2.1. The market

Let $\mathcal{X}$ be the set of bounded real-valued measurable functions on a given measurable space $\left(\Omega,\mathcal{F}\right)$ , and let $\mathcal{X}_+$ denote its nonnegative elements. There are $n\in\mathbb{N}$ agents wishing to share their endowed risks among themselves without any central authority involvements. For each $i \in \mathcal{N} \,:\!=\, \{1,\ldots,n\}$ , let $X_i \in \mathcal{X}_+$ denote the nonnegative loss of the ith agent. We consider a one-period, risk-sharing economy, where all risks are realized at the end of the period. The risk-sharing mechanism in this market is as follows.

For each $i \in \mathcal{N}$ , agent i pays ex ante the contribution amount $\pi_i\in\mathbb{R}$ to a pool. At the end of the period, the aggregate loss $S\,:\!=\, \sum_{i=1}^{n}X_i$ is covered by the aggregate amount $\sum_{i=1}^{n}\pi_i$ in the pool. The residual aggregate loss to be shared among the agents is thus given by $\widetilde S \,:\!=\, \sum_{i=1}^{n}X_i-\sum_{i=1}^{n}\pi_i$ . If $\widetilde S \geq 0$ , the pool is subject to a residual aggregate loss; whereas if $\widetilde S \leq 0$ , the pool has a monetary surplus to be shared among the agents at the end of the period. Let $Y_i$ be the shared loss by the ith agent, $i\in\mathcal{N}$ , from the residual aggregate loss $\widetilde S$ . For each $i \in \mathcal{N}$ , the end-of-period, post-transfer risk exposure of agent i is thus given by $Y_i+\pi_i$ .

We assume that the market is a comonotone risk-sharing market, as previously studied by Boonen et al. (Reference Boonen, Liu and Wang2021). This is a market in which the admissible contracts are only those that are comonotone with the aggregate initial risk. Under this restriction, the contracts all satisfy the no-sabotage condition of Carlier and Dana (Reference Carlier and Dana2003, Reference Carlier and Dana2005), which guarantees that no agent has an incentive to misreport their actual realized loss. This is one possible justification for the existence of such a market, since preventing ex-post moral hazard is in the best interest of every agent. An alternative justification arises from the situation where each agent’s preference is monotone with respect to the convex order. In this case, it is well known that Pareto optima are comonotone (e.g., Ghossoub and Zhu, Reference Ghossoub and Zhu2024a).

The set of ex ante admissible decision variables is therefore given by

\begin{align*} \mathcal{A}\,:\!=\, \left\{\left(\left\{Y_i\right\}_{i=1}^{n},\left\{\pi_i\right\}_{i=1}^{n}\right)\in\mathcal{X}_+^n\times\mathbb{R}^{n}\,:\,\sum_{i=1}^{n}(Y_i+\pi_i)=\sum_{i=1}^{n}X_i,\,\{Y_i\}_{i=1}^n\mbox{ is comonotone}^{1}\right\}.\end{align*}

The preferences of each agent i, for $i\in\mathcal{N}$ , are represented by given risk measures $\rho_i$ . Consequently, each agent i measures their pre-transfer risk exposure by $\rho_i\left(X_i\right)$ , and their post-transfer risk exposure by $\rho_i\left(Y_i+\pi_i\right)$ . We recall some properties of risk measures below.

We will assume throughout that all risk measures are monotone.

2.2. Pareto optimality in peer-to-peer arrangements

First, we show that all comonotone allocations are translations of a suitable non-decreasing function of the aggregate endowment. To this end, define the set $\mathcal{G}$ by the following.

\begin{align*}\mathcal{G}\,:\!=\, \left\{\left\{g_i\right\}_{i=1}^{n}\,\middle|\,g_i \,:\, \mathbb{R}_+\rightarrow\mathbb{R}_+\text{ non-decreasing Borel-measurable},\text{ and }\sum_{i=1}^{n}g_i\left(\cdot\right)=\text{Id}\right\}.\end{align*}

Note that if $\{g_i\}_{i=1}^n\in\mathcal{G}$ , then each $g_i$ is 1-Lipschitz. Furthermore, since each $g_i$ is non-negative, we have $g_i(0)=0$ for all $i\in\mathcal{N}$ .

Lemma 2.5. Let $(\{Y_i\}_{i=1}^n,\{\pi_i\}_{i=1}^n)\in\mathcal{A}$ . Then there exist functions $\{g_i\}_{i=1}^n\in\mathcal{G}$ and constants $\{c_i\}_{i=1}^n\in\mathbb{R}^n$ such that

\[Y_i+\pi_i=g_i(S)+c_i\quad\forall i\in\mathcal{N}, \]

and $\sum_{i=1}^nc_i=0$ .

In the case where every agent uses a translation invariant risk measure, we have a characterization of PO in terms of solutions of a minimization problem. Let $\mathcal{S}$ denote the set of all solutions to the problem

(2.1) \begin{equation} \inf_{(\{Y_i\}_{i=1}^n,\{\pi_i\}_{i=1}^n)\in\mathcal{IR}} \ \sum_{i=1}^{n}\rho_i(Y_i+\pi_i) .\end{equation}

3.1. Robust distortion risk measures

In this section, we provide an explicit characterization of Pareto-optimal allocations when agents use robust DRMs. These are defined below.

Definition 3.1. A risk measure $\varrho\,:\,\mathcal{X}\to\mathbb{R}$ is a robust distortion risk measure if there exists a set of distortion functions $\mathcal{T}$ and a probability measure $\mathbb{P}$ on $(\Omega,\mathcal{F})$ such that for all random variables Z on the probability space $(\Omega,\mathcal{F},\mathbb{P})$ ,

\[ \varrho(Z)=\sup_{T\in\mathcal{T}}\int Z\,dT\circ\mathbb{P} . \]

Robustness in the context of Definition 3.1 refers to the consideration of different distortion functions by the decision maker, not different probability distributions on the underlying measurable space. Our definition is in line with that of Wang and Xu (Reference Wang and Xu2023), who argue that these risk measures can be motivated by the ambiguity present in each agent’s preference. We note that the use of the term robust distortion risk measure is not consistent throughout the literature. For instance, Bernard et al. (Reference Bernard, Pesenti and Vanduffel2023) use this term to refer to the case where the robustness relates to uncertainty in the distribution of the risk. In contrast, Definition 3.1 assumes that the distribution of a given risk is fixed with respect to a reference probability measure.

Lemma 3.2. For a robust distortion risk measure with respect to a set of distortion functions $\mathcal{T}$ , we have

\[ \sup_{T\in\mathcal{T}}\int Z\,dT\circ\mathbb{P}=\sup_{T\in\overline{\mathrm{co}}(\mathcal{T})}\int Z\,dT\circ\mathbb{P}, \]

where $\overline{\mathrm{co}}(\mathcal{T})$ is the closed convex hull of $\mathcal{T}$ with respect to pointwise convergence.

We now assume that for each $i\in\mathcal{N}$ , the risk measure $\rho_i$ is a robust distortion risk measure. We allow for each agent to have heterogeneous beliefs, represented by different probability measures $\mathbb{Q}_i$ on the same measurable space. Specifically, we assume that for each $i\in\mathcal{N}$ , there exists a set of distortion functions $\mathcal{T}_i$ such that

(3.1) \begin{equation} \rho_i(Z)=\sup_{T_i\in\mathcal{T}_i}\int Z\,dT_i\circ\mathbb{Q}_i, \end{equation}

for all risks $Z\in\mathcal{X}$ . By Lemma 3.2, we may assume without loss of generality that each $\mathcal{T}_i$ is convex and closed with respect to pointwise convergence.

Note that DRMs are translation invariant. For translation-invariant risk measures, a characterization of Pareto-optimal allocations is given by Proposition 2.6. However, in the case of robust DRMs, a more explicit characterization of the set $\mathcal{S}$ is possible, as shown by the following result.

In other words, Theorem 3.3 provides a necessary condition for an allocation to be Pareto-optimal allocation. This result is similar to the characterization obtained by Ghossoub and Zhu (Reference Ghossoub and Zhu2024a, Theorem 4.15). However, a key difference is that the present result of Theorem 3.3 does not require concavity of the distortion function (or equivalently, the convexity of the risk measure). The present result also holds under a setting of heterogeneous beliefs, as represented by the different probability measures $\mathbb{Q}_i$ . These stronger conclusions can be obtained in the setting of this paper because the admissible allocations are constrained to be comonotone, which was not the case in Ghossoub and Zhu (Reference Ghossoub and Zhu2024a). The assumption that the risk-sharing market is comonotone is necessary for Theorem 3.3, since there does not necessarily exist a comonotone Pareto-optimal allocation when risk measures are not convex.

Theorem 3.3 can also be seen as an extension of previous characterizations of Pareto optima for DRMs: see Liu (Reference Liu2020, Theorem 3.3) and Ghossoub and Zhu (Reference Ghossoub and Zhu2024a, Corollary 5.3). Indeed, if the set $\mathcal{T}_i$ is a singleton, then a robust distortion risk measure reduces to an ordinary distortion risk measure. The main technical difficulty introduced by robustness is that a solution $(T_1^*,\ldots,T_n^*)$ to problem (3.2) must be obtained, and there is no closed-form expression for such a solution. Nonetheless, once $(T_1^*,\ldots,T_n^*)$ is determined, the characterization for Pareto optima can be obtained explicitly in terms of $(T_1^*,\ldots,T_n^*)$ .

In particular, Pareto-optimal allocations are obtained as translations of suitable functions of the aggregate risk. Part (ii) of Theorem 3.3 decomposes the optimal risk allocation for agent i into a constant $c_i^*$ and a risky portion $g_i^*(S)$ , which is normalized so that $g_i^*(0)=0$ . Additionally, for any fixed $\{g_i^*\}_{i=1}^n$ given by Theorem 3.3, it is always possible to choose $\{c_i^*\}_{i=1}^n\in\mathbb{R}^n$ such that $\{g_i^*(S)+c_i^*\}_{i=1}^n\in\mathcal{IR}$ . One such choice is given by

(3.3) \begin{equation} c_i^*\,:\!=\, \rho_i(X_i)-\rho(g_i^*(S)), \quad i\in\{1,\ldots,n-1\}, \quad\quad c_n^*\,:\!=\, -\sum_{i=1}^{n-1}c_i^* . \end{equation}

Then by translation invariance of each $\rho_i$ , we have $\rho_i(X_i)-\rho(g_i^*(S)+c_i^*)=0$ for $i\in\{1,\ldots,n\}$ . Furthermore,

\begin{align*} \rho_n(X_n)-\rho_n(g_n^*(S)+c_n^*)&=\rho_n(X_n)-\rho_n(g_n^*(S))+\sum_{i=1}^{n-1}(\rho_i(X_i)-\rho(g_i^*(S)))\\ &=\left(\sum_{i=1}^n\rho_i(X_i)\right)-\left(\sum_{i=1}^n\rho_i(g_i^*(S))\right)\ge0, \end{align*}

where the last inequality follows since $\{g_i^*(S)+c_i^*\}_{i=1}^n\in\mathcal{S}$ and the no-risk-sharing arrangement $\{X_i\}$ is individually rational.

More generally, the role of the constants $c_i^*$ is to determine the distribution of the aggregate welfare gain from P2P insurance among the agents in the market. We define the aggregate welfare gain as the difference between the aggregate valuation of the individuals’ initial risks, given by $\sum_{i=1}^n\rho_i^*(X_i)$ , and the aggregate valuation of the agents’ post-transfer risks, given by $\sum_{i=1}^n\rho_i(g_i^*(S)+c_i^*)$ :

\begin{align*} W\,:\!=\, \sum_{i=1}^n\left(\rho_i(X_i)-\rho_i(g_i^*(S)+c_i^*)\right)=\sum_{i=1}^n\left(\rho_i(X_i)-\rho_i(g_i^*(S))\right). \end{align*}

Note that $\sum_{i=1}^n\rho_i(g_i^*(S)+c_i^*)$ is the value of problem (2.1), and therefore does not depend on the choice of the Pareto-optimal allocation $\{g_i^*(S)+c_i^*\}_{i=1}^n$ . Hence, the aggregate welfare gain W is well-defined, and $W\ge0$ . Now let $\{w_i\}_{i=1}^n\in\mathbb{R}_n$ be such that $\sum_{i=1}^nw_i=W$ and $w_i\ge0$ for all $i\in\mathcal{N}$ . Fix $\{g_i^*\}_{i=1}^n$ satisfying Theorem 3.3 and define

\[ c_i^*\,:\!=\, \rho_i(X_i)-\rho(g_i^*(S))-w_i, \quad \forall i\in\mathcal{N} . \]

By construction, we have $\sum_{i=1}^nc_i^*=\left(\sum_{i=1}^n\rho_i(X_i)\right)-\left(\sum_{i=1}^n\rho(g_i^*(S))\right)-W=0$ . The allocation $\{g_i^*(S)+c_i^*\}$ is the risk-sharing arrangement that provides a welfare gain of $w_i$ to each agent i, in the sense that

\[ \rho_i(X_i)-\rho(g_i^*(S)+c_i^*)=w_i\ge0 . \]

Hence, $\{g_i^*(S)+c_i^*\}\in\mathcal{IR}$ . Note that the above example (3.3) corresponds to the arrangement that allocates the entire welfare gain to agent n: that is, $(w_1,\ldots,w_n)=(0,0,\ldots,W)$ .

3.2. Common parametric families of distortion functions

When each set of distortion functions $\mathcal{T}_i$ is a singleton, there is no robustness in the risk measure of each agent, and we recover the simpler setting of P2P risk-sharing with DRMs. Suppose that $\mathcal{T}_i=\{T_i\}$ for all $i\in\mathcal{N}$ . As a special case of Theorem 3.3, we obtain a closed-form solution for Pareto optima that is known in the literature (e.g., Liu, Reference Liu2020, Theorem 3.3). This result is restated below.

Corollary 3.4. Suppose that for each $i\in\mathcal{N}$ , the risk measure $\rho_i$ is given by a distortion risk measure. Then an allocation $(\{Y_i^*\}_{i=1}^n,\{\pi_i^*\}_{i=1}^n)$ is Pareto optimal if and only if

\[ Y_i^*+\pi^*_i=g_i^*(S)+c_i^*, \]

where $\{c_i^*\}_{i=1}^n\in\mathbb{R}^n$ is chosen such that $\sum_{i=1}^nc_i^*=0$ and $\{g_i^*(S)+c_i^*\}_{i=1}^n\in\mathcal{IR}$ , and $\{g_i^*\}_{i=1}^n\in\mathcal{G}$ can be written in terms of the integrals of suitable functions $h_i$ . Specifically, for each $i\in\mathcal{N}$ , we can write $g_i^*(x)=\int_0^xh_i(z)\,dz$ , where each $h_i \,:\,\mathbb{R}_+\to[0,1]$ is a function such that for almost every $x\in\mathbb{R}_+$ ,

\[ \sum_{i\in L_x}h_i(x)=1\mbox{ and }\sum_{i\in L_x^C}h_i(x)=0, \]

where

\begin{align*} L_x&\,:\!=\, \left\{i\in\mathcal{N}\,:\, T_i(\mathbb{Q}_i(S\unicode{x003E}x))=\min_{j\in\mathcal{N}}\{T_j(\mathbb{Q}_j(S\unicode{x003E}x))\}\right\}, \\ L_x^C&=\mathcal{N}\setminus L_x . \end{align*}

In this subsection, we examine the consequences of Corollary 3.4 when agents’ preferences are determined by common parametric families of probability distortion functions. First, suppose that each $T_i$ is an inverse S-shaped distortion function from a parametric family, and let $\alpha_i$ denote the value of the parameter that determines $T_i$ . It can be verified that certain values of the parameter $\alpha_i$ are “more S-shaped,” which indicates a larger deviation from risk neutrality. To facilitate an economically meaningful comparison, we provide the following definition, which can be interpreted as a proxy for the level of risk aversion for a distortion function.

Definition 3.5 (Probabilistic Risk Aversion Index). For a given distortion $T_i$ assumed to be twice differentiable, we define the Probabilistic Risk Aversion Index of $T_i$ to be

\[ PRA_i(t)\,:\!=\, -\frac{T_i^{\prime\prime}(t)}{T_i^\prime(t)} . \]

This index is similar to the index of ambiguity aversion proposed by Carlier and Dana (Reference Carlier and Dana2008). Indeed, if $T_2=g\circ T_1$ where g is a concave function, then it can be verified that $PRA_2\ge PRA_1$ . Hence, $T_2$ is more risk averse than $T_1$ , which is consistent with the notion that the risk aversion of DRMs is due to concavity of the distortion function (see Yaari Reference Yaari1987 for more on this interpretation). Alternatively, we may use an index of relative probabilistic risk aversion, as defined below.

Definition 3.6 (Relative Probabilistic Risk Aversion Index). For a given distortion $T_i$ assumed to be twice differentiable, we define the Relative Probabilistic Risk Aversion Index of $T_i$ to be

\[RPRA_i(t)\,:\!=\, t\times PRA_i(t)=-\frac{t\,T_i^{\prime\prime}(t)}{T_i^\prime(t)} .\]

One example of inverse S-shaped distortion functions is the parametric family of Prelec-1 distortion functions introduced by Prelec (Reference Prelec1998), who showed that these distortion functions are inverse-S shaped with a unique inflection point at $(e^{-1},e^{-1})$ . These distortion functions are of the form

\[T_i(t)=\exp({-}({-}\ln(t))^{\alpha_i}), \]

where $\alpha_i\in(0,1)$ .

In the following, suppose that each $\rho_i$ is a distortion risk measure with respect to a Prelec-1 distortion function with parameter $\alpha_i$ . Let $j,k\in\mathcal{N}$ such that $\alpha_j=\max_{i\in\mathcal{N}}\alpha_i$ and $\alpha_k=\min_{i\in\mathcal{N}}\alpha_i$ .

Proposition 3.7. For $t\in(0,e^{-1})$ , the probabilistic risk aversion index $PRA_i(t)$ is larger for smaller values of the parameter $\alpha_i$ . In particular,

\[PRA_k(t)\ge PRA_i(t)\ge PRA_j(t) .\]

This result can be interpreted in the following manner. Recall that in the context of DRMs, each distortion function $T_i$ is applied to the survival function $\mathbb{P}(S\unicode{x003E}x)$ . Hence, the argument of the distortion $t\in(0,e^{-1})$ represents the probability of a tail event. Proposition 3.7 then implies that agent k is most averse to tail events, provided that the probability of this tail event is no more than $e^{-1}\approx0.3679$ .

Proposition 3.8. Suppose each $\rho_i$ is distortion risk measure given by a Prelec-1 distortion function with parameter $\alpha_i$ , and let $\alpha_j=\max_{i\in\mathcal{N}}\alpha_i$ and $\alpha_k=\min_{i\in\mathcal{N}}\alpha_i$ . Then a Pareto-optimal allocation $(\{Y_i^*\}_{i=1}^n,\{\pi_i^*\}_{i=1}^n)$ is given by

\[Y_i^*+\pi_i^*=g_i^*(S)+c_i^*, \]

where $\{c_i\}_{i=1}^n\in\mathbb{R}^n$ is chosen so that the allocation is $\mathcal{IR}$ , and

\begin{align*} g_k^*(x)&=\min\{x,d^*\}, \\ g_j^*(x)&=\max\{x-d^*,0\}, \\ g_i^*(x)&=0,\mbox{ for }i\in\mathcal{N}\setminus\{j,k\} . \end{align*}

where $d^*=\mathrm{VaR}_{e^{-1}}(S)$ .

Consequently, when all agents use a Prelec-1 distortion function, a possible PO allocation is a translate of the contract where agent k provides full coverage of the aggregate risk up to a limit $d^*$ , and the excess aggregate loss beyond the level $d^*$ is covered by agent j. By Proposition 3.7, agent k exhibits the most risk aversion toward tail events. This is reflected by the form of Pareto optima in this case, since agent k does not cover losses beyond the deductible. Instead, the tail risk is assumed by agent j, who exhibits the least risk aversion toward tail events.

Prelec (Reference Prelec1998) also defines a two-parameter family of distortion functions (Prelec-2 distortion functions), by

\[T_i(t)=\exp({-}\beta_i({-}\ln(t))^{\alpha_i}), \]

where $\alpha_i\in(0,1)$ and $\beta_i\unicode{x003E}0$ . It can be verified that these distortion functions have an inflection point at $(e^{-1},e^{-\beta_i})$ . In this case, we have a result similar to that of Proposition 3.7.

Hence, if there exists an agent k such that $\alpha_k=\min_{i\in\mathcal{N}}\alpha_i$ and $\beta_k=\min_{i\in\mathcal{N}}\beta_i$ , then this agent exhibits the most risk aversion to tail events. A similar result to that of Proposition 3.8 is given below.

Part (i) of Proposition 3.10 is a similar result to that of Proposition 3.8 in the case where every agent has the same second parameter. That is, the agent with the largest aversion to tail events provides coverage up to a deductible. Part (ii) states that if every agent uses the same first parameter $\alpha$ , then the agent with the least aversion to tail events provides full insurance.

Similar results can be obtained for the family of S-shaped distortion functions first introduced by Kahneman and Tversky (Reference Kahneman and Tversky1979), Tversky and Kahneman (Reference Tversky and Kahneman1992):

\[T_i(t)=\frac{t^{\gamma_i}}{\left(t^{\gamma_i}+(1-t)^{\gamma_i}\right)^{1/\gamma_i}}, \]

where $\gamma_i\in(0.279,1]$ . Let $\gamma_j=\max_{i\in\mathcal{N}}\gamma_i$ and $\gamma_k=\min_{i\in\mathcal{N}}\gamma_i$ .

Remark 3.11. It can be numerically verified that for any $t\in(0,\varepsilon)$ for a sufficiently small $\varepsilon\unicode{x003E}0$ , the probabilistic risk aversion index $PRA_i(t)$ is larger for smaller values of the parameter $\gamma_i$ . The probabilistic risk aversion index is plotted in Figure 1(b). In particular,

\[PRA_k(t)\ge PRA_i(t)\ge PRA_j(t) .\]

Again, since small values of t correspond to tail events, we see that agent k exhibits the most aversion to the risk of tail events, whereas agent j exhibits the least aversion.

Figure 1.

Kahneman–Tversky inverse S-shaped distortion functions.

It can be shown using numerical methods that there exists $t^*\in(0,1)$ for which

\begin{align*} j&\in L_x, \mbox{ if }\,\mathbb{P}(S\unicode{x003E}x)\le t^*, \\ k&\in L_x, \mbox{ if }\,\mathbb{P}(S\unicode{x003E}x)\unicode{x003E}t^* .\end{align*}

Distortion functions for different values of the parameter $\gamma$ are shown in Figure 1(a). It is clear from the figure that the minimum of the distortion functions is obtained by either the agent with the largest parameter (agent j) or by the agent with the smallest parameter (agent k). Thus, by the same arguments as those of Proposition 3.8, it follows that a PO allocation of a similar form exists. That is, consider the contract such that agent k provides full coverage of the aggregate risk up to a limit $d^*$ , and the excess aggregate loss beyond the level $d^*$ is covered by agent j. Then a suitable translate of this contract is PO.

4.1. A centralized insurance market

First, we revisit the scenario of a centralized insurance market, in which a single monopolistic insurer is the sole provider of insurance. In the context of flood risk in the United States, this insurer is the National Flood Insurance Program (NFIP)Footnote ² overseen by the Federal Emergency Management Agency (FEMA). Under the NFIP, the federal government of the United States is ultimately responsible for financial gains and losses.

A comprehensive numerical study of flood risk insurance in this centralized market is performed by Boonen et al. (Reference Boonen, Chong and Ghossoub2024), using a public dataset of claims statistics dating back to the year 1978. They find that by aggregating the flood risk across geographically distant regions, a centralized market is able to realize a significant welfare gain. In the following, we provide a similar but updated analysis of a centralized market. First, the public NFIP dataset has recently been updated to include the amount of financial loss for each flood event, instead of only including the claim amount. As a result, we no longer need to use the claim amount as a proxy for the loss amount. Additionally, Ghossoub and Zhu (Reference Ghossoub and Zhu2024b) are able to characterize Pareto optima without requiring that each agent uses a coherent risk measure. As a result, we are able to identify Pareto-optimal contracts under more general conditions. Finally, in our example, the risk measure of the central agent will be given by the expected shortfall (ES), which is the most popular coherent risk measure in practice. We recall the following standard definitions:

Definition 4.1. The value-at-risk (VaR) at level $\alpha\in(0,1)$ of a random variable $Z\in\mathcal{X}$ under the probability measure $\mathbb{P}$ is

\[\mathrm{VaR}_\alpha^{\mathbb{P}}(Z)\,:\!=\, \inf_{t\in\mathbb{R}}\left\{\mathbb{P}(Z\unicode{x003E}t)\le\alpha\right\} .\]

Definition 4.2. The ES at level $\alpha\in(0,1)$ of a random variable $Z\in\mathcal{X}$ under the probability measure $\mathbb{P}$ is

\[\mathrm{ES}_\alpha^{\mathbb{P}}(Z)\,:\!=\, \frac{1}{\alpha}\int_0^\alpha\mathrm{VaR}_u^{\mathbb{P}}(Z)\,du .\]

In the context of centralized insurance markets with n agents and a central insurer, an insurance contract is a pair $\left(\{I_i\}_{i=1}^n,\{\pi_i\}_{i=1}^n\right)\in\mathcal{I}^n\times\mathbb{R}^n$ , where $\mathcal{I}$ denotes an admissible set of indemnity functions (usually monotone and 1-Lipschitz). Here, $I_i$ represents the amount of coverage (i.e., the indemnity) that agent i receives from the central insurer, in exchange for a premium of the amount $\pi_i$ . In other words, agent i cedes the risk $I_i(X_i)$ to the central insurer. We denote the retained risk of agent i by $R_i(X_i)\,:\!=\, X_i-I_i(X_i)$ .

The concepts of individual rationality and Pareto optimality for centralized markets are defined similarly to IR and PO in decentralized markets (see Definition 2.4). In particular, IR allocations for centralized markets also incentivize the central insurer to participate. A characterization of Pareto- optimal contracts in a centralized market is provided by the following result, which extends the result of Ghossoub and Zhu (Reference Ghossoub and Zhu2024b, Corollary 3.7) to the case of robust DRMs.

For ease of comparison with Boonen et al. (Reference Boonen, Chong and Ghossoub2024) and Ghossoub and Zhu (Reference Ghossoub and Zhu2024b), we will adopt their model parameters for the remainder of this section. Specifically, we will assume that the distribution of the financial losses arising from floods in a given month adheres to the historical data. Specifically, we assume a discrete probability space $(\Omega,\mathcal{F},\mathbb{P})$ , where $\Omega$ has 553 states, corresponding to each of the months from January 1978 to January 2024. The measure $\mathbb{P}$ assigns an equal probability to each of the states in $\Omega$ . The risk measures used by each agent are DRMs on this probability space. That is, for all $Z\in\mathcal{X}$ and $i\in\{1,\ldots,n\}$ ,

\[\rho_i(Z)=\int Z\,dT_i\circ\mathbb{P}, \]

where each $T_i$ is a distortion function. Let $\nu_i \,:\!=\, T_i\circ\mathbb{P}$ .

We now apply the result of Proposition 4.3 to obtain Pareto-optimal contracts in the centralized insurance market. We consider a market with three agents: the States of California ( $i=1$ ), New York ( $i=2$ ), and Texas ( $i=3$ ). The correlation between the monthly losses of these States is given by the matrix

\[ \left[ \begin{array}{c@{\quad}c@{\quad}c} 1&-0.0094&-0.0109\\ -0.0094&1&-0.0044\\ -0.0109&-0.0044&1 \end{array} \right], \]

which indicates very little correlation. Some summary statistics for the monthly losses for these States are shown in Table 1, in which all values are dollar amounts. We see that the financial losses are right-skewed and that it is possible to experience losses far larger than the average.

Table 1.

Summary statistics for monthly losses due to floods.

We assume that the distortion functions $T_1$ and $T_2$ are both inverse S-shaped distortion functions, as in Kahneman and Tversky (Reference Kahneman and Tversky1979), Tversky and Kahneman (Reference Tversky and Kahneman1992). Specifically, for $i=1,2$ , we take

\[ T_i(t)=\frac{t^{\gamma_i}}{(t^{\gamma_i}+(1-t)^{\gamma_i})^\frac{1}{\gamma_i}}, \]

with $\gamma_1=0.4$ and $\gamma_2=0.5$ . A parameter value of approximately $0.5$ has recently been estimated by Rieger et al. (Reference Rieger, Wang and Hens2017) to be descriptive of actual behavior. We assume a power distortion for the remaining agent. Specifically,

\[ T_3(t)=t^{\gamma_3}, \]

with $\gamma_3=0.4$ . Finally, we assume that the central insurer uses the Expected Shortfall with parameter $\alpha=15\%$ . While this parameter may seem large, this is to allow for the central insurer’s risk measure to be display enough risk tolerance that it will accept a significant portion of risk from the agents in the market. If this parameter is too low, then the central insurer would be too conservative and avoid taking on any flood risk, leading to a market where no insurance is possible. Indeed, it can be verified that under these parameters, no insurance is provided if $\alpha\approx2.5\%$ .

The retained risk of each agent in a Pareto-optimal contract is shown in Figure 2. Note that in this figure and subsequent figures, the amount of the premium is not shown, for visual clarity and to facilitate easier comparison between plots. In other words, each plot has been normalized so that an experienced loss of $\$0$ corresponds to a retention of $\$0$ as well. We see that in each State prefers to cede a portion of their tail risk to the central insurer, while choosing to retain some of the smaller financial losses. This suggests that a deductible contract is best in this situation, with losses beyond the deductible fully covered by the insurer. The value of this deductible is approximately $1.9047\times10^7$ for California, $4.0645\times10^7$ for New York, and $5.1287\times10^7$ for Texas.

Figure 2.

Centralized Pareto-optimal retention, CA/NY/TX.

As a measure of the effectiveness of this contract, we calculate the aggregate welfare gain that results from this arrangement. The welfare gain for each agent i is the difference between their evaluation of the initial risk and their evaluation of the post-contract position:

\[ \rho_i(X_i)-\rho_i(R_i(X_i)+\pi_i) . \]

On the other hand, the welfare gain for the central insurer is the sum of premia received less their evaluation of their assumed risk:

\[ \sum_{i=1}^n\pi_i-\mathrm{ES}_\alpha^{\mathbb{P}}\left(\sum_{i=1}^nI_i(X_i)\right) . \]

The aggregate welfare gain is defined to be the sum of the welfare gain of each agent in the market, including the central insurer. Under our assumptions, the aggregate welfare gain is maximized in a Pareto-optimal allocation (see, e.g., Boonen et al. Reference Boonen, Chong and Ghossoub2024, Theorem 2.1). In the example above, the maximum aggregate welfare gain achievable through centralized insurance is $7.9785\times10^8$ . The average gain per agent is in this example is $1.9946\times10^8$ . Note that including the centralized insurer, there are four agents in the market in this example.

However, while a significant welfare gain is possible in a centralized market, Ghossoub and Zhu (Reference Ghossoub and Zhu2024b) warn that a monopolistic insurer with a first-mover’s advantage can absorb the entirety of the welfare gain by increasing premium prices. In particular, the central insurer has an incentive to increase prices and ultimately leave each policyholder with no welfare gain at an equilibrium. The amount of premium paid by each State in a Stackelberg equilibrium in our example is shown in Table 2. These amounts are extremely high, even exceeding the value-at-risk at the 5% level. In this case, the entire aggregate welfare gain of $7.9785\times10^8$ is allocated only to the central insurer.

Table 2.

Premium paid in a stackelberg equilibrium, CA/NY/TX.

4.2. Peer-to-peer risk sharing

As a partial solution to the concerns arising from the Stackelberg setting, the present paper proposes a P2P risk-sharing scheme among the agents themselves without the presence of a central agency. In the context of flood risk, this is the scenario where the individual State governments of California, New York, and Texas choose to pool their flood risk and each assume a portion of the aggregate financial loss.

Applying the result of Theorem 3.3 to the present example yields the Pareto-optimal allocation shown in Figure 3. The States of California and New York end up with all of the variability of the risk, since the risk measure of Texas is more risk averse under our chosen parameters. However, this does not mean that Texas is receiving free insurance, since each State must pay a premium for participation in the risk-sharing pool. Both Figures 3 and 2 show that Texas prefers retaining less risk than the other agents.

Figure 3.

Decentralized Pareto-optimal distribution, CA/NY/TX.

Figure 4.

Retained Loss, CA/NY/TX.

A comparison between the centralized and decentralized insurance schemes is shown in Figure 4, which also includes the scenario of no insurance. The vertical axis represents the retained monthly loss in dollars, and the horizontal axis represents the 553 states of the world $\omega\in\Omega$ , which are sorted by the value of $S(\omega)$ , the realized aggregate loss in state $\omega \in \Omega$ . For ease of comparison, the plots have been normalized so that each agent is indifferent between each retention structure. In the case of centralized insurance, we see that each agent pays a premium to insure their risk beyond a deductible. This results in significantly less variance in retained risk, at a cost of a monthly premium. On the other hand, the structure of the Pareto-optimal risk-sharing arrangement is different for each agent. The plots show that California’s retention is similar to that under centralized insurance, in the sense that variation is reduced and extreme losses are insured. However, extreme losses are transferred to New York, who accepts a payment of a premium as compensation. Since there is no exogenous insurer in the decentralized market, all flood risk must be retained among the three agents in the market. The plot in Figure 4 shows that New York is the agent that accepts the extreme tail risk in this example. Finally, Texas elects to pay a premium to transfer all the flood risk to the other two agents, and so its retention structure is a flat horizontal line. This suggests that Texas is the most risk-averse in this scenario.

By Theorem 3.3, both the aggregate welfare gain and the average welfare gain are maximized by Pareto-optimal allocations. Hence, evaluating the effectiveness of the P2P risk-sharing scheme is possible by using the average welfare gain as a metric. Recall that the welfare gain for agent i is defined as the improvement in the risk-sharing scheme over the status quo:

\[ \rho_i(X_i)-\rho_i(Y_i+\pi_i) . \]

The average welfare gain for the P2P risk-sharing scheme is the arithmetic average of the welfare gains of the three agents in the market. In this example, the average welfare gain is $1.9512\times10^8$ . While this is a significant gain, this is not quite as large as the average gain of $1.9946\times10^8$ that is possible from a centralized insurance market with the monopolistic insurer. Hence, we see the P2P risk-sharing scheme as a compromise compared to the centralized structure. While a larger welfare gain is ultimately possible in the centralized setting, this carries the risk that the welfare gain is absorbed by only the monopolistic insurer through aggressive pricing schemes; again, see Ghossoub and Zhu (Reference Ghossoub and Zhu2024b). On the other hand, the decentralized market admits no such risk, since the welfare gain is necessarily distributed among the agents participating in the risk-sharing pool. However, the trade-off is that the maximum possible average welfare gain is potentially lower in the P2P market.

In Table 3, we show the average welfare gain per agent in both centralized and decentralized markets for varying values of the parameter $\gamma_3$ in the distortion function of the third agent $T_3$ . It can be verified that power distortion functions have a constant relative index of probabilistic risk aversion. As $\gamma_3$ increases, the RPRA of the third agent decreases, and the hence the third agent eventually prefers to retain more of the flood risk in the market. In the extreme case, we see that the P2P market can actually dominate that of the centralized market in terms of possible welfare gain, since the decentralized scheme allows for the third agent to insure others’ risk whereas the centralized scheme does not. Hence, it is possible for the decentralized market’s average welfare gain to exceed that of the centralized market. However, in practical settings, we would expect that the policyholders are more risk averse than the central authority, which would lead to an advantage for the centralized insurance market in general.

Table 3.

Average welfare gain with varying parameter $\gamma_3$ .