${\mathrm{e}}^{-2s}q(s)$ , where q(s) is a polynomial in s.

Consider now $\mathbb{E}[\widetilde{X}_{1}^{2}]$ . We have

\begin{equation*} \mathbb{E}[\widetilde{X}_{1}^{2}] = \int_{0}^{s}x^{2} \bigg[{-}\frac{{\mathrm{d}}}{{\mathrm{d}} x}\overline{F}_{1}(x)(\overline{F_{2}\ast F_{3}})(s - x)\bigg] \,{\mathrm{d}} x + \int_{s}^{+\infty }x^{2}f_1(x) \,{\mathrm{d}} x. \end{equation*}

Integrating by parts and letting L(x) denote the primitive of $2x\overline{F}_{1}(x)$ which satisfies $L(+\infty) = 0$ , it remains to calculate

\begin{equation*} -L(0)(\overline{F_{2} \ast F_{3}})(s) - \int_{0}^{s}L(x)(f_{2} \ast f_{3})(s - x) \,{\mathrm{d}} x. \end{equation*}

Then, after a certain number of steps, it is found that the contribution to $\varepsilon p_{1}(s)$ is given by $\varepsilon(\!-\!{s^{4}}/{6} + \mathrm{l.o.t.}(s))$ , where l.o.t. stands for lower-order terms.

In the case of $\mathbb{E}[\widetilde{X}_{2}^{2}]$ the contribution to $\varepsilon p_{1}(s)$ is only given by

\begin{equation*} \int_{0}^{s}x^{2}\bigg[{-}\frac{{\mathrm{d}}}{{\mathrm{d}} x}\overline{F}_{2}(x) (\overline{F_{1} \ast F_{3}})(s - x)\bigg] \,{\mathrm{d}} x, \end{equation*}

and standard calculations lead to $\varepsilon(\!-\!{s^{4}}/{12} + \mathrm{l.o.t.}(s))$ , which, in $G(\varepsilon, s)$ , becomes $\varepsilon({s^{4}}/{12} + \mathrm{l.o.t.}(s))$ .

Consider now $\mathbb{E}[\widetilde{X}_{1}\widetilde{X}_{3}]$ and $\mathbb{E}[\widetilde{X}_{2}\widetilde{X}_{3}]$ . Then,

\begin{equation*} \mathbb{E}[\widetilde{X}_{1}\widetilde{X}_{3}] = \int_{0}^{+\infty}\int_{0}^{+\infty}xy\frac{\partial}{\partial x\partial y}K(x,y) \,{\mathrm{d}} x\,{\mathrm{d}} y \end{equation*}

since $K(x,y) = \mathrm{Prob}(\widetilde{X}_{3} \leq x,\,\widetilde{X}_{1}\leq y)$ . Therefore, by (6), $({\partial}/{\partial x\partial y})K(x,y) = ({\partial}/{\partial x\partial y})H(x,y)$ , where

\begin{equation*} H(x,y) = \mathrm{Prob}(\widetilde{X}_{3} > x, \widetilde{X}_{1} > y) = \overline{F}_{3}(x)\overline{F}_{1}(y)\overline{F}_{2}(s-x-y), \end{equation*}

with $x,y>0$ and $\overline{F}_{2}(s-x-y)<1$ when $x+y<s$ . Hence, we have to calculate $\mathbb{E}[\widetilde{X}_{1}\widetilde{X}_{3}]$ on the triangle $T = \{0<x<s,0<y<s-x\}$ , i.e.

\begin{equation*} \iint\limits_{T}xy\frac{\partial}{\partial x\partial y}H(x,y) \,{\mathrm{d}} x\,{\mathrm{d}} y. \end{equation*}

By the same arguments, in estimating $\mathbb{E}[\widetilde{X}_{2}\widetilde{X}_{3}]$ , we have to compute

\begin{equation*} \iint\limits_{T}xz\frac{\partial}{\partial x\partial z}R(x,z) \,{\mathrm{d}} x\,{\mathrm{d}} z, \end{equation*}

where $R(x,z) = \overline{F}_{3}(x)\overline{F}_{2}(z)\overline{F}_{1}(s-x-z)$ . By a suitable change of variables we obtain

\begin{equation*} \iint\limits_{T}xz\frac{\partial}{\partial x\partial z}R(x,z) \,{\mathrm{d}} x\,{\mathrm{d}} z = \iint\limits_{T}xy\frac{\partial}{\partial x\partial y}H(x,y) \,{\mathrm{d}} x\,{\mathrm{d}} y, \end{equation*}

so that the difference $\iint\limits_{T}xy\frac{\partial}{\partial x\partial y}H(x,y) \,{\mathrm{d}} x\,{\mathrm{d}} y - \iint\limits_{T}xz\frac{\partial}{\partial x\partial z}R(x,z) \,{\mathrm{d}} x\,{\mathrm{d}} z$ is zero. Hence, the contribution to $\varepsilon p_{1}(s)$ of higher order in s coming from $\mathbb{E}[\widetilde{X}_{1}\widetilde{X}_{3}] - \mathbb{E}[\widetilde{X}_{2}\widetilde{X}_{3}]$ is given by $\int_{0}^{s}xf_{3}(x) \,{\mathrm{d}} x \int_{s-x}^{+\infty}yf_{1}(y) \,{\mathrm{d}} y$ , which yields, after simple steps, $\varepsilon(\!-\!{s^{4}}/{12} + \mathrm{l.o.t.}(s))$ . In conclusion, $\varepsilon p_{1}(s) = \varepsilon(\!-\!{s^{4}}/{6} + \mathrm{l.o.t.}(s))$ . Hence, no matter how high s is and how small $\varepsilon$ is accordingly, $G(\varepsilon, s) < 0$ and Theorem 1 cannot be applied.

5.2. A heavy-tail example with inversion

Consider three independent non-negative variables (risks) $X_{1}, X_{2}, X_{3}$ , whose distributions are

\begin{equation*} F_{1}(x) = 1 - \frac{1}{4}{\mathrm{e}}^{-kx^{2}} + (h - 2){\mathrm{e}}^{-kx^{2}} - \frac{h - {5}/{4}}{(1 + hx)^{3}} , \quad F_{2}(x) = F_{3}(x) = 1 - \frac{1}{4}{\mathrm{e}}^{-kx^{2}} - \frac{{3}/{4}}{(1 + 2x)^{3}} ,\end{equation*}

where $h = 2 + \varepsilon$ , with $k > 0$ high enough and $\varepsilon > 0$ as small as we will need.

Proposition 2. Using the same notation as Section 5.1, there exist pairs $(\overline{\varepsilon},\overline{s})$ satisfying the conditions of Theorem 1 and therefore inverting the majorization of the Shapley values for the tail variance and standard deviation games.

Proof (outline). First of all, $\varepsilon$ can be chosen so small as to ensure $f_{1}(x) = F_{1}^{\prime}(x) > 0$ for $x \geq 0$ . Then, straightforward computations show that, for k sufficiently large and $\varepsilon$ sufficiently small, $\mathrm{Var}[X_{1}] - \mathrm{Var}[X_{2}] > {\varepsilon}/{257}$ , implying, in terms of Shapley values, $\phi_{1}(\nu_{0}^\varepsilon) > \phi_{2}(\nu_{0}^\varepsilon) = \phi_{3}(\nu_{0}^\varepsilon)$ , where $\nu_0^\varepsilon$ denotes the value function for the tail variance game depending on the parameter $\varepsilon$ and the threshold $s=0$ . On the other hand, we observe that

\begin{align*} 1 - \frac{5}{4h} & = \frac{3}{8} + \frac{5}{16}\varepsilon + \mathrm{h.o.t.}(\varepsilon), \\[5pt] \frac{1}{h} - \frac{5}{4h^{2}} & = \frac{3}{16} + \frac{1}{16}\varepsilon + \mathrm{h.o.t.}(\varepsilon), \\[5pt] \frac{1}{h^{2}} - \frac{5}{4h^{3}} & = \frac{1}{10} - \frac{1}{64}\varepsilon + \mathrm{h.o.t.}(\varepsilon). \end{align*}

Then, adopting the same notation as the previous proposition, we have to estimate the sign of $G(\varepsilon, s)$ when $s > 0$ is sufficiently high and $\varepsilon > 0$ is accordingly sufficiently small, knowing that $G(0, s) = 0$ for all $s \geq 0$ and $G(\varepsilon, 0) > 0$ for sufficiently small values of $\varepsilon > 0$ .

By steps analogous to those followed in the previous proposition, we are first led to calculate the contribution to $\varepsilon p_{1}(s)$ given by $\mathbb{E}[\widetilde{X}_{1}^{2}]$ . Precisely, we need to compute

\begin{equation*} -L(0)(\overline{F_{2} \ast F_{3}})(s) - \int_{0}^{s}L(x)(f_{2} \ast f_{3})(s - x) \,{\mathrm{d}} x, \end{equation*}

where L(x) is the primitive of $2x\overline{F}_{1}(x)$ satisfying $L(\!+\!\infty) = 0$ . In particular, applying the mean value theorem, we have to estimate

\begin{equation*} \int_{0}^{s}\bigg(\frac{1}{h} - \frac{5}{4h^{2}}\bigg)\frac{2}{1+hx}(f_{2} \ast f_{3})(s - x) \,{\mathrm{d}} x = \bigg(\frac{1}{h} - \frac{5}{4h^{2}}\bigg)\frac{2}{1+h\overline{z}}(F_{2} \ast F_{3})(s), \end{equation*}

where $\overline{z} \in (0,s)$ . This implies that, calling $\varepsilon a(s)$ the contribution of $\mathbb{E}[\widetilde{X}_{1}^{2}]$ to $\varepsilon p_{1}(s)$ , a(s) tends to zero whenever $s \rightarrow +\infty$ not faster than $s^{-1}$ . In order to evaluate its sign, let us apply the mean value theorem the other way around,

\begin{align*} & \int_{0}^{s}\bigg(\frac{1}{h} - \frac{5}{4h^{2}}\bigg)\frac{2}{1+hx}(f_{2} \ast f_{3})(s - x) \,{\mathrm{d}} x \\[5pt] & \quad = \bigg(\frac{1}{h^{2}} - \frac{5}{4h^{3}}\bigg)(f_{2} \ast f_{3})(\,\widehat{z}\,) 2\bigg(\!\ln\!(1 + 2s) + \ln\!\bigg(1 + \varepsilon\frac{s}{1+2s}\bigg)\bigg), \end{align*}

so that, when s is sufficiently high, $a(s) < 0$ . It can be checked that the other contributions to $\varepsilon p_{1}(s)$ tend to zero when $s \rightarrow +\infty$ at least as $s^{-2}$ , except possibly the one given by $\int_{0}^{s}xf_{3}(x) \,{\mathrm{d}} x\int_{s-x}^{+\infty}yf_{1}(y) \,{\mathrm{d}} y$ , yielding

\begin{equation*} \int_{0}^{s}\frac{9}{4}\bigg(\frac{1}{(1+2x)^{3}} - \frac{1}{(1+2x)^{4}}\bigg) \bigg(\frac{3}{2}\frac{1-{5}/{4h}}{(1+h(s-x))^{2}} - \frac{1-{5}/{4h}}{(1+h(s-x))^{3}}\bigg) \,{\mathrm{d}} x. \end{equation*}

Hence, the integrand function tends to zero not faster than $({1-{5}/{4h}})/{(1+hs)^{2}}$ , which can be written as

\begin{align*}\bigg(\frac{3}{8} + \frac{5}{16}\varepsilon\bigg) \frac{1}{(1 + 2s)^{2}}\bigg(1 - 2\varepsilon\frac{s}{1 + 2s}\bigg) + \mathrm{h.o.t.}(\varepsilon).\end{align*}

Then, when s is sufficiently high, the above integral is smaller than

\begin{align*}b - c\varepsilon\frac{1}{16}\frac{s}{(1 + 2s)^{2}},\end{align*}

where b and c are suitable positive numbers.

It follows that, for s sufficiently high and $\varepsilon$ sufficiently small, $G(\varepsilon, s) < 0$ , so that, since $G(\varepsilon,0) > 0$ , there exist pairs $(\overline{\varepsilon},\overline{s})$ satisfying $G(\overline{\varepsilon},\overline{s}) = 0$ . Therefore, Theorem 1 can be applied and the inverted majorization occurs.

6. Implications for tail risk management

The implications of the above results concern the actuarial allocation of the tail loss of a portfolio of risks into individual risk contributions. Allocating the tail variance into risk contributions is essential for the individual fair pricing of insurance/reinsurance contracts. In this regard the Shapley value provides a useful attribution method for sharing the insurance premium among the holders of the risks in the portfolio.

Consider a portfolio of non-negative risks $(X_{1},X_{2},\ldots,X_{n})$ . Let us assume for the moment that risks are independent and consider the limit case $s=0$ . Then, adopting the Shapley values for the standard deviation game, the highest risks are proportionally allocated a smaller fraction than they would receive when allocating the variance. In other words, these Shapley values induce a ‘solidarity’ effect among the risks in the portfolio, since the conjecture of [Reference Colini-Baldeschi, Scarsini and Vaccari8] holds true. Thus, asking whether the majorization holds for any $s>0$ is equivalent to asking whether the Shapley values for the tail standard deviation game always induce a stronger solidarity effect than those for the variance game. However, our example shows that in general the solidarity of the two Shapley values depends on the threshold s and on the tail fatness. The implication is that the ranking of the individual risk contributions can be changed by allocating either the tail variance or the tail standard deviation. In the next section we show that well-known premium principles are Shapley values for tail games and, thus, the ranking of the premiums to be individually paid might be changed simply depending on the chosen tail game.

6.1. Tail mean-variance games and premium principles

We define the game $\varepsilon_{s}(J) = \mathbb{E}[S_{J} \mid S > s]$ as the tail mean game. By the linearity of the conditional expectation, $\varepsilon_{s}(I\cup J) = \varepsilon_{s}(I) + \varepsilon_{s}(J)$ for all $I,J\subset N$ such that $I\cap J = \varnothing$ . A game satisfying this condition is called an additive game [Reference Colini-Baldeschi, Scarsini and Vaccari8]. It follows that $\varepsilon_{s}$ is an additive game and hence the Shapley value is equal to $\phi_{i}(\varepsilon_{s}) = \mathbb{E}[X_{i} \mid S > s]$ . In the actuarial literature, the Shapley value for the tail mean $\phi_{i}(\varepsilon_{s})$ is well known as the conditional tail expectation (CTE) allocation rule [Reference Denault10, Reference Dhaene, Henrard, Landsman, Vandendorpe and Vanduffel12, Reference Dhaene, Tsanakas, Valdez and Vanduffel13, Reference Kim and Kim22, Reference Tasche and Dev35].

Consider now the value function

(7) \begin{equation} \gamma_{s}(J) = \varepsilon_{s}(J) + k\nu_{s}(J),\end{equation}

where k is a positive constant. Using the tail mean and the tail variance games, we construct the game $\gamma_{s}$ in (7) as a linear combination of the two games. We call the game $\gamma_{s}$ the tail mean-variance game. This choice is motivated by analogy with the tail mean-variance risk model introduced in [Reference Landsman23]. We can characterize the Shapley values for the game $\gamma_{s}$ .

Proposition 3. For the value function defined in (7),

(8) \begin{equation} \phi_{i}(\gamma_{s}) = \mathbb{E}[X_{i} \mid S > s] + k\cdot\mathrm{Cov}[X_{i},S \mid S > s]. \end{equation}

Proof. Consider the tail mean game $\varepsilon_{s}$ and the tail variance game $\nu_{s}$ . Their related Shapley values are $\phi_{i}(\varepsilon_{s}) = \mathbb{E}[X_{i} \mid S > s]$ and $\phi_{i}(\nu_{s}) = \mathrm{Cov}[X_{i},S \mid S > s]$ respectively. By the definition of $\gamma _{s}$ in (7) and the linearity of the Shapley value, the proof is concluded.

We can connect the result of Proposition 3 to premium principles presented in the actuarial literature. For the random variable $X_{i}$ with $i=1,2,\ldots,n$ , [Reference Furman and Landsman16] defined the tail covariance premium $\mathrm{TCP}_{s}(X_{i} \mid S)$ as

\begin{equation*} \mathrm{TCP}_{s}(X_{i} \mid S) = \mathbb{E}[X_{i} \mid S > s] + a \cdot \mathrm{Cov}[X_{i},S \mid S > s],\end{equation*}

where a is a non-negative constant. Since the covariance is expressed in a different currency unit, [Reference Wang39] introduced the tail covariance premium adjusted, $\mathrm{TCPA}_{s}(X_{i} \mid S)$ as

\begin{equation*} \mathrm{TCPA}_{s}(X_{i} \mid S) = \mathbb{E}[X_{i} \mid S > s] + a\frac{\mathrm{Cov}[X_{i},S \mid S > s]}{\sqrt{\mathrm{Var}[S \mid S > s]}}.\end{equation*}

From (8) it is clear that the premium principles $\mathrm{TCP}_{s}(X_{i} \mid S)$ and $\mathrm{TCPA}_{s}(X_{i} \mid S)$ are Shapley values with the choice $k=a$ and $k = a\cdot(\mathrm{Var}[S \mid S > s])^{-1/2}$ , respectively. More precisely, by the linearity property of the Shapley value we can write

\begin{align*} \mathrm{TCP}_{s}(X_{i} \mid S) & = \phi_{i}(\varepsilon_{s} + a \cdot \nu_{s}) , \\[5pt] \mathrm{TCPA}_{s}(X_{i} \mid S) & = \phi_{i}(\varepsilon_{s} + a \cdot \nu_{s} \cdot (\mathrm{Var}[S \mid S > s])^{-1/2}).\end{align*}

This connection constitutes an important theoretical justification for the use of these two premium principles. In particular, the fairness of the two allocation methods is an appealing property derived from the Shapley value axiomatization.

6.2. Peer-to-peer reinsurance pricing

The peer-to-peer insurance scheme is a recent form of participating insurance in which a community of policyholders agree to share the first layer of losses that hit participants [Reference Clemente and Marano7, Reference Denuit11]. While the monetary transfers (i.e. loss coverage and/or partial premium refunds) among the policyholders in the community take place ex post, there is the need to protect ex ante the insurance community from large losses which cannot be distributed among participants. Thus, a reinsurance contract must be purchased in advance to protect the community. The problem is then how to calculate a fair reinsurance premium for every participant.

We propose the Shapley value for the tail variance and standard deviation games as a reliable answer to this question. Specifically, consider a reinsurance company which is asked to indemnify the losses S given the fact that they exceed a threshold s. The total reinsurance premium paid to the reinsurance company to be shared by policyholders can be quantified according to the variance principle,

(9) \begin{equation} P_\mathrm{V}(S,s) = \mathbb{E}[S - s \mid S > s] + \alpha_\mathrm{V} \cdot \mathrm{Var}[S - s \mid S > s],\end{equation}

or the standard deviation principle,

(10) \begin{equation} P_\mathrm{SD}(S,s) = \mathbb{E}[S - s \mid S > s] + \alpha_\mathrm{SD}\sqrt{\mathrm{Var}[S - s \mid S > s]}.\end{equation}

In the standard deviation principle $P_\mathrm{SD}$ the two terms $\mathbb{E}[S-s \mid S>s]$ and $\sqrt{\mathrm{Var}[S-s \mid S>s]}$ are expressed in the same monetary unit ( $\$ + \$ $ ), while the variance principle includes terms of different units ( $\$ + \$^{2}$ ). The Shapley values for the tail variance and standard deviation games represent an ideal attribution method to allocate the reinsurance premiums (9) and (10) among the peer-to-peer community members. Precisely, for the ith policyholder, where $i=1,2,\ldots,n$ , we find the Shapley values $\phi_{i}(P_\mathrm{V}(S,s))$ and $\phi_{i}(P_\mathrm{SD}(S,s))$ respectively, with the former expressed in closed form as proved by Proposition 3. Note that, if the coefficients in $\alpha_\mathrm{V}$ and $\alpha_\mathrm{SD}$ in (9) and (10) were set to zero as well as s, then we would find $\phi_{i}(P_\mathrm{V}(S,0)) = \phi_{i}(P_\mathrm{SD}(S,0)) = \mathbb{E}[X_{i} \mid S>0]$ , which is exactly the CTE allocation method proposed in [Reference Denuit11] to price the peer-to-peer insurance premiums without a reinsurance purchase.

Finally, [Reference Albrecher, Beirlant and Teugels2] stated that ‘measuring variability by variance or standard deviation gives just different weight to “additional” risk and in that sense is a matter of taste and choice’ [Reference Albrecher, Beirlant and Teugels2, p. 220]. The results of our work show that, if the reinsurance company allocates either the total variance premium (9) or the total standard deviation premium (10) among policyholders using the Shapley value, it might generate different rankings concerning the individual paid premiums across the community members. Investigating the mechanism and the conditions by which the threshold level and the tail fatness make the two rankings change is an open question for future research.