Bilateral Risk Sharing with Heterogeneous Beliefs and Exposure Constraints

This paper studies bilateral risk sharing under no aggregate uncertainty, where one agent has Expected-Utility preferences and the other agent has Rank-Dependent Utility preferences with a general probability distortion function. We impose exogenous constraints on the risk exposure for both agents, and we allow for any type or level of belief heterogeneity. We show that Pareto-optimal risk-sharing contracts can be obtained via a constrained utility maximization under a participation constraint of the other agent. This allows us to give an explicit characterization of optimal risk-sharing contracts. In particular, we show that an optimal risk-sharing contract contains allocations that are monotone functions of the likelihood ratio, where the latter is obtained from Lebesgue's Decomposition Theorem.


Introduction
Bilateral risk-sharing is a risk transfer and reallocation mechanism popularized by the prevalence of over-the-counter trading, that is, direct trading between two parties without the supervision of an exchange. This paper examines a situation in which a decision maker (DM), such as a risk manager, is able to construct a contract to transfer risk to another institution (the counterparty). Since the seminal work of Borch [13] and Wilson [50], risk sharing is an important problem in economics, finance, and actuarial science. Our focus here is on bilateral risk sharing initiated by the DM, who seeks to maximize his/her expected utility (EU) under subjective probabilities. The counterparty is allowed to have different beliefs regarding the underlying risk's probability distribution, and is endowed with a flexible class of rank-dependent utility (RDU) preferences (Quiggin [40,41,42]) generated by a probability distortion. We do not impose restrictions on the shape of the probability distortion, and we allow in particular for both strongly risk-averse (in the sense of aversion to mean-preserving spreads) and strongly risk-seeking preferences 1 . We also allow for preferences that overweight extreme (good and bad) risks via an inverse-S-shaped distortion, as in Cumulative Prospect Theory (Tversky and Kahneman [48]). Both the DM and the counterparty are subject to constraints on their risk-absorbing capacity. In other words, we impose ex ante liability constraints that state that even when a firm assigns a low or zero probability to certain events, it is not able to pay a nearly infinite amount of capital to the other agent.
While the notion of a competitive equilibrium is a popular tool and solution concept in risksharing problems, it requires the assumption that individuals regard the prices of goods or consumption bundles as being independent of their own choices, and is thus suitable for markets with many agents (e.g., Arrow and Debreu [6]). If there are only two agents, this assumption is very strong. Consequently, we focus instead on the weaker concept of Pareto optimality. We show that Pareto-optimal risk-sharing contracts can be obtained by a maximization of the DM's objective function under a participation constraint of the counterparty.
Risk sharing received considerable attention in markets with aggregate uncertainty, since the seminal work of Borch [13], Wilson [50], Gerber [25], Bühlmann and Jewell [15], and Kaluszka [36]. In such markets, trading a state-contingent payoff is interpreted as hedging rather than betting. Our focus is on risk sharing without aggregate risk, which is the case when the sum of all risk in the market is risk-free before and after the risk-sharing contract is determined (as in Billot et al. [7,8] or Chateauneuf et al. [18], for instance). In such markets, trading a statecontingent payoff is interpreted as betting rather than hedging. Our key contribution in this paper is to examine the effect of belief heterogeneity on risk sharing. The two agents may have different beliefs due to asymmetric information, or difficulty in estimating the distribution with limited data. Heterogeneity of beliefs gained considerable interest in economics and finance. For a non-exhaustive list of references that study heterogeneous beliefs in financial markets, we refer to Gollier [32], David [23], Chen et al. [19], and Simsek [44,45]. Previous approaches to risk sharing with heterogeneous beliefs include Wilson [50] and Boonen et al. [11], but these approaches rely on EU preferences and agreements about zero-probability events. Moreover, if all agents are endowed with cash-invariant utilities with different reference probabilities, existence of Pareto-optimal risk-sharing contracts is studied by Acciaio and Svindland [3]. Our approach is more general, as it allows for a very general form of disagreement about probabilities. We allow in particular for singularity between the beliefs, that is, disagreement about zero-probability events. In optimal (re)insurance contract design, optimal insurance with divergent beliefs is studied by Boonen [9], Ghossoub [30,28,29], and Chi [21], for instance. These approaches impose particular conditions on the type or level of disagreement about probabilities, unlike this paper.
The literature on risk sharing has hitherto examined situations in which the preferences of all agents belong to the same class. For instance, Borch [13], Wilson [50], Gerber [25], Bühlmann and Jewell [15], Kaluszka [36], and Aase [1,2] study the EU case, while Jouini et al. [35] and Ludkovski and Young [37] study dual utilities (as in Yaari [53]), or more generally, law-invariant monetary utility functions. Moreover, Tsanakas and Christofides [47], Xia and Zhou [51], Jin et al. [34], and Boonen et al. [12] study the case of RDU. As an exception, Boonen [10] studies Pareto-optimal risk sharing with both expected and dual utilities. All of these approaches impose assumptions that ensure that the optimal contracts are comonotonic 2 . For instance, some authors assume that the beliefs regarding the underlying distribution are homogeneous and the distortions are convex. Moreover, all these optimal solutions are implicit, and aim either at characterizing existence of a solution or at providing a solution algorithm. In this paper, we provide an explicit description of an optimal bilateral risk-sharing contract. Interestingly, this contract is not necessarily comonotonic, but it is a monotone function of the likelihood ratio. Existence of this likelihood ratio follows from Lebesgue's Decomposition Theorem.
If agents have homogeneous beliefs and are both averse to mean-preserving spreads, then we show that it is always optimal for both agents not to hold a risky position after risk is transferred from the DM to the counterparty. Billot et al. [7,8] and Chateauneuf et al. [18] show that this also holds in risk sharing with multiple priors, when the agents' sets of priors have a non-empty intersection. We show that optimality of a deterministic position after risk-sharing needs no longer be true when either (i) beliefs are heterogeneous, or (ii) the counterparty is endowed with a general (non-convex) probability distortion. We characterize optimal risk-sharing contracts for any type or level of belief heterogeneity and any probability distortion function, and we provide an explicit description of the optimal risk-sharing contract for the DM subject to a participation constraint of the counterparty. It has a simple two-part structure: the DM receives a maximal wealth transfer on an event to which the counterparty assigns zero probability, and an explicit solution on the complement of this event. Therefore, the risk-sharing contract can depend on a "sunspot" if the agents disagree on the likelihood of a sunspot occurrence. This concept of a sunspot as an extrinsic random variable is introduced by Cass and Shell [17], and the existence of sunspot equilibria with Choquet-Expected Utility (CEU) is first shown by Tallon [46]. The rest of this paper is organized as follows. Section 2 discusses the agents' preferences and introduces the constrained risk-sharing problem. Section 3 examines the relationship between the constrained risk-sharing problem and Pareto optimality. Section 4 provides a solution, as well as a description of optimal risk-sharing contracts. Section 5 examines the special cases of dual-utility preferences and expected-utility preferences for the counterparty, respectively. Section 6 concludes. Related analysis and the proofs are presented in the Appendices.

2.1.
Preferences of the Agents. Let pS, Σq be a measurable space, and let B pΣq be the vector space of all bounded, R-valued, and Σ-measurable functions on pS, Σq.
We assume that there are two agents who seek a risk-sharing arrangement. We propose a maximization of the utility of one agent under a generic participation constraint of the other agent. As we will show in Section 3, this formulation is consistent with the criterion of Pareto optimality in our setting.
The DM is subject to an original risk X 1 P B pΣq and the counterparty is subject to a risk X 2 P B pΣq, where the realizations are interpreted as losses. A key assumption in this paper is that there is no aggregate risk (as in Billot et al. [7,8], Chateauneuf et al. [18], or Ghirardato and Siniscalchi [26]), which implies that X 1`X2 " c P R. Trading is therefore seen as betting rather than as hedging. For instance, consider a situation where two traders meet and engage in Over-The-Counter trading. The background wealth (initial endowments) of the agents consist of all previous bilateral contracts in their portfolios. Thus, as previous contracts have been bought/sold by the two traders, the initial endowments sum up to zero. Now, suppose that these traders wish to reallocate risk in order to gain jointly from belief heterogeneity. In the literature on risk-sharing, it is common to focus on homogeneous beliefs, and to consider an exogenously given aggregate risk. In such situations, Pareto-optimal risk allocations are typically comonotonic with this aggregate risk (e.g., Boonen et al. [12]). This does not allow for betting (e.g., put options) against the aggregate risk. In order to isolate the effect of heterogeneous beliefs on Pareto-optimal risk allocations, we keep the aggregate risk deterministic.
The riskX 1 is subtracted from the initial wealth of the DM, and the riskX 2 is subtracted from the initial wealth of the counterparty, both without financial frictions. The DM has initial wealth W 1 0 P R, and his/her total state-contingent wealth after risk sharing is the random variable W P B pΣq defined by W psq :" W 1 0´X 1 psq , @s P S.
As in Ghossoub [30,29], we assume that the DM has preferences over future random wealth admitting a subjective EU representation. The DM's preferences induce a utility functionû 1 : R Ñ R and a subjective probability measure P on pS, Σq. The DM is risk averse, such that his/her utility functionû 1 satisfies the following commonly used assumption.
Assumption 2.1. The DM's utility functionû 1 is increasing, 3 strictly concave, continuously differentiable, and satisfies the Inada conditions lim We assume that the DM maximizes The DM will seek to maximize this objective function under a participation constraint of the counterparty. The counterparty's preferences induce a probability measure Q on pS, Σq and a utility functionû 2 : R Ñ R. We assume that the counterparty distorts this probability measure Q using a distortion function T . Thus, the counterparty is endowed with RDU preferences (Quiggin [40,41,42]), which admit a representation in terms of a Choquet integral 4 . Expected utility and dual utility are special cases of RDU preferences, and these two special cases will be discussed in more detail in Section 5. We assume that the participation constraint of the counterparty is given by where W 2 0 P R is the counterparty's initial non-random wealth, and V 0 P R is the counterparty's reservation utility. As in the ε-constraint method for multi-objective optimization (see, e.g., Cohon [22] and Miettinen [38]), our aim is to span the set of Pareto-optimal contracts by allowing V 0 to be flexible 5 . The probability distortion function T and utility functionû 2 satisfy the following assumption.
If the distortion function T is convex andû 2 is concave, Chew et al. [20] show that the counterparty is averse to mean-preserving spreads. In parts of the literature (e.g., Amarante et al. [5]), a RDU preference representation is sometimes seen as a special case of CEU, in which the agent's non-additive measure (sometimes called a capacity) υ is a distortion of a probability measure (υ " T˝µ, for some probability measure µ). In this case, convexity (resp. concavity) of the distortion function T yields convexity (resp. concavity) of the capacity ν. In CEU, a convex capacity reflects ambiguity-aversion, while a concave capacity reflects ambiguity-seeking behavior.
2.2. The Constrained Demand Problem. To ensure existence of optimal solutions, we impose a lower limit on the ex post wealth for both the DM and the counterparty. Specifically, we assume that (2.3) W 1 0´X 1 psq ě a, W 2 0´X 2 psq ě b, @s P S, where a, b P R are exogenously given. Note that this is equivalent to assuming exogenous upper bounds on risk exposureX i , for both the DM and the counterparty. Infinite losses are not feasible in reality due to limited liability, even when an agent assigns zero-probability to these events. 4 For any Z P B pΣq, the Choquet integral ofû 2 pZq with respect to T˝Q is defined as Moreover, for any Z P B pΣq and C P Σ, we define Agents are not able to pay out more than a given amount. These bounds are not necessarily the same for the two agents. The wealth of the DM cannot be smaller than a, so that we impose W 1 0´X 1 psq ě a for all s P S. Likewise, the wealth of the counterparty cannot fall below b, so that W 2 0´X 2 psq ě b for all s P S. The DM's problem is that of finding a risk-sharing contract that maximizes his/her subjective expected utility of terminal wealth, subject to the participation constraint of the counterparty, and to the constraints that the ex post wealth satisfy the given lower bounds in eq. (2.3). This is formalized in the following problem.
Define Y :"X 1´c , L :" b´W 2 0 , and R :" W 1 0´a´c , and define the utility functions u 1 and u 2 by u 1 pxq :"û 1 pW 1 0´c`x q and u 2 pxq :"û 2 pW 2 0`x q for x P R. We can then rewrite eq. (2.3) as and ifû 1 andû 2 satisfy Assumptions 2.1 and 2.2, then so do u 1 and u 2 , respectively. Substituting this into Problem 2.3 yields the following problem reformulation.
Problem 2.4 is the main problem that we study in this paper. Note that the risk Y may have a negative realization; and a lower realization yields a higher wealth for the DM. We will refer to the risk Y as a risk-sharing contract. To rule out trivial situations, we will make the following assumption.
Indeed, if V 0 ą u 2 pRq or L ą R, then Problem 2.4 has no feasible solution satisfying the participation constraint in eq. (2.2), and thus no risk-sharing contract is optimal. Moreover, if V 0 ă u 2 pLq, then any feasible contract satisfies the participation constraint, and thus the solutions are identical to the solutions when V 0 " u 2 pLq. Note that V 0 ě ş u 2 p´X 2 q dT˝Q ensures individual rationality for the counterparty, and ż u 1 p´Y q dP ě ż u 1 pc´X 1 q dP ensures individual rationality for the DM. We do not impose individual rationality constraints ex ante. This setup also allows us to introduce deterministic financial transaction costs, that thus do not depend on the risk-sharing contract Y . These deterministic financial transaction costs need then to be added to V 0 .
Electronic copy available at: https://ssrn.com/abstract=3345149 2.3. Singularity and the Likelihood Ratio. By Lebesgue's Decomposition Theorem [4, Theorem 10.61] there exists a unique pair pP ac , P s q of (non-negative) finite measures on pS, Σq such that P " P ac`Ps , where (i) P ac ! Q, that is, for all B P Σ, Q pBq " 0 ùñ P ac pBq " 0.
(ii) P s K Q, that is, there exists some A P Σ such that Q`SzA˘" P s pAq " 0, which then implies that P ac`S zA˘" 0 and Q pAq " Q pSq " 1.
Therefore, by the Radon-Nikodým Theorem [4,Theorem 13.20] there exists a Q-a.s. unique h P L 1 pS, Σ, Qq such that h : S Ñ r0,`8q and P ac pCq " h can be interpreted as a likelihood ratio: h " dP ac dQ , which is also known as Radon-Nikodým derivative. In the rest of this paper, we fix the set A and the random variable h.

The Constrained Demand Problem and Pareto Optimality
In this section, we examine the relationship between the constrained demand problem in Problem 2.4 and Pareto optimality. We first define the concept of Pareto optimality for our constrained risk-sharing problem. The next result shows that for every Pareto-optimal risk-sharing contract Y˚, there exists some V 0 P R such that Y˚maximizes the objective function given in eq. (2.1) under the participation constraint given in eq. (2.2).  (ii) If Q ! P ac , then for a given V 0 P " u 2 pLq , u 2 pRq ‰ any solution to Problem 2.4 is Pareto optimal; (iii) If Q ! P ac and Y˚P B pΣq solves Problem 2.4 for a given V 0 P " While the first result is well-known in operations research and optimization (see Theorem 3.2.2 in Miettinen [38]), the two other results necessitate the handling of the intrinsic singularity between the two probability measures. Note that the condition Q ! P ac , which is equivalent to the mutual absolute continuity of Q and P ac (denoted by Q " P ac ), is weaker than the condition Q ! P . If P K Q (and so Q ! P ac does not hold), we will show in Section 4 that there exist solutions to Problem 2.4 that are not Pareto optimal (see Propositions 4.2 and 4.3). Remark 3.3. As a consequence of Theorem 3.2, we obtain that if Q ! P ac , then finding solutions to Problem 2.4 for all V 0 P " u 2 pLq , u 2 pRq ‰ is equivalent to finding all Pareto-optimal risk-sharing contracts. Note that Q ! P ac holds when P " Q, Q ! P , or Q " P (i.e., Q and P are equivalent 6 ). Since Pareto optimality does not distinguish the DM and counterparty other than via their preferences, it is also a solution of the dual approach of maximizing the counterparty's utility under a participation constraint of the DM. Thus, if Q " P , all solutions of Problem 2.4 are also solutions obtained under the dual approach.

Optimal Risk-Sharing Contracts
Our main result, Theorem 4.5, provides an explicit description of an optimal risk-sharing contract, for any level of belief divergence. It is precisely the belief divergence and the probability distortion that create room for risk sharing. We show this in the following proposition. Proposition 4.1 implies that if agents share beliefs, and if the counterparty distorts this common belief via a convex distortion function, then the optimal allocation is the full insurance allocation given byX 1 " u´1 2 pV 0 q`c andX 2 "´u´1 2 pV 0 q.
Next, we examine another extreme case where P and Q are mutually singular (i.e., h " 0). Proposition 4.2. If Assumptions 2.1, 2.2, and 2.5 hold, and if P K Q, then for any L ďŶ ď R such that ż A u 2´Ŷ¯d T˝Q ě V 0 , the risk-sharing contract Y˚:" L1 SzA`Ŷ 1 A is optimal in Problem 2.4. Hence, in particular, P pY˚" Lq " 1.
When the beliefs are mutually singular, it is optimal for the DM to receive the maximum share from the counterparty (that is to receive´L) for events to which the DM assigns full probability. Proposition 4.2 characterizes a collection of solutions to Problem 2.4. This collection of solutions is larger than the collection of Pareto-optimal risk-sharing contracts, as shown in the following proposition. Proposition 4.3. If Assumptions 2.1 and 2.2 hold, and if P K Q, then any Pareto-optimal risk-sharing contract is given by where Y 1 " L, P -a.s., and Y 2 " R, Q-a.s. In other words, Y˚" L, P -a.s., and Y˚" R, Q-a.s. Proposition 4.3 states that if P K Q, solutions to Problem 2.4 are not necessarily Pareto optimal. In particular, Proposition 4.2 selects a collection of solutions to Problem 2.4, and when Y 1 A " R1 A , Q-a.s., these solutions are Pareto optimal. Note that Theorem 3.2(ii) states that every solution to Problem 2.4 is Pareto optimal when Q ! P ac , e.g., when P and Q are equivalent probability measures.
Next, we state the main result of this paper. To do so, we need some definitions. For the likelihood ratio h, we denote by F h,Q the cumulative distribution function with respect to the probability measure Q, defined by F h,Q ptq :" Q`ts P S : h psq ď tu˘, @t P R, , @t P r0, 1s .
For a real-valued function f on a convex subset of R containing the interval r0, 1s, the convex envelope of f on the interval r0, 1s is defined as the greatest convex function g on r0, 1s such that g pxq ď f pxq, for each x P r0, 1s. The construction of the convex envelope with an inverse-S shaped distortion is studied by Ghossoub [31] and Wang et al. [49]. We make the following assumption.
Assumption 4.4. The likelihood ratio h " dP ac dQ is a continuous random variable on the probability space pS, Σ, Qq (i.e., Q˝h´1 is non-atomic) such that F´1 h,Q is increasing and positive on r0, 1s.
This assumption states that the likelihood ratio must, in particular, be a continuous random variable with respect to probability measure Q, but not necessarily with respect to probability measure P . ‚ λ˚is chosen such that Moreover, the function g˚is non-increasing. Example 4.6. Suppose that the modified utility functions of the two agents are of the exponential type: for some α i ą 0, where i " 1, 2 and α 1 ‰ α 2 . Then, for i " 1, 2, bothû i and u i display constant absolute risk aversion, and the absolute risk aversion is the constant function given by We define the aggregate risk aversion in the market as α 1`α2 . Moreover, in this case, m pxq " e pα 1`α2 q x , for x P R, and thus m´1 pyq " ln pyq α 1`α2 , for y ą 0.
Therefore, when α 1`α2 Ñ 0, it follows that the optimal risk-sharing solution Y˚is very dispersed on the event A, based on the sign of δ 1´φ`1´F h,Q phq˘¯. Hence, if the aggregate risk aversion in the market is small, then there will be more betting based on the heterogeneity of beliefs via the likelihood ratio h, and vice versa.

Two Special Cases of the Preferences of the Counterparty
In this section, we discuss two special cases. First, we consider the case where the counterparty is endowed with EU preferences (no distortion), and second we consider the case where the counterparty is endowed with dual utility preferences (linear utility).

EU Preferences.
The first special case is the case in the absence of a probability distortion. In this case, T ptq " t, for all t P r0, 1s. Then, Ψ ptq " 1´T`1´φ´1 ptq˘" φ´1 ptq for all t P r0, 1s. Moreover, φ 1 ptq " F´1 h,Q p1´tq ą 0, and so φ is increasing. By Assumption 4.4, F´1 h,Q is increasing and thus φ is concave. The Inverse Function Theorem implies that φ´1 is convex and increasing. Consequently, Ψ is convex and increasing on r0, 1s. Hence, δ " Ψ on r0, 1s, yielding As a direct consequence of the above argument, we obtain the following description of optimal risk-sharing contracts by Theorem 4.5. ‚ λ˚is chosen such that For instance, if u 1 and u 2 are both of the exponential type, then Corollary 5.1 implies that the risk-sharing contract Note that by Assumptions 2.1 and 2.2, it follows that lim yÑ8 m´1pyq " 8. So, for all s P S such that hpsq " 0, it follows that Y˚psq " R. So, an optimal risk-sharing contract is such that maximal risk is shifted to the DM. However, the DM assigns zero probability to these events. The counterparty, on the other hand, may assign a non-zero probability to these events.

Dual-Utility Preferences.
Our second special case is the case where the counterparty is endowed with dual utility preferences. Then, it holds that u 1 2 pxq " 1 for all x P R, and thus mpxq " So, it holds that m´1pyq "´pu 1 1 q´1 pyq, and hence we readily obtain the following result from Theorem 4.5. ‚ g˚phq :"´pu 1 1 q´1´λ˚δ 1´φ`1´F h,Q phq˘¯¯, which depends on the state of the world only through the likelihood ratio h " dPac dQ ; ‚ δ is the convex envelope on r0, 1s of the function Ψ defined by Ψ ptq :" 1´T`1´φ´1 ptq˘, for all t P r0, 1s, where φ ptq :" dx, for all t P r0, 1s; and, ‚ λ˚is chosen such that ż u 2 pY˚q dT˝Q " V 0 .

Conclusion
This paper examined the problem of bilateral risk sharing with no aggregate uncertainty and when there is a fixed lower and upper bounds on the shifted loss. We assumed that the counterparty and the decision maker may disagree about the likelihoods associated with the state space. The decision maker maximizes a subjective expected-utility functional, and the counterparty is endowed with rank-dependent utility preferences, with a general probability Electronic copy available at: https://ssrn.com/abstract=3345149 distortion function and a possibly different probability measure from that of the decision maker. We allowed for any type or level of belief heterogeneity.
We showed that a constrained maximization under a participation constraint yields all Paretooptimal risk-sharing contracts. We provided a full description of an optimal risk-sharing contract. It has a simple two-part structure: maximal risk is shifted on an event to which the counterparty assigns zero probability, and an explicit formulation on the complement of this event. If the counterparty is endowed with expected or dual utility, simpler solutions are obtained.

Appendix A. Equimeasurable Rearrangements
Let pS, G, µq be a probability space and let V P L 8 pS, G, µq be a continuous random variable (i.e., µ˝V´1 is non-atomic). For each Z P L 8 pS, G, µq, let F Z,µ ptq " µ`ts P S : Z psq ď tud enote the cumulative distribution function of Z with respect to the probability measure µ, and let F´1 Z,µ ptq be the left-continuous inverse of the distribution function F Z,µ (that is, the quantile function of Z w.r.t. µ), defined by , @t P r0, 1s .
For instance, if Y 1 , Y 2 P B pΣq, and if Y 2 is of the form Y 2 " I˝Y 1 , for some Borel-measurable function I, then Y 2 is comonotonic with Y 1 if and only if the function I is non-decreasing.
(iv) If Z˚is any other element of L 8 pS, G, µq that has the same distribution as Y under µ and that is anti-comonotonic with V , then Z˚" r Y µ , µ-a.s. r Y µ is called the non-increasing µ-rearrangement of Y with respect to V . Since µ˝V´1 is nonatomic, it follows that F V,µ pV q has a uniform distribution over p0, 1q [24, Lemma A.25]. Letting U :" F V,µ pV q, it follows that U is a random variable on the probability space pS, Σ, µq with a uniform distribution on p0, 1q and that V " F´1 V,µ pU q , µ-a.s..
A function L : R 2 Ñ R is said to be supermodular if for any x 1 , x 2 , y 1 , y 2 P R with x 1 ď x 2 and y 1 ď y 2 , one has (A.2) L px 2 , y 2 q`L px 1 , y 1 q ě L px 1 , y 2 q`L px 2 , y 1 q .

Equation (
A.2) then implies that a function L : R 2 Ñ R is supermodular if and only if the function η pyq :" L px`z, yq´L px, yq is non-decreasing on R, for any x P R and z ě 0.
(1) If g : R Ñ R is concave and a P R, then the function L 1 : R 2 Ñ R defined by L 1 px, yq " g pa´x`yq is supermodular. Moreover, if g is strictly concave, then L 1 is strictly supermodular.
Proposition A.4 (Hardy-Littlewood-Pólya Inequality [16]). Let Y P L 8 pS, G, µq, and let r Y µ be the non-increasing µ-rearrangement of Y with respect to V . If the function L is supermodular then provided the integrals exist.
Appendix B. Proof of Theorem 3.2 The first statement follows almost directly from Theorem 3.2.2 in Miettinen [38], where the state space is finite. We provide a short self-contained proof for general probability spaces. Suppose that Y˚is Pareto optimal. Then, L ď Y˚ď R, and so ż u 2 pY˚q dT˝Q P " u 2 pLq , u 2 pRq ‰ .
Suppose that Y˚does not solve Problem 2.4 with V 0 " ż u 2 pY˚q dT˝Q. Then, there exist Y P BpΣq satisfying eq. (2.4) that solves Problem 2.4 with V 0 . Therefore, ż u 1 p´Y q dP ą ż u 1 p´Y˚q dP and Thus, Y is a Pareto improvement over Y˚, contradicting the Pareto optimality of Y˚.
We continue with the second statement. Suppose that Q ! P ac , fix V 0 P " u 2 pLq , u 2 pRq ‰ , and let Y˚be a solution to Problem 2.4. Suppose that Y˚is not Pareto optimal, so that there exist Y P BpΣq satisfying eq. (2.4) with ż u 1 p´Y q dP ě ż u 1 p´Y˚q dP, and ż u 2 pY q dT˝Q ě ż u 2 pY˚q dT˝Q, with at least one strict inequality. Then, in particular, Y is feasible for Problem 2.4. If ż u 1 p´Y q dP ą ż u 1 p´Y˚q dP , this contradicts the optimality of Y˚for Problem 2.4. Assume then that ż u 2 pY q dT˝Q ą ż u 2 pY˚q dT˝Q, and let ε be such that ż u 2 pY´εq dTQ " ż u 2 pY˚q dT˝Q. Then ε ą 0, by strict monotonicity of u 2 . If Y " L, Q-a.s., then Hence, Q pY ą Lq ą 0 and so P pY ą Lq ě P ac pY ą Lq ą 0, since Q ! P ac . LetȲ :" max pL, Y´εq P B pΣq. Then L ďȲ ď R, Y´ε ďȲ ď Y , and 7 P pȲ ă Y q ą 0. Therefore, since u 1 is increasing, it follows 7 It is immediate to verify that ts P S : Y psq " Lu Ď ts P S :Ȳ psq " Y psqu. Now, if s P S is such that Y psq " Y psq but Y psq ą L, then L ăȲ psq " maxtY psq´ε, Lu, and henceȲ psq " Y psq´ε ă Y psq, a contradiction. Consequently, ts P S :Ȳ psq " Y psqu Ď ts P S : Y psq " Lu, and hence ts P S :Ȳ psq " Y psqu " Moreover, sinceȲ ě Y´ε it follows that where the second inequality follows from the feasibility of Y˚for Problem 2.4. Consequently,Ȳ is feasible for Problem 2.4, which contradicts the optimality of Y˚for Problem 2.4.
We conclude with the proof of the third statement. Suppose that Q ! P ac , fix V 0 P " u 2 pLq , u 2 pRq ‰ , and let Y˚be a solution to Problem 2.4. Suppose that Hence, Q pY ą Lq ą 0 and so P pY ą Lq ě P ac pY ą Lq ą 0, since Q ! P ac . Let ε be such that ş u 2 pY˚´εq dT˝Q " V 0 . Then ε ą 0. LetȲ :" max pL, Y˚´εq P B pΣq. Then L ďȲ ď R, Y˚´ε ďȲ ď Y˚, and P pȲ ă Y˚q ą 0. Consequently, and soȲ is feasible for Problem 2.4. Moreover, since u 1 is increasing, it follows that ż u 1`´Ȳ˘d P " ż rȲ "Y˚s u 1 p´Y˚q dP`ż rȲ ăY˚s contradicting the optimality of Y˚for Problem 2.4. Therefore, ż u 2 pY˚q dT˝Q " V 0 . l ts P S : Y psq " Lu. Therefore, ts P S :Ȳ psq ă Y psqu " ts P S : Y psq ą Lu. Since P pY ą Lq ą 0, it follows that P pȲ ă Y q ą 0.

Appendix C. Proof of Proposition 4.1
It is well-known that a concave utility function yields risk-aversion in EU, so that due to the Jensen's inequality we have for all Y P BpΣq. Chew et al. [20] show that in RDU, a concave utility function with a convex probability distortion function yields aversion to mean-preserving spreads. Hence, in particular, concavity of u 2 and convexity of T imply that for all Y P BpΣq Thus, if Y solves Problem 2.4, then so does Y˚:" ż u 2 pY q dP . The rest follows from Theorem First, for any A P Σ and Y P B pΣq, let Proof. Since, Q pAq " 1 and since Q is monotone, non-negative, and additive, we have for each B P Σ, T˝Q pBq " T`Q pBq˘" T˜Q´`B X A˘Y`B X`SzA˘˘¯" T´Q`B X A˘`Q`B X`SzA˘˘¯" T´Q`B X A˘¯" T˝Q pB X Aq .
(D.1) Therefore, it follows from eq. (D.1) that T˝Q´B Y`SzA˘¯" T˝Qˆ´B Y`SzA˘¯X A˙" T˝Q pB X Aq " T˝Q pBq .
Consequently, for any Y P B pΣq, setting B t :" ts P S : Y psq ą tu for all t P R, it follows that Electronic copy available at: https://ssrn.com/abstract=3345149 If P K Q, then P " P s , P ac " 0, and h " 0. In this case, P pAq " 0 and Q pAq " 1. Choose anyŶ P B pΣq such that L ďŶ ď R and ż A u 2´Ŷ¯d T˝Q ě V 0 , and define Y˚P B pΣq by Then L ď Y˚ď R by construction, and Proposition D.1 implies that Hence, Y˚is feasible for Problem 2.4. Suppose, by way of contradiction, that Y˚is not optimal for Problem 2.4. Then, there exist Z P B pΣq such that L ď Z ď R, ż u 2 pZq dT˝Q ě V 0 , and ż u 1 p´Zq dP ą ż u 1 p´Y˚q dP . Since L ď Z and P`SzA˘" 1, it follows that which is a contradiction. Hence, Y˚is optimal for Problem 2.4.
l Appendix E. Proof of Proposition 4.3 Let Z P B pΣq be such that L ď Z ď R. Then Y˚psq " R ě Zpsq, for Q-a.e. s P A, and u 1`´Y˚p sq˘" u 1 p´Lq ě u 1`´Z psq˘, for P -a.e. s P SzA. Since P pSzAq " 1, it follows that Moreover, it follows from Proposition D.1 and QpAq " 1 that Thus, there cannot exist a Z P BpΣq such that L ď Z ď R with ż u 1 p´Zq dP ě ż u 1 p´Y˚q dP and ż u 2 pZq dT˝Q ě ż u 2 pY˚q dT˝Q, with at least one strict inequality. Consequently, Y˚is Pareto optimal.
Electronic copy available at: https://ssrn.com/abstract=3345149 Furthermore, suppose thatŶ P B pΣq is another Pareto-optimal risk-sharing contract. Then L ďŶ ď R. Hence, in particular, ż u 1´´Ŷ¯d P ď u 1 p´Lq. Moreover, Suppose, by way of contradiction, that P pN q ą 0, where N :" ts P S : L ăŶ psqu. Then ż N u 1´´Ŷ¯d P ă ż N u 1 p´Lq dP , by definition of N and strict monotonicity of u 1 . Thus, Since the deterministic contract L is feasible, this contradicts the Pareto optimality ofŶ . Hence, Y " L, P -a.s. Now, suppose, by way of contradiction, that Q´ts P A : R ąŶ psqu¯ą 0. Then, there exists a t 0 ă u 2 pRq such that Q`ts P A : u 2 pRq ą tu˘" 1 ą Qˆts P A : u 2´Ŷ psq¯ą tu˙, @t P rt 0 , u 2 pRqq.
Then by strict monotonicity of T and u 2 , we have T˜Qˆts P A : u 2´Ŷ psq¯ą tu˙¸ď T´Q`ts P A : u 2 pRq ą tu˘¯, @t P R, and T˜Qˆts P A : u 2´Ŷ psq¯ą tu˙¸ă T´Q`ts P A : u 2 pRq ą tu˘¯, @t P rt 0 , u 2 pRqq.
T˜Qˆts P A : u 2´Ŷ psq¯ą tu˙¸dt T´Q`ts P A : u 2 pRq ą tu˘¯´1 Since Y " R is feasible, this contradicts the Pareto optimality ofŶ . Hence,Ŷ " R, Q-a.s. l Appendix F. Proof of Theorem 4.5 Consider the following two problems: Since the function u 1 is continuous by Assumption 2.1, it is bounded on any closed and bounded subset of R. Therefore, since the range of Y is bounded, the supremum of each of the above two problems is finite when their feasibility sets are non-empty. Moreover, the constant function Y " u´1 2 pV 0 q is feasible for Problem F.1 by Assumption 2.5, and so Problem F.1 has a non-empty feasibility set.
Lemma F.3. The random variable Y˚:" L1 SzA is optimal for Problem F.2.
Proof. The feasibility of Y˚" L1 SzA for Problem F.2 is clear. To show optimality, let Y be any feasible solution for Problem F.2. Then for each s P SzA, L ď Y psq ď R. Therefore, since u 1 is increasing, we have for each s P SzA, u 1`´Y psq˘ď u 1 p´Lq " u 1`´Y˚p sq˘. Thus, ż SzA u 1 p´Y q dP ď ż SzA u 1 p´Y˚q dP " u 1 p´Lq P`SzA˘.
Lemma F.4. If Y1 is optimal for Problem F.1, then Y˚:" Y1 1 A`L 1 SzA is optimal for Problem 2.4.
Proof. By the feasibility of Y1 for Problem F.1, we have L ď Y1 ď R and ż u 2 pY1 q dT˝Q ě V 0 .
Therefore, L ď Y˚ď R, and Proposition D.1 yields ż u 2 pY˚q dT˝Q " Hence, Z is feasible for Problem F.2. Thus, by Lemma F.3, u 1 p´Lq P`SzA˘" ş SzA u 1 p´Lq dP ě ż SzA u 1 p´Zq dP, and so we obtain Consequently, Y˚is optimal for Problem 2.4.
For all Z P B pΣq, we have where the second-to-last equality follows from the fact that ş SzA Z dP s " ş SzA Z dP , since P ac`S zA˘" 0, and the last equality follows from the fact that Q`SzA˘" 0. Consequently, for all Z P B pΣq, Zh dQ. Hence, we can re-write Problem F.1 as: We will solve Problem F.1 using a quantile reformulation. For that purpose, let Q˚denote the collection of all Q-quantile functions f that satisfy L ď f ptq ď R for all t P p0, 1q, i.e.,
sup f PQ˚# ż 1 0 u 1`´f ptq˘F´1 h,Q p1´tq dt : Lemma F.7. If f˚is optimal for Problem F.6, then Y˚" f˚`1´F h,Q phq˘is optimal for Problem F.5 and anti-comonotonic with h.
Proof. First, note that since Q˝h´1 is non-atomic by Assumption 4.4, it follows that F h,Q phq has a uniform distribution over p0, 1q [24,Lemma A.25], that is, Q`ts P S : F h,Q phq psq ď tu˘" t for each t P p0, 1q. Letting r U :" F h,Q phq, it follows that r U is a random variable on the probability space pS, Σ, Qq with a uniform distribution on p0, 1q and that h " F´1 h,Q´r U¯, Q-a.s.
Let f˚be optimal for Problem F.6 and Y˚" f˚´1´r U¯. Then, since f˚P Q˚, it follows that F´1 Y˚,Q " f˚and L ď Y˚ď R. Moreover, by monotonicity of u 2 , Fubini's Theorem yields ż u 2 pY˚q dT˝Q " where the inequality follows from the feasibility of f˚for Problem F.6. Hence, Y˚is feasible for Problem F.5.
To show optimality of Y˚for Problem F.5, let Y by any other feasible solution for Problem F.5 and F´1 Y,Q its quantile function. Then r Y :" F´1 Y,Q´1´r U¯is the non-increasing rearrangement of Y with respect to h, and so r Y is feasible solution for Problem F.5, by properties of the rearrangement (Proposition A.2). Moreover, since the function u 1 is increasing, it follows that the map L : R 2 Ñ R defined by L px, yq :"´u 1 p´yq x is supermodular (see Example A.3). Consequently, by Proposition A.4, it follows that Since Y is feasible for Problem F.5, it follows that F´1 Y,Q is feasible for Problem F.6. Therefore, where the second inequality follows from the optimality of f˚for Problem F.6. Therefore, Y˚is optimal for Problem F.5.
sup qPQ˚# ż 1 0 u 1`´q ptq˘dt : Lemma F.9. If q˚is a solution of Problem F.8, then f˚:" q˚˝φ is a solution of Problem F.6.
Proof. Suppose q˚is optimal for Problem F.8. Since q˚P Q˚, it follows that f˚P Q˚. Moreover, using the variable z " v´1 ptq, where v " φ´1 and φ are as defined above, we have where the last inequality follows from the feasibility of q˚for Problem F.8. Therefore, f˚is feasible for Problem F.6.
To show optimality of f˚for Problem F.6, let f be any other feasible solution for Problem F.6, and let q :" f˝v. Since f is feasible for Problem F.6, it is non-decreasing, left-continuous, and satisfies, for all t P p0, 1q, L ď f ptq ď R. Therefore, since v is increasing and continuous (by the Inverse Function Theorem), q is non-decreasing, left-continuous, and satisfies, for all t P p0, 1q, L ď q ptq " f`v ptq˘ď R. Therefore, q P Q˚. Furthermore, Electronic copy available at: https://ssrn.com/abstract=3345149 where the last inequality follows from the feasibility of f for Problem F.6. Thus, q is feasible for Problem F.8, and hence ż 1 0 u 1`´q ptq˘dt ď ż 1 0 u 1`´q˚p tq˘dt. Thus, using the variable z " v´1 ptq, we obtain Therefore, f˚is optimal for Problem F.6.
In light of Lemma F.9, we turn our attention to solving Problem F.8. In order to do that, we will use a similar methodology to the one used by Xu [52], but adapted to the present setting. First, we recall the following result, due to He et al. [33,Appendix A].

Moreover,
(5) If f is increasing, then so is g; (6) If f is continuously differentiable on p0, 1q, then g is continuously differentiable on p0, 1q.
The following lemma is a direct consequence of Theorem 1 of Moriguti [39].
Lemma F.11 (Moriguti [39]). Let δ be the convex envelope of Ψ on r0, 1s. Then for any q P Q˚, Now, consider the following problem.

BILATERAL RISK SHARING WITH HETEROGENEOUS BELIEFS AND EXPOSURE CONSTRAINTS 25
We first solve Problem F.12 and then show that the solution is also optimal for Problem F.8.
Proof. Let q˚P Q˚be such that the two conditions above are satisfied. Then q˚is feasible for Problem F.12. To show optimality, let q P Q˚be any feasible solution for Problem F.12. Then, by definition of q˚, it follows that for each t, u 1`´q˚p tq˘´u 1`´q ptq˘ě λ " δ 1 ptq u 2`q ptq˘´δ 1 ptq u 2`q˚p tq˘ı .
Proof. Assumptions 2.1 and 2.2 imply that u 1 and u 2 are increasing and continuously differentiable. Moreover, u 1 1 is decreasing and u 1 2 is non-increasing. Therefore, the function m is continuously differentiable and m 1 pxq "´u Hence, the function m is continuous and increasing. Consequently, it is invertible, and its inverse is also increasing, by the Inverse Function Theorem. Thus, for each λ ą 0, the convexity and continuity of δ imply that the function gλ " m´1 pλδ 1 q is continuous and increasing. Therefore, for each λ ą 0, qλ P Q˚.
Consequently, by Lemma F.4, Y˚1 A`L 1 SzA is optimal for Problem 2.4.
Finally, the fact that g˚is non-increasing is a direct consequence of the convexity of δ, the fact that λ˚ą 0, and the fact that m, φ, and F h,Q are all increasing (see the proof of Lemma F.14 and Assumption 4.4). This concludes the proof of Theorem 4.5.