Preference robust state-dependent distortion risk measure on act space and its application in optimal decision making

Abstract

Decision-making under uncertainty involves three fundamental components: acts, states of nature, and consequences, as first introduced by Savage (The foundations of statistics, Wiley, New York, 1954). An act is uniquely determined by a number of random consequences that are associated with different states of nature. If the consequences are identical across all states of nature, then the act is state-independent. Prior research on distortion risk measures (DRMs) has primarily focused on state-independent acts. In this paper, we extend the research to state-dependent acts by introducing a state-dependent DRM (SDRM) under the Anscombe–Aumann’s framework (Anscombe and Aumannin in Ann Math Stat 34(1):199–205, 1963). The proposed SDRM is the weighted average of DRMs at each state, where the weights are determined by the decision maker’s (DM’s) subjective probabilities of the states. In situations where there is incomplete information about the DM’s true distortion function and/or the true subjective probabilities of the states, we introduce a preference robust SDRM (PRSDRM) for acts. The PRSDRM is based on the worst-case state-dependent distortion function and the worst-case subjective probabilities over a dependent joint ambiguity set constructed with partially available information. To compute the PRSDRM numerically, we show that when the distortion functions are concave, it can be formulated as a biconvex program and further as a convex program by changing some variables. As a motivation and application, we use the PRSDRM for decision-making problems and propose an alternating iterative algorithm for solving it. Finally, we conduct numerical experiments to assess the performance of our proposed model and computational scheme.

Appendices

Appendix A Supplementary materials

1.1 Some examples of DRMs

Example A.1

In this example, we assume that X is a non-negative random variable.

(i) Value at Risk (VaR). Let $g_{\nu }(t)={\textbf{1}}_{(1-\nu ,1]}(t)$, where ${\textbf{1}}_{[\nu ,1]}(t)$ is the indicator function and $\nu \in (0,1)$. Observe that $F_X(x)\in [0,\nu )$ iff $S_X(x)\in (1-\nu ,1]$, that is, $x\in [0,F_X^{\leftarrow }(\nu ))$ iff $g_{\nu }(S_X(x))=1$. Consequently, it follows from (2.1) that

$$\begin{aligned} \rho _{g_{\nu }}(X)= \int _0^{F_X^{-1}(\nu )} dx=F_X^{-1}(\nu )=:\text{ VaR}_{\nu }(X). \end{aligned}$$

(ii) Conditional Value at Risk $\text{(CVaR) }$ (also called Expected Shortfall (Tasche 2002)). Let $g_{\alpha }(t)=\min \left\{ \frac{t}{1-\alpha },1\right\}$ with $\alpha \in (0,1)$. From (2.1), using integration by parts of Lebesgue-Stieltjes integral (see, for instance, Merkle et al. (2014)), we have

$$\begin{aligned} \rho _{g_{\alpha }}(X)= & {} \int _0^{F_X^{\leftarrow }(\alpha )} 1 dx +\int _{F_X^{\leftarrow }(\alpha )}^{\infty }\frac{S_X(x)}{1-\alpha }dx\nonumber \\= & {} F_X^{\leftarrow }(\alpha ) + \frac{1}{1-\alpha }\left[ S_X(x)x\big |_{F_X^{\leftarrow }(\alpha )_-}^\infty -\int _{F_X^{\leftarrow }(\alpha )}^{\infty }xdS_X(x)\right] \nonumber \\= & {} \frac{1}{1-\alpha }\left\{ {\mathbb {E}}[X{\textbf{1}}_{X\ge F_{X}^{\leftarrow }(\alpha )}] -F_X^{\leftarrow }(\alpha )[ P(X<F_X^{\leftarrow }(\alpha ))-\alpha ] \right\} \nonumber \\=: & {} \text{ ES}_\alpha (X) =\text{ CVaR}_\alpha (X), \end{aligned}$$

(A.1)

where $(\cdot )_- = \lim _{\delta \downarrow 0} (\cdot -\delta )$ and the third equality is due to the fact that $\lim _{x\rightarrow \infty } xg(S_X(x))=0$ and $-\int _{F_X^{\leftarrow }(\alpha )}^{\infty }xdS_X(x)=\mathbb E[X{\textbf{1}}_{X\ge F_X^{\leftarrow }(\alpha )}]$. The last equality is perhaps known, we refer readers to Wang and Xu (2023) for the detailed discussion about this. The relation can also be derived from (2.2), where we have

$$\begin{aligned} \rho _{g_{\alpha }}(X)=\int _0^1 F_X^{\leftarrow }(t)d{\tilde{g}}_{\alpha }(t)=\frac{1}{1-\alpha } \int _{\alpha }^1F_X^{\leftarrow }(t)dt=:\text{ CVaR}_{\alpha }(X), \end{aligned}$$

(A.2)

which is shown in Acerbi (2002).

From an insurance premium point of view, it might be more advantageous to have a non-flat tail distortion function because the whole loss distribution will be utilized, see, for instance, Wang (1995, 1996) and Wang et al. (1997) on premium principles.

(iii) Proportional hazards transform risk measure (Wang 1995). Let $g_{\gamma }(t)=t^{\frac{1}{\gamma }}$ for $\gamma >1$. Then

$$\begin{aligned} \rho _{g_{\gamma }}(X)=\int _0^{\infty } S_X(x)^{\frac{1}{\gamma }}dx=:\rho _{\textrm{PH}}(X). \end{aligned}$$

(iv) Gini’s risk measure (Denneberg 1990b). Let $g_s(t)=t-s(t^2-t)$ for $s\in (0,1)$. Then from (2.2),

$$\begin{aligned} \rho _{g_s}(X)= & {} \int _0^1 F_X^{\leftarrow }(1-t)dg_s(t)=\int _0^1 F_X^{\leftarrow }(1-t)[1-s(2t-1)]dt\\= & {} \int _0^1 F_X^{\leftarrow }(1-t)dt - s \int _0^1 F_X^{\leftarrow }(1-t)(2t-1)dt\\= & {} {\mathbb {E}}[X]+s\int _0^1 F_X^{\leftarrow }(t)(2t-1)dt= {\mathbb {E}}[X]+\frac{1}{2}s{\mathbb {E}}[|X-X'|]=:\text{ Gini}_s(X), \end{aligned}$$

where $X'$ is an independent copy of X. The second last equality is due to the fact that ${\mathbb {E}}[|X-X'|]=2\int _0^1 (2t-1)F_X^{\leftarrow }(t)dt$ (can be calculated easily with Fubini’s Theorem, or see, for instance, Furman et al. (2017)), which is called the Gini’s mean difference and is closely related to stochastic dominance, see, for instance, Yitzhaki (1982).

1.2 Proof of Theorem 4.1

Proof

We use Lemma 4.1 to prove the result and do so in four steps.

Step 1. By the definition of PRSDRM of a state-dependent act ${\varvec{X}}$ and the structure of $({\mathcal {G}},{\mathcal {Q}}_{\epsilon })_1$, we have

$$\begin{aligned} {\mathcal {T}}_{({\mathcal {G}},{\mathcal {Q}}_{\epsilon })_1}({\varvec{X}})= \sup _{(g,Q)\in ({\mathcal {G}},{\mathcal {Q}}_{\epsilon })_1}{\mathcal {T}}_{Q}^g({\varvec{X}})=\sup _{Q\in {\mathcal {Q}}_{\epsilon }} \sup _{(g,Q)\in ({\mathcal {G}},Q)_1}{\mathcal {T}}_{Q}^g({\varvec{X}}), \end{aligned}$$

where $({\mathcal {G}},Q)_1=({\mathcal {G}},Q)_{pair}\cap ({\mathcal {G}},Q)_{ce}\cap ({\mathcal {G}}_{coh}\times Q)$ and $Q\in {\mathcal {Q}}_{\epsilon }$. Let $\varphi (Q)=\sup _{(g,Q)\in ({\mathcal {G}},Q)_1} {\mathcal {T}}_{Q}^g({\varvec{X}})$. Then ${\mathcal {T}}_{{\mathcal {Q}}_{\epsilon }}^{{\mathcal {G}}_1}({\varvec{X}})=\sup _{Q\in {\mathcal {Q}}_{\epsilon }}\varphi (Q)$. It suffices to derive the tractable reformulation of $\varphi (Q)$ for a specified probability distribution $Q=(q_1,q_2,\ldots ,q_n)\in {\mathcal {Q}}_{\epsilon }$. Throughout the proof, we use ${\mathcal {G}}_1(Q)=\{g\in {\mathcal {G}}:(g,Q)\in ({\mathcal {G}},Q)_1\}$ to emphasize the dependence of the ambiguity set of state-dependent distortion functions on the probability distribution Q of the belief of the states of nature. Similar notation for ${\mathcal {G}}_{pair}(Q)$, ${\mathcal {G}}_{ce}(Q)$, and ${\mathcal {G}}_{coh}(Q)$ represent $({\mathcal {G}},Q)_{pair}$, $({\mathcal {G}},Q)_{ce}$ and ${\mathcal {G}}_{coh}\times Q$ respectively.

Step 2. Let ${\overline{g}}\in {\mathcal {G}}_1(Q)$, $\xi _j^s\in \Xi ^s$ and $v_j^s={\overline{g}}(1-\xi _j^s,s)$ for $j=0,1,\ldots ,J_s$ and $s=1,\ldots ,n$. Then by the definition of distortion function, we have $v_0^s={\overline{g}}(1,s)=1$ and $v_{J_s}^s={\overline{g}}(0,s)=0$ for $s=1,\ldots ,n$. Let $\tau _i^s={\overline{g}}(1-\pi _i^s,s)$ for $i=1,\ldots ,n_s$ and $s=1,\ldots ,n$. Then $\tau _0^s={\overline{g}}(1,s)=1$ and $\tau _{n_s}^s={\overline{g}}(0,s)=0$ for $s=1,\ldots ,n$. For the fixed $v^s=(v_0^s,v_1^s,\ldots ,v_{J_s}^s)$, define

$$\begin{aligned} {\mathcal {G}}^s(v^s):=\{g\in {\mathcal {G}}^s:g(1-\xi _j^s,s)=v_j^s\;\text {for}\; j=0,1,\ldots ,J_s\}, \end{aligned}$$

where ${\mathcal {G}}^s$ represents all distortion functions for the state s, and for the fixed $v=(v^1,\ldots ,v^n)$, define

$$\begin{aligned} {\mathcal {G}}(v)=\{g\in {\mathcal {G}}:g(1-\xi _j^s,s)=v_j^s\;\text {for}\; j=0,1,\ldots ,J_s;s=1,\ldots ,n\}. \end{aligned}$$

Thus, ${\mathcal {G}}(v)=\times _{s=1}^n {\mathcal {G}}^s(v^s)$. By the definition, ${\mathcal {G}}(v)\in {\mathcal {G}}$ and ${\overline{g}}\in {\mathcal {G}}(v)$. Then, ${\mathcal {G}}(v)\ne \emptyset$. On the other hand, for any $g\in {\mathcal {G}}$, there exists ${\tilde{v}}=({\tilde{v}}^1,\ldots ,{\tilde{v}}^n)$ with ${\tilde{v}}^s\in {\mathrm{I\!R}}^{J_s+1}$ for $s=1,\ldots ,n$ such that $g\in {\mathcal {G}}({\tilde{v}})$. Thus,

$$\begin{aligned} {\mathcal {G}}=\bigcup _{v} {\mathcal {G}}(v)\;\text {and}\; {\mathcal {G}}_1(Q)=\bigcup _{v} ({\mathcal {G}}_1(Q)\cap {\mathcal {G}}(v)) \end{aligned}$$

and consequently we can write $\varphi (Q)$ as

$$\begin{aligned} \varphi (Q)={\mathcal {T}}_Q^{{\mathcal {G}}_1}({\varvec{X}}) :=\sup _{v} {\mathcal {T}}_Q^{{\mathcal {G}}(v)}({\varvec{X}}) \quad \text{ s.t. } \quad {\mathcal {G}}(v)\cap {\mathcal {G}}_1(Q)\ne \emptyset . \end{aligned}$$

Note that ${\mathcal {G}}(v)$ defines the set of all functions in ${\mathcal {G}}$ whose values on $\Xi ^s$ are $v^s=(v_0^s,v_1^s,\ldots ,v_{J_s}^s)$ for $s=1,\ldots ,n$, whereas ${\mathcal {G}}_{ce}(Q)$ is a set of specific functions in ${\mathcal {G}}$ which satisfy certainty equivalent conditions and their values are determined on a subset of $\Xi ^s$ for $s=1,\ldots ,n$. Moreover, since ${\mathcal {G}}(v)$ is determined by v, then for fixed v, either v satisfies the certainty equivalent conditions or not, which implies that either ${\mathcal {G}}(v)$ is a subset of ${\mathcal {G}}_{ce}(Q)$ or is disjoint from it. The same is true for the set ${\mathcal {G}}_{pair}(Q)$. Since ${\mathcal {G}}_1(Q)={\mathcal {G}}_{pair}(Q)\cap {\mathcal {G}}_{ce}(Q)\cap {\mathcal {G}}_{coh}(Q)$, it follows that

$$\begin{aligned} \varphi (Q)=\sup _{v}{} & {} {\mathcal {T}}_Q^{{\mathcal {G}}(v)\cap {\mathcal {G}}_{coh}(Q)}({\varvec{X}}) \end{aligned}$$

(A.4a)

$$\begin{aligned} \text{ s.t. }{} & {} {\mathcal {G}}(v)\cap {\mathcal {G}}_{coh}(Q)\ne \emptyset , {\mathcal {G}}(v)\subset {\mathcal {G}}_{pair}(Q), {\mathcal {G}}(v)\subset {\mathcal {G}}_{ce}(Q). \end{aligned}$$

(A.4b)

Step 3. Let ${\varvec{Y}}$ be a state-dependent act with consequence $Y_s$ being a finite discretely distributed non-negative random loss such that ${\mathbb {P}}(Y_s=y_i^s)=q_i^s$ for $i=1,\ldots ,L_s$ where $y_1^s<y_2^s<\cdots <y_{L_s}^s$ for $s=1,\ldots ,n$. We want to represent ${\mathcal {T}}_Q^g({\varvec{Y}})$ via $\{(1-\xi _j^s,v_j^s)\}_{j=0,1}^{J_s}$ for $s=1,\ldots ,n$ if the breakpoints of the quantile function of $Y_s$ are contained in $\Xi ^s$ for $s=1,\ldots ,n$. Since the quantile function of $Y_s$ is a step-like function with

$$\begin{aligned} F_{Y_s}^{\leftarrow }(t)=y_{l+1}^s,\;\text{ for }\; t\in (\pi _l^s,\pi _{l+1}^s]\;\text{ and }\;l=0,1,\ldots ,L_s, \end{aligned}$$

where $y_0^s=0$, $\pi _0^s=0$, $\pi _{L_s+1}^s=1$ and $\pi _l^s=\sum _{j\le l}q_j^s$ for $l=1,\ldots ,L_s$ and $s=1,\ldots ,n$. Moreover, since $\pi _l^s\in \Xi ^s$ for $l=1,\ldots ,L_s+1$ and $s=1,\ldots ,n$ and $\xi _j^s$ is the j-th smallest element in set $\Xi ^s$ for $s=1,\ldots ,n$, then by (4.2) we have

$$\begin{aligned} {\mathcal {T}}_Q^g({\varvec{Y}})= & {} \sum _{s=1}^n\sum _{j=1}^{L_s} q_s(y_j^s-y_{j-1}^s)g(1-\pi _{j-1}^s,s)\nonumber \\= & {} \sum _{s=1}^n\sum _{j=1}^{J_s}q_s[F_{Y_s}^{\leftarrow }(\xi _j^s)-F_{Y_s}^{\leftarrow }(\xi _{j-1}^s)]v_{j-1}^s. \end{aligned}$$

(A.5)

Consequently, ${\mathcal {T}}_Q^g({\varvec{W}}_k),{\mathcal {T}}_Q^g({\varvec{G}}_m)$ and ${\mathcal {T}}_Q^g({\varvec{B}}_m)$ have the similar representations of (A.5) for $k=1,\ldots ,K$ and $m=1,\ldots ,M$.

Step 4. We are now ready to reformulate problem (A.4). Let $h_j^s(1-\xi )=v_j^s+\beta _j^s[(1-\xi )-(1-\xi _j^s)]$ be a support function of the graph of $g(\cdot ,s)$ at point $(1-\xi _j^s,v_j^s)$ for $j=0,1,\ldots ,J_s$ and $s=1,\ldots ,n$. Note that here $v_0^s,v_1^s,\ldots ,v_{J_s}^s$ and $1-\xi _0^s,1-\xi _1^s,\ldots ,1-\xi _{J_s}^s$ for $s=1,\ldots ,n$ are both in nonincreasing order. Let ${\tilde{\xi }}_j^s=1-\xi _j^s$ for $j=0,1,\ldots ,J_s$ and $s=1,\ldots ,n$. Relabel the sequence ${\tilde{\xi }}_0^s,{\tilde{\xi }}_1^s,\ldots ,{\tilde{\xi }}_{J_s}^s$ in nondecreasing order and denote them by ${\hat{\xi }}_j^s$ for $j=0,1,\ldots ,J$ and $s=1,\ldots ,n$. Then ${\hat{\xi }}_j^s={\tilde{\xi }}_{J-j}^s$ for $j=0,1,\ldots ,J_s$ and $s=1,\ldots ,n$. Let ${\hat{v}}_j^s:=g({\hat{\xi }}_j^s,s)=g({\tilde{\xi }}_{J-j}^s,s)=g(1-\xi _{J-j}^s,s):=v_{J-j}^s$ for $j=0,1,\ldots ,J$ and $s=1,\ldots ,n$. Consequently, we can apply Lemma 4.1 with $({\hat{\xi }}_j^s, {\hat{v}}_j^s)$ for $j=0,1,\ldots ,J$ and $t_{j-1}^s=1-\pi _{j-1}^s$ for$j=1,\ldots ,L_s$ for each s. By Lemma 4.1 and (4.2), the objective function and constraint ${\mathcal {G}}(v)\cap {\mathcal {G}}_{coh}\ne \emptyset$ in program (A.4) can be reformulated as

$$\begin{aligned} {\mathcal {T}}_Q^{{\mathcal {G}}(v)\cap {\mathcal {G}}_{coh}(Q)}({\varvec{X}})=\sup _{\tau ,\beta }{} & {} \sum _{s=1}^n \sum _{i=1}^{n_s}(x_i^s-x_{i-1}^s)q_s\tau _{i-1}^s\\ \text{ s.t. }{} & {} (4.5d)-(4.5h). \end{aligned}$$

Constraint ${\mathcal {G}}(v)\subset {\mathcal {G}}_{pair}(Q)$ can be represented as (4.5c) and constraint ${\mathcal {G}}(v)\subset {\mathcal {G}}_{ce}(Q)$ by (4.5b). Consequently,

$$\begin{aligned} \varphi (Q)=\sup _{v,\beta ,\tau }{} & {} \sum _{s=1}^n \sum _{i=1}^{n_s}(x_i^s-x_{i-1}^s)q_s\tau _{i-1}^s \\ \text{ s.t. }{} & {} (4.5b)-(4.5h). \end{aligned}$$

Finally, based on the arguments in Step 1, ${\mathcal {T}}_{({\mathcal {G}},{\mathcal {Q}}_{\epsilon })_1}({\varvec{X}})$ is the optimal value of the following program:

$$\begin{aligned} \sup _{v,\beta ,\tau ,q}{} & {} \sum _{s=1}^n \sum _{i=1}^{n_s}(x_i^s-x_{i-1}^s)q_s\tau _{i-1}^s \\ \text{ s.t. }{} & {} (4.5b)-(4.5j). \end{aligned}$$

As for the bi-convexity of program (4.5), the bi-convexity of constraints is directly from Proposition 3.1 and the bi-linear property of the objective function is straightforward. $\square$

1.3 Construction of ambiguity set for the special case ${\mathcal {G}}_1\times {\mathcal {Q}}_{\epsilon }$

In the case when the ambiguity set of g and the ambiguity set of Q are independent, the construction of joint ambiguity set $({\mathcal {G}},{\mathcal {Q}}_{\epsilon })_1$ may be simplified. This is primarily because the distortion function at each state can be elicited only based on a DM’s preference information at such state.

For each state $s\in S$, let ${\mathcal {G}}^s$ denote the ambiguity set of plausible distortion functions of the DM with partially available information at state s. The ambiguity set of state-dependent distortion function can be represented as ${\mathcal {G}}=\times _{s\in S}{\mathcal {G}}^s$. Specifically, we introduce the following classes of distortion functions for the specified state $s\in S$. For each $s\in S$, let $\{G_m^s,B_m^s\}_{m=1}^{M_s}$ be a set of comparable lotteries. The set ${\mathcal {G}}^s_{pair}$ denotes the set of all state-dependent distortion functions at state s, for $m=1,\ldots ,M_s$, i.e.

$$\begin{aligned} {\mathcal {G}}^s_{pair}:=\{g_s\in {\mathcal {G}}^s:\rho _{g_s}(G_m^s)\le \rho _{g_s}(B_m^s) \;\text {for}\; m=1,\ldots ,M_s \}. \end{aligned}$$

For each $s\in S$, let $\{W_k^s\}_{k=1}^{K_s}$ be a list of lotteries with an associated set of “confidence” intervals $[{\underline{w}}_k^s,{\overline{w}}_k^s]$ for the “certainty equivalent” of each $W_k^s$. The set ${\mathcal {G}}_{ce}^s$ denotes the set of all state dependent distortion functions at state s, which evaluate the risk of each $W_k^s$ to be larger than ${\underline{w}}_k^s$ and lower than ${\overline{w}}_k^s$, i.e.,

$$\begin{aligned} {\mathcal {G}}_{ce}^s:=\{g_s\in {\mathcal {G}}^s: {\underline{w}}_k^s\le \rho _{g_s}(W_k^s)\le {\overline{w}}_k^s ,\;\text{ for }\;k=1,\ldots ,K_s\}. \end{aligned}$$

Let ${\mathcal {G}}_{coh}^s$ be the set of all concave state-dependent distortion function at the specified state $s\in S$. Then, we consider the following subset of the general ambiguity set ${\mathcal {G}}_1\times {\mathcal {Q}}_{\epsilon }$:

• ${\mathcal {G}}_1\times {\mathcal {Q}}_{\epsilon }:= \left[ \times _{s\in S}({\mathcal {G}}_{pair}^s\cap {\mathcal {G}}_{ce}^s\cap {\mathcal {G}}_{coh}^s)\right] \times {\mathcal {Q}}_{\epsilon }$.

The following example illustrates the fundamental difference between the general ambiguity set $({\mathcal {G}}_1,{\mathcal {Q}}_{\epsilon })$ and the special ambiguity set ${\mathcal {G}}_1\times {\mathcal {Q}}_{\epsilon }$ by comparing the difference between $({\mathcal {G}}_{pair},Q)$ and $(\times _{s\in S}{\mathcal {G}}_{pair}^s)\times Q$ for $Q\in {\mathcal {Q}}_{\epsilon }$ based on Example 3.1.

Example A.2

We now reconsider Example 3.1 with the states of nature $S=\{s_1,s_2\}$ and the number of the pairs of pairwise comparisons is $M=2$. A DM is asked to compare the following two sets of state-dependent acts:

$$\begin{aligned} {\varvec{G}}_1= & {} {\left\{ \begin{array}{ll} G_1^1=[1,\frac{1}{4}; 10,\frac{3}{4}] &{} \text{ for } \; s=s_1,\\ G_1^2=[2,\frac{1}{2};\;\;8,\frac{1}{4}] &{} \text{ for } \; s=s_2, \end{array}\right. }\\ \quad \text {and}\quad {\varvec{B}}_1= & {} {\left\{ \begin{array}{ll} B_1^1=[2,\frac{1}{5};12,\frac{3}{5}] &{} \text{ for } \; s=s_1,\\ B_1^2=[2,\frac{1}{5};10,\frac{3}{5}] &{} \text{ for } \; s=s_2, \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} {\varvec{G}}_2= & {} {\left\{ \begin{array}{ll} G_2^1=[1,\frac{1}{5}; 10,\frac{4}{5}] &{} \text{ for } \; s=s_1,\\ G_2^2=[3,\frac{1}{2};\;\;9,\frac{1}{4}] &{} \text{ for } \; s=s_2, \end{array}\right. }\\ \quad \text {and}\quad {\varvec{B}}_2= & {} {\left\{ \begin{array}{ll} B_2^1=[2,\frac{1}{4};12,\frac{1}{2}] &{} \text{ for } \; s=s_1,\\ B_2^2=[2,\frac{1}{5};10,\frac{3}{5}] &{} \text{ for } \; s=s_2. \end{array}\right. } \end{aligned}$$

Assume that the specified probability distribution of the DM’s belief of the states of nature $\{s_1,s_2\}$ is $Q=(\frac{1}{4},\frac{3}{4})$. In the following, we consider the differences between the two cases of pairwise comparison discussed in the preceding section: one is to compare the state-dependent acts ${\varvec{G}}_m$ and ${\varvec{B}}_m$ for $m=1,2$, the other is to compare the consequences $G_m^s$ and $B_m^s$ at each state $s=1,2$ for $m=1,2$.

Case 1. Assume that the DM prefers ${\varvec{G}}_m$ to ${\varvec{B}}_m$ for $m=1,2$, i.e. ${\mathcal {T}}_Q^g({\varvec{G}}_m)\le {\mathcal {T}}_Q^g({\varvec{B}}_m),\;\text{ for }\; m=1,\ldots ,2$. Thus, the general ambiguity set

$$\begin{aligned}{} & {} ({\mathcal {G}}_{pair},Q) \\{} & {} \quad =\left\{ (g,Q): \begin{aligned}&g=(g_1,g_2)\;\text {such that}\; g_1(\cdot )\;\text {and}\;g_2(\cdot )\;\text {are distortion functions}\;\text {and}\;Q=(\tfrac{1}{4},\tfrac{3}{4})\;\\&\text {with}\;\tfrac{1}{4}g_1(1)+\tfrac{9}{4}g_1(\tfrac{3}{4})+\tfrac{3}{2}g_2(\tfrac{3}{4})+\tfrac{9}{2}g_2(\tfrac{1}{4})\le \tfrac{1}{2}g_1(\tfrac{4}{5})+\tfrac{5}{2}g_1(\tfrac{3}{5})+\tfrac{3}{2}g_2(\tfrac{4}{5})+6g_2(\tfrac{3}{5}),\\&\text {and}\;\;\tfrac{1}{4}g_1(1)+\tfrac{9}{4}g_1(\tfrac{4}{5}) +\tfrac{9}{4}g_2(\tfrac{3}{4})+\tfrac{9}{2}g_2(\tfrac{1}{4}) \le 2g_1(\tfrac{3}{4})+\tfrac{5}{2}g_1(\tfrac{1}{2})+\tfrac{3}{2}g_2(\tfrac{4}{5})+6g_2(\tfrac{3}{5}). \end{aligned} \right\} . \end{aligned}$$

Case 2. Assume that the DM prefers $G_m^s$ to $B_m^s$ at each state for $m=1,2$, i.e. $\rho _{g_s}(G_m^s)\le \rho _{g_s}(B_m^s)$ at $s=1,2$ for $m=1,2$. Thus, the special ambiguity set

$$\begin{aligned}{} & {} {\mathcal {G}}_{pair}\times Q \\{} & {} \quad =\left\{ g\times Q: \begin{aligned}&g=(g_1,g_2)\;\text {such that}\; g_1(\cdot )\;\text {and}\;g_2(\cdot )\;\text {are distortion functions}\;\text {and}\;Q=(\tfrac{1}{4},\tfrac{3}{4})\\&\text {with}\; g_1(1)+9g_1(\tfrac{3}{4})\le 2g_1(\tfrac{4}{5})+10g_1(\tfrac{3}{5}),\; 2g_2(\tfrac{3}{4})+6g_2(\tfrac{1}{4})\le 2g_2(\tfrac{4}{5})+8g_2(\tfrac{3}{5}),\; \\&\text {and}\;\;g_1(1)+9g_1(\tfrac{4}{5})\le 2g_1(\tfrac{3}{4})+10g_1(\tfrac{1}{2}), \; 3g_2(\tfrac{3}{4})+6g_2(\tfrac{1}{4})\le 2g_2(\tfrac{4}{5})+8g_2(\tfrac{3}{5}). \end{aligned} \right\} . \end{aligned}$$

It is easy to observe that ${\mathcal {G}}_{pair}\times Q\subset ({\mathcal {G}}_{pair},Q)$.

Before ending this subsection, we give some properties of the defined ambiguity sets ${\mathcal {G}}_1\times {\mathcal {Q}}_{\epsilon }$, which will be beneficial for us to analyze the properties of the optimization programs of the tractable reformulation for the proposed PRSDRM based on ${\mathcal {G}}_1\times {\mathcal {Q}}_{\epsilon }$.

Proposition A.1

${\mathcal {G}}_1\times {\mathcal {Q}}_{\epsilon }$ is a convex set.

Proof

Given $(g_i,Q_i)\in {\mathcal {G}}_1\times {\mathcal {Q}}_{\epsilon }$ with $Q_i=(q_1^i,q_2^i,\ldots ,q_n^i)$ for $i=1,2$, let

$$\begin{aligned} (g_{\lambda },Q_{\lambda }):=\lambda (g_1,Q_1)+(1-\lambda )(g_2,Q_2)=(\lambda g_1+(1-\lambda )g_2, \lambda Q_1 +(1-\lambda ) Q_2) \end{aligned}$$

for an arbitrarily chosen $\lambda \in [0,1]$. It is easy to see that $g_{\lambda }$ is a state-dependent distortion function and $Q_{\lambda }$ is a probability distribution on S. Moreover, since $\phi$ is convex, then

$$\begin{aligned} \sum _{i=1}^n \phi \left( \frac{\lambda q_i^1+(1-\lambda )q_i^2}{q_i^0}\right) \le \lambda \sum _{i=1}^n \phi \left( \frac{q_i^1}{q_i^0}\right) +(1-\lambda ) \sum _{i=1}^n \phi \left( \frac{q_i^2}{q_i^0}\right) \le \epsilon . \end{aligned}$$

Consequently, $Q_{\lambda }\in {\mathcal {Q}}_{\epsilon }$. In the following, we show that $g_{\lambda }\in {\mathcal {G}}_1$. Since $g_i\in {\mathcal {G}}_1$, then by definition, we have that for each $s\in S$,

$$\begin{aligned} (g_i)_s\in {\mathcal {G}}_{pair}^s,\;(g_i)_s\in {\mathcal {G}}_{ce}^s,\;\text {and}\;(g_i)_s\in {\mathcal {G}}_{coh}^s. \end{aligned}$$

By the definition of DRM, we have that for any lottery X and $s\in S$

$$\begin{aligned} \rho _{(g_{\lambda })_s}(X)=\rho _{\lambda (g_1)_s +(1-\lambda ) (g_2)_s}(X)=\lambda \rho _{(g_1)_s}(X)+(1-\lambda )\rho _{(g_2)_s}(X). \end{aligned}$$

Consequently, $(g_{\lambda })_s\in {\mathcal {G}}_{pair}^s$ and $(g_{\lambda })_s\in {\mathcal {G}}_{ce}^s$ for $s\in S$. Moreover, since $(g_i)_s$ is a concave function on [0, 1] for $i=1,2$, then $(g_{\lambda })_s$ is also a concave function on [0, 1], i.e. $(g_{\lambda })_s\in {\mathcal {G}}_{coh}^s$. To summarize, we have showed that for each $s\in S$, $(g_{\lambda })_s\in {\mathcal {G}}_{pair}^s\cap {\mathcal {G}}_{ce}^s\cap {\mathcal {G}}_{coh}^s$, i.e. $g_{\lambda }\in \times _{s\in S} \left( {\mathcal {G}}_{pair}^s \cap {\mathcal {G}}_{ce}^s\cap {\mathcal {G}}_{coh}^s\right) ={\mathcal {G}}_1$. Finally, by the independence of the ambiguity sets ${\mathcal {G}}_1$ and ${\mathcal {Q}}_{\epsilon }$, we conclude that $(g_{\lambda },Q_{\lambda })\in {\mathcal {G}}_1\times {\mathcal {Q}}_{\epsilon }$. $\square$

1.4 Tractable reformulations for the special case

The tractable reformulation of PRSDRM under the special construction of the ambiguity set ${\mathcal {G}}_1\times {\mathcal {Q}}_{\epsilon }$ can also be simplified. Here we give the details.

1.4.1 Tractable reformulation of PRSDRM under ${\mathcal {G}}_1\times {\mathcal {Q}}_{\epsilon }$

In this subsection, we consider tractable reformulation of PRSDRM when the ambiguity set of state dependent distortion functions is constructed as follows:

$$\begin{aligned} {\mathcal {G}}_1\times {\mathcal {Q}}_{\epsilon }= & {} ({\mathcal {G}}_{pair}\cap {\mathcal {G}}_{ce}\cap {\mathcal {G}}_{coh})\times {\mathcal {Q}}_{\epsilon } = \left[ \times _{s\in S}({\mathcal {G}}_{pair}^s\cap {\mathcal {G}}_{ce}^s\cap {\mathcal {G}}_{coh}^s)\right] \nonumber \\{} & {} \times {\mathcal {Q}}_{\epsilon }=:\left[ \times _{s\in S}{\mathcal {G}}_1^s\right] \times {\mathcal {Q}}_{\epsilon }, \end{aligned}$$

(A.9)

where ${\mathcal {G}}_1^s={\mathcal {G}}_{pair}^s\cap {\mathcal {G}}_{ce}^s\cap {\mathcal {G}}_{cave}^s$ for $s\in S$. Then

$$\begin{aligned} {\mathcal {T}}_{{\mathcal {Q}}_{\epsilon }}^{{\mathcal {G}}_1}({\varvec{X}}): =\sup _{(g,Q)\in {\mathcal {G}}_1\times {\mathcal {Q}}_{\epsilon }} {\mathbb {E}}_{Q}[\rho _{g_s}(X_s)]. \end{aligned}$$

(A.10)

The program (A.10) is a semi-infinite optimization program and hard to solve in general. However, the following proposition says that for such construction of the ambiguity set ${\mathcal {G}}_1\times {\mathcal {Q}}_{\epsilon }$, the PRSDRM for a state-dependent act can be calculated by a two-stage optimization program.

Proposition A.2

For each $s\in S$, let $\psi _1(s)=\sup _{g_s\in {\mathcal {G}}_1^s}\rho _{g_s}(X_s)$. Then ${\mathcal {T}}_{{\mathcal {Q}}_{\epsilon }}^{{\mathcal {G}}_1}({\varvec{X}})=\sup _{Q\in {\mathcal {Q}}_{\epsilon }}{\mathbb {E}}_Q[\psi _1(s)]$.

Proof

Since the ambiguity sets ${\mathcal {G}}_1$ and ${\mathcal {Q}}_{\epsilon }$ are independent, then the PRSDRM of a state-dependent act ${\varvec{X}}$ under ${\mathcal {G}}_1\times {\mathcal {Q}}_{\epsilon }$ is equivalent to the optimal value of the following optimization problem:

$$\begin{aligned} {\mathcal {T}}_{{\mathcal {Q}}_{\epsilon }}^{{\mathcal {G}}_1}({\varvec{X}}) =\sup _{(g,Q)\in {\mathcal {G}}_1\times {\mathcal {Q}}_{\epsilon }} {\mathbb {E}}_{Q}[\rho _{g_s}(X_s)] =\sup _{Q\in {\mathcal {Q}}_{\epsilon }}\sup _{g\in {\mathcal {G}}_1}{\mathbb {E}}_{Q}[\rho _{g_s}(X_s)]. \end{aligned}$$

Moreover, since ${\mathcal {G}}_1=\times _{s\in S}{\mathcal {G}}_1^s$ and for each $s\in S$ the ambiguity set ${\mathcal {G}}_1^s$ only depends on the information of state s, then

$$\begin{aligned} \sup _{g\in {\mathcal {G}}_1}{\mathbb {E}}_{Q}[\rho _{g_s}(X_s)]={\mathbb {E}}_Q\left[ \sup _{g\in {\mathcal {G}}_1^s}\rho _{g_s}(X_s)\right] . \end{aligned}$$

This completes the proof. $\square$

Note that for each $s\in S$, $\sup _{g_s\in {\mathcal {G}}_1^s}\rho _{g_s}(X_s)$ is also a semi-infinite program in general. However, when $X_s$ is discretely distributed, it can be computed by solving a finite dimensional linear program, see, e.g. (Wang and Xu 2023, Theorem 4.1). Once $\psi _1(s)$ is determined for each $s\in S$, the program $\sup _{Q\in {\mathcal {Q}}_{\epsilon }}{\mathbb {E}}_Q[\psi _1(s)]$ would be a convex program and can be easily solved by convex optimization solver, see, e.g. CVX in Matlab.

In the followup discussions, for each $s\in S$, we will use $\Xi ^s$ to denote the set of all breakpoints of the quantile functions of $W_k^s$, $G_m^s$ and $B_m^s$ for $k=1,\ldots ,K_s$ and $m=1,\ldots ,M_s$ and label them in the increasing order of the values, i.e. we will use $\xi _j^s$ to denote the j-th smallest element of set $\Xi ^s$ and let $\xi _0^s=0$. For clarity of exposition, scenarios in $\Omega$ will be indexed by i and elements in $\Xi ^s$ will be indexed by j. Thus, the size of the optimization problem $\sup _{g_s\in {\mathcal {G}}_1^s}\rho _{g_s}(X_s)$ will be determined by the number of pairwise comparisons $M_s$, the number of certainty equivalent lotteries $K_s$, the size of scenarios $|\Omega |=n_s$, and the size of the total breakpoints $J_s=|\Xi ^s|$. The next proposition showed by Wang and Xu (2023) states that $\sup _{g_s\in {\mathcal {G}}_1^s}\rho _{g_s}(X_s)$ can be computed by solving a finite dimensional linear program of reasonable size when $X_s$ is finite discretely distributed (without loss of generality, we may assume that $X_s$ has $n_s$ different realizations) as it involves $2J_s+n_s+3$ variables and $K_s+M_s+n_sJ_s+n_s+3J_s+5$ constraints at most (not counting the non-negative constraints).

Proposition A.3

Let $X_s$ be a finite discretely distributed non-negative random loss with ${\mathbb {P}}(X_s=x_i^s)=p_i^s$ where $x_1^s<x_2^s<\cdots <x_{n_s}^s$. Then $\psi _1(s)$ is the optimal value of the following linear program:

$$\begin{array}{llr}\sup \limits_{v,\beta ,\tau }{} & {} \sum \limits_{i=1}^{n_s} (x_i^s-x_{i-1}^s) \tau _{i-1} & ({\text{A}}.11{\text{a}})\\ \\ \text{ s.t. }{} & {} {\underline{w}}_k^s\le \sum \limits_{j=1}^{J_s}[F_{W_k}^{\leftarrow }(\xi _j^s)-F_{W_k}^{\leftarrow }(\xi _{j-1}^s)] v_{j-1} \le {\overline{w}}_k^s,\;\text{ for }\; k=1,\ldots ,K_s, & ({\text{A}}.11{\text{b}})\\ \\{} & {} \sum \limits_{j=1}^{J_s}[F_{G_m^s}^{\leftarrow }(\xi _j^s)-F_{G_m^s}^{\leftarrow }(\xi _{j-1}^s)] v_{j-1}\le \sum \limits_{j=1}^{J_s}[F_{B_m^s}^{\leftarrow }(\xi _j^s)\nonumber \\{} & {} \quad \quad -F_{B_m^s}^{\leftarrow }(\xi _{j-1}^s)]v_{j-1},\; m=1,\ldots ,M_s, & ({\text{A}}.11{\text{c}})\\ \\{} & {} v_j+\beta _j(\xi _j^s-\pi _i)\ge \tau _i,\;\text{ for }\; i=0,1,\ldots ,n_s;j=0,1,\ldots ,J_s, & ({\text{A}}.11{\text{d}}) \\ \\{} & {} v_j+\beta _j(\xi _j^s-\xi _{j+1}^s)\ge v_{j+1},\;\text{ for }\; j=0,1,\ldots ,J_s-1, & ({\text{A}}.11{\text{e}}) \\ \\{} & {} v_j+\beta _{j+1}(\xi _j^s-\xi _{j+1}^s)\le v_{j+1},\;\text{ for }\; j=0,1,\ldots ,J_s-1, & ({\text{A}}.11{\text{f}}) \\ \\{} & {} \beta _j\ge 0,\;\text{ for }\; j=0,1,\ldots ,J_s, & ({\text{A}}.11{\text{g}}) \\ \\{} & {} v_0=1,v_{J_s}=0,\tau _0=1,\tau _{n_s}=0, \\ \end{array}$$

where $\pi _0=0$ and $\pi _i=\sum _{l\le i}p_l^s$ for $i=1,\ldots ,n_s$.

Proof

The proof is similar to that of Wang and Xu (2023, Theorem 4.1), we skip the details. $\square$

To summarize the above discussion, we have the following theorem:

Theorem A.1

Let ${\varvec{X}}$ be a state-dependent act with consequences $X_s$ being discretely distributed non-negative random loss with ${\mathbb {P}}(X_s=x_i^s)=p_i^s$ where $x_1^s<x_2^s<\cdots <x_{n_s}^s$ for $s\in S$. Then ${\mathcal {T}}_{{\mathcal {Q}}_{\epsilon }}^{{\mathcal {G}}_1}({\varvec{X}})$ is the optimal value of the following convex program:

$$\begin{array}{llr}\sup \limits_{q}{} & {} \sum \limits_{i=1}^{n} q_i \psi _1(s_i) & ({\text{A}}.12{\text{a}})\\ \\ \mathrm{{s.t.}}{} & {} \sum \limits_{i=1}^n q_i=1, \sum \limits_{i=1}^n \phi \left( \frac{q_i}{q_i^0}\right) q_i^0\le \epsilon ,\;\text {and}\;q_i\ge 0,\;\text {for}\;i=1,\ldots ,n. & ({\text{A}}.12{\text{b}}) \end{array}$$

1.5 Proof of Proposition 5.1

Proof

Let $(g,Q)\in ({\mathcal {G}},{\mathcal {Q}}_{\epsilon })_1$ be fixed and $\tau _i^s=g(1-\frac{i}{n_s},s)$ for $i=0,1,\ldots ,n_s$ and $s=1,\ldots ,n$. To ease the notation, we set $\tau _{-1}^s:=\tau _0^s$ for $s=1,\ldots ,n$. Based on (5.2), we define

$$\begin{aligned} v(z,\tau , q)= \sum _{s=1}^n\sum _{i=1}^{n_s} q_s \gamma _i^s\textrm{CVaR}_{\beta _i^s}(f_s(z,\xi _s)), \end{aligned}$$

where $\beta _i^s=\frac{i-1}{n_s}$, $\gamma _i^s=(\phi _i^s-\phi _{i-1}^s)(n_s-i+1)$, $\phi _i^s=\tau _{i-1}^s-\tau _i^s$ for $i=1,\ldots ,n_s$ and $s=1,\ldots ,n$. In other words, we have

$$\begin{aligned} \gamma _i^s=(2\tau _{i-1}^s-\tau _{i}^s-\tau _{i-2}^s)(n_s-i+1),\;\text {for}\; i=1,\ldots ,n_s \; \text {and}\; s=1,\ldots ,n. \end{aligned}$$

Since $f_s(z,\xi _s)$ is a convex function in z for every fixed $\xi _s$ and the operator $\textrm{CVaR}_{\beta _i^s}(\cdot )$ is non-decreasing and convex, then $\textrm{CVaR}_{\beta _i^s}(f_s(z,\xi _s))$ is a convex function of z. Moreover, since $g\in {\mathcal {G}}_{coh}$, then $\gamma _i^s\ge 0$ for $i=1,\ldots ,n_s$ and $s=1,\ldots ,n$. Thus, $v(z,\tau ,q)$ is a convex function in z for every fixed $\tau$ ( depending on g) and q. On the other hand, for fixed $z\in Z$, $v(z,\tau , q)$ is linear in both $\tau$ and q. Unfortunately, v is not linear in $(\tau ,q)$ jointly which prevents us from applying the existing minimax saddle point results in the proof. This motivates us to do some transformation of the variables. Specifically, we set ${\tilde{\tau }}_i^s=q_s\tau _i^s$ and $\tilde{\gamma }_i^s=(2{\tilde{\tau }}_{i-1}^s-{\tilde{\tau }}_{i}^s-{\tilde{\tau }}_{i-2}^s)(n_s-i+1)$ for $i=1,\ldots ,n_s\; \text {and}\; s=1,\ldots ,n$. Consequently, we can write $v(z,\tau , q)$ as

$$\begin{aligned} {\tilde{v}}(z,{\tilde{\tau }})=\sum _{s=1}^n\sum _{i=1}^{n_s} {\tilde{\gamma }}_i^s\textrm{CVaR}_{\beta _i^s}(f_s(z,\xi _s)). \end{aligned}$$

Note that ${\tilde{v}}(z,{\tilde{\tau }})$ is a convex function in z for every fixed ${\tilde{\tau }}$ ( depending on g and q). On the other hand, for fixed $z\in Z$, ${\tilde{v}}(z,{\tilde{\tau }})$ is linear in ${\tilde{\tau }}$. By Remark 4.1, we know that the joint ambiguity set $({\mathcal {G}},{\mathcal {Q}}_{\epsilon })_1$ can be represented by the constraints in Problem (4.7). Let $\Upsilon$ denote the feasible set of the problem and $\Upsilon _{{\tilde{\tau }}}$ the projection of the set $\Upsilon$ on the space of ${\tilde{\tau }}$ variable. Since $\Upsilon$ is convex and compact, then $\Upsilon _{{\tilde{\tau }}}$ is also convex and compact. Thus, we can recast problem (5.1) as

$$\begin{aligned} \min _{z\in Z}\max _{{\tilde{\tau }} \in \Upsilon _{{\tilde{\tau }}}} {\tilde{v}}(z,{\tilde{\tau }}). \end{aligned}$$

(A.13)

Note that set $\Upsilon _{{\tilde{\tau }}}$ is independent of z. By Fan (1953, Theorem 1 (ii)),

$$\begin{aligned} \min _{z\in Z} \max _{{\tilde{\tau }}\in \Upsilon _{\tau }} {\tilde{v}}(z,{\tilde{\tau }}) = \max _{{\tilde{\tau }}\in \Upsilon _{\tau }} \min _{z\in Z} {\tilde{v}}(z,{\tilde{\tau }}), \end{aligned}$$

(A.14)

which, by Karlin (1959, Corollary 1.3.1), is sufficient and necessary for the existence of a saddle point. Let $(z^*,{\tilde{\tau }}^*)$ denote the saddle point. Then

$$\begin{aligned} {\tilde{v}}(z^*,{\tilde{\tau }}) \le {\tilde{v}}(z^*,{\tilde{\tau }}^*) \le {\tilde{v}}(z,{\tilde{\tau }}^*). \end{aligned}$$

(A.15)

We are now ready to show the convergence of the sequence $\{(z^j,{\tilde{\tau }}^j)\}$ generated by Algorithm 5.1. For $j=1,2,\ldots$, it follows from Step 2 of Algorithm 5.1,

$$\begin{aligned} {\tilde{v}}(z^{j-1},{\tilde{\tau }}^j)\ge {\tilde{v}}(z^{j-1},{\tilde{\tau }}),\;\text {for all}\; {\tilde{\tau }}\in \Upsilon _{{\tilde{\tau }}}. \end{aligned}$$

(A.16)

Likewise, it follows from Step 3 of the algorithm

$$\begin{aligned} {\tilde{v}}(z^j,{\tilde{\tau }}^j) \le {\tilde{v}}(z,{\tilde{\tau }}^j),\;\text {for all}\; z\in Z. \end{aligned}$$

(A.17)

In the case when the algorithm terminates in finite steps, we have $z^{j-1}=z^j$ and ${\tilde{\tau }}^{j-1}={\tilde{\tau }}^j$ for some j and consequently $(z^j,{\tilde{\tau }}^j)$ satisfies (A.15).

Next, we consider the case that the algorithm generates an infinite sequence $\{(z^j,{\tilde{\tau }}^j)\}$. Let $({\hat{z}},\hat{{\tilde{\tau }}})$ be a cluster point of $\{(z^j,{\tilde{\tau }}^j)\}$. By taking a subsequence if necessary, we assume for the simplicity of notation that $(z^j,{\tilde{\tau }}^j)\rightarrow ({\hat{z}},\hat{{\tilde{\tau }}})$ as $j\rightarrow \infty$. Assume for the sake of a contradiction that $({\hat{z}},\hat{{\tilde{\tau }}})$ is not a solution to the problem (A.13). Then $({\hat{z}},\hat{{\tilde{\tau }}})$ would violate one of the inequalities in (A.15). Consider the case that the second inequality of (A.15) is violated. Then there exists $z_0\in Z$ such that

$$\begin{aligned} {\tilde{v}}({\hat{z}},\hat{{\tilde{\tau }}}) > {\tilde{v}}(z_0,\hat{{\tilde{\tau }}}). \end{aligned}$$

Since v is continuous, the inequality means that for j sufficiently large,

$$\begin{aligned} {\tilde{v}}(z^j,{\tilde{\tau }}^j)> {\tilde{v}}(z_0,{\tilde{\tau }}^j), \end{aligned}$$

which contradicts (A.17). Likewise, we can show that $({\hat{z}},\hat{{\tilde{\tau }}})$ satisfies the first inequality in (A.15). This completes the proof. $\square$

Appendix B Data generation

2.1 Sample data generation

In this part, we consider how to generate data set of the random loss of $m=3$ cities in $n=2$ states, i.e. to generate the random matrix $\varvec{\xi }=(\xi _1,\xi _2)\in {\mathrm{I\!R}}^{3\times 2}$. Based on Zhang et al. (2020), we also assume that there are three possible underlying loss scenarios corresponding to different levels of terrorist attacks: reduced loss, standard loss and increased loss, which means the size of sample space is $|\Omega |=3$. Moreover, we assume that the random loss for city j at state s is generated from a given distribution. Specifically, we assume that the random loss $\xi _j^s$ is decomposable into a state-based risk factor $\vartheta _s \sim U(0,100\times (2+s))$ common to all cities for $s=1,2$ and a city-based risk factor $\zeta _j \sim U(100\times (1+j\times 3\%), 200\times (1+j\times 5\%))$ specific to city j for $j=1,2,3$, where U means the uniform distribution over a specified range. Thus, we set

$$\begin{aligned} \xi _j^s=\vartheta _s+\zeta _j, \;\text {for}\; j=1,2,3\;\text {and}\;s=1,2. \end{aligned}$$

Note that for each s and j, after generating three losses from $\xi _j^s$, we may sort them in non-decreasing order and assign the smallest loss to the scenario reduced loss, the second smallest loss to the scenario standard loss, and the largest loss to the scenario increased loss. Moreover, we also assume that the probability to each scenario is 1/3. Table 3 presents the generated sample dataset.

Table 3 The random loss $\xi _j^s(\omega )$ for $j=1,2,3$, $s=1,2$ and $|\Omega |=3$

2.2 Elicited data generation

Potential questionnaires for eliciting a DM’s risk preference are given as follows.

• Q1. Which risky state-dependent act in the m-th pair of acts does the DM prefer for $m=1,\ldots ,M$?

  The preferred act is denoted by ${\varvec{G}}_m$ and the other is denoted by ${\varvec{B}}_m$. Note that when generating such acts, we require that the difference between the SDRMs of each pair of acts is greater than the specified threshold denoted by $\kappa$ in order to make sure that the compared acts having significant differences and the DM can easily choose which one he prefers to. In our setting, we set $\kappa =1$. The decision is made based on the values of SDRMs for each acts induced by the true state-dependent distortion function $g_1^*$ and the true probability distribution of the belief of the states of nature, and the smaller is preferred.

• Q2. What is the smallest amount of cash, denoted by $w_k^+$, that the DM would decline to pay instead of being exposed to the risk of ${\varvec{W}}_k$ and what is the largest amount of cash, denoted by $w_k^-$, that the DM would be willing to commit instead of being exposed to the risk of ${\varvec{W}}_k$ for $k=1,\ldots ,K$? In this kind of questionnaire, we assume that $w_k^+$ and $w_k^-$ are determined based on the values of SDRMs for each acts induced by the true distortion function $g_1^*$ and the true probability distribution of the belief of the states of nature as follows:

  $$\begin{aligned} w_k^+=(1+r){\mathcal {T}}_{Q^*}^{g_1^*}({\varvec{W}}_k),\quad w_k^-=(1-r){\mathcal {T}}_{Q^*}^{g_1^*}({\varvec{W}}_k), \end{aligned}$$

  where r is randomly generated from the uniform distribution on $[0,1\%]$.

• Q3. Experts reach a consensus that the probability distribution of the states of nature is $Q^*$, does the DM agree with such judgement to some extent?

  Based on the DM’s answers, we may set the level of $\epsilon$ in ${\mathcal {Q}}_{\epsilon }$ based on the DM’s confidence level to experts’ judgement. For example, if the DM fully agrees with such a judgement, then $\epsilon =0$; if the DM has $\alpha \%$ confidence to agree with the experts’ judgement, then we may set $\epsilon =1-\alpha \%$, for example, we may choose $\alpha \in \{99,97.5,95\}$, i.e. $\epsilon \in \{0.01,0.025,0.05\}$.

In order to make the elicited data comparable with the random loss of $\xi _j^s$, we generate the loss $x_1,x_2$ from the uniform distribution over

$$\begin{aligned} {}[r_1\times \min _{s,j,\omega }\xi _j^s(\omega ), r_2\times \max _{s,j,\omega }\xi _j^s(\omega )] \end{aligned}$$

where $r_1<1<r_2$. In our experiment, we set $r_1=0.5$ and $r_2=1.2$ and then the range to generate the realization of $x_1$ and $x_2$ is [188.87, 695.95]. Moreover, $p_1,p_2$ are also randomly generated from the uniform distribution on (0, 1). We give an example of the generated data for preference elicitation in Table 4.

Table 4 An example of generated data for preference elicitation