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Equilibrium in risk-sharing games

The large majority of risk-sharing transactions involve few agents, each of whom can heavily influence the structure and the prices of securities. This paper proposes a game where agents' strategic sets consist of all possible sharing securities and pricing kernels that are consistent with Arrow-Debreu sharing rules. First, it is shown that agents' best response problems have unique solutions. The risk-sharing Nash equilibrium admits a finite-dimensional characterisation and it is proved to exist for arbitrary number of agents and be unique in the two-agent game. In equilibrium, agents declare beliefs on future random outcomes different than their actual probability assessments, and the risk-sharing securities are endogenously bounded, implying (among other things) loss of efficiency. In addition, an analysis regarding extremely risk tolerant agents indicates that they profit more from the Nash risk-sharing equilibrium as compared to the Arrow-Debreu one.


Introduction
The design of securities that optimally share risky positions of agents has been a subject of an ongoing research. Starting from the seminal works of [Bor62], [Arr63], [BJ79] and [Buh84], the existence and characterisation of welfare risk allocations has been extensively studied. However, market frictions such as asymmetric information, transaction costs and limited market participation spur agents to act strategically and prevent markets from reaching maximum efficiency. In the financial risk sharing literature, the impact of asymmetric or private information in risk-sharing has been addressed under both static and dynamic models (see among others [NN94], [MR00], [Par04], [Axe07], [Wil11]). One the other hand, the importance of frictions like transaction costs has be highlighted in [AG91] (see also [CRW12]).
This work aims to contribute to the risk-sharing literature by focusing on how limited participation motivates strategic behaviour from the part of acting agents. The vast majority of real-world sharing transactions involves only a few participants, each of whom could influence the way risk is going to be shared. A standard example is the case of insurance companies wishing to share their insurance portfolio through securities, the design of which depends on the risk that each insurer is willing to share. Since such sharing consists of only few companies (typically, two), it is reasonable to argue that the structure and valuation of securities will result as the outcome of a symmetric game played among the participating companies, with the strategic set being the risk that each player is willing to share (as opposed to their actual exposure). We propose a novel way of modelling strategic actions from the part of agents, inducing a Nash game that results in an equilibrium sharing of risk. In our framework, all agents are risk averse but heterogeneous with respect to risk profiles and risky positions.
Our main contributions and insights are summarised below. The entire structure and valuation of the risk-sharing securities are endogenously derived as an outcome of agents' strategic behaviour, under constant absolute risk-aversion (CARA) preferences. To the best of our knowledge, this is the first work that models the way agents choose the risk they are willing to share and studies whether such strategic behaviour results in equilibrium.
Our results demonstrate how the game leads to risk-sharing inefficiency and security mispricing. In particular, we demonstrate endogenous limited liability of securities, a feature that, while usually suboptimal, is in fact observed in practice. Although the agents' set of strategic choices is infinite-dimensional, one of our main contribution is to show that Nash equilibrium admits a finite-dimensional 1 characterisation. In the important case of two participating agents, we show existence of a unique Nash equilibrium; this existence and uniqueness is also observed for more agents in any numerical experiment we have worked out. Our notion of Nash risk-sharing equilibrium highlights the importance of agents' risk tolerance level. More precisely, we show that agents with sufficiently low risk aversion prefer the risk-sharing game than the outcome of an Arrow-Debreu risk-sharing equilibrium that would have resulted from absence of strategic behaviour. Even more interestingly, the result is valid irrespective of their actual risky position. It follows that even risk-averse agents, as long as their risk-aversion is sufficiently low, may prefer thin risk-sharing markets, which in turn results in loss of risk-sharing welfare.
Our model is introduced in Section 1, and consists of a two-period economy with uncertainty, containing possibly infinite 2 states of the world. In the economy we consider a finite number of agents, each of whom is endowed with an accumulated (up to the terminal time) random endowment. Agents seek to trade securities that will result in mutual 1 The dimension is actually one less than number of participating agents. 2 The generality in the state space (which results in an infinite-dimensional problem) is essential in our framework, since we do not wish to enforce any restrictive assumption on the probability distribution or the support of agents' random endowments-in fact, we shall enforce only minimal integrability conditions. Allowing infinite state space is important in risk-sharing applications, since in general agents' random endowments do not have a-priori bounds. reduction of their risk exposures, via mutual increase of their expected utilities. At the first stage of the game, agents agree on rules which map submitted random endowments to sharing securities. These rules are the ones that would efficiently share any bundle of submitted random endowments, i.e., they are supposed to be consistent with the optimal sharing ones coming from Arrow-Debreu equilibrium. It is well known (see e.g., [Bor62]) that under CARA preferences the sharing securities are linear functions of the agents' endowments. 3 At a second stage, agents propose securities and valuation rules, aiming to maximise their own expected utility. Given the agreed rules of the first stage, proposing risk-sharing securities and a valuation kernel is in fact equivalent to agents reporting the risk they are willing to share. Knowledge of the aggregate risk that counterparties are willing to share may result in a readjustment of how much risk an agent himself wishes to share, which could be only part of his actual risky position. In effect, agents form a Nash game by responding to other agents' submitted endowments; the fixed point of this game (if it exists) is called Nash risk sharing equilibrium. 4 The first step of analysing Nash risk-sharing equilibria is to address the well-posedness of an agent's best-response problem, which is addressed in Section 2. Agents have motive to exploit other agents' reported hedging needs and drive the sharing transaction as to maximise their own utility. Each agent's strategic choice set consists of all possible risky positions to be shared, and the optimal is called best endowment response. Although this infinite-dimensional maximisation problem involves a non-concave functional (see Appendix B), it admits a unique (modulo constants) solution. 5 It is shown that the risk that an agent is willing to share coincides with his actual risk exposure only when his position cannot be improved by any transaction with other agents. Furthermore, securities corresponding to best endowment responses have bounded liability. In fact, the arguments and results of the best response problem receive extra attention and discussion in the paper, since they demonstrate in particular how an agent can exploit asymmetric information on the risk of his counterparties (consider, for instance, the design of a reinsurance contract where only one insurance company knows the risk exposure of the other). 3 The assumption of agreement in risk-sharing rules is consistent with practice. In fact, securities such as reinsurance contracts and income swaps are standardised functions of transaction inputs, which in our case are the endowments that agents are willing to share. Such agreement is also transaction-cost efficient, and could make a potential subsequent sell of the security easier for the agents. 4 The combination of strategic and competitive stages is widely used in the literature of financial innovation and risk sharing under a variety of different concepts (see among others, [Dj89], [Pes95], [Bis98], [Bra05] and [RZ09]). 5 We use a bare-hands approach in establishing the best response for this infinite-dimensional problem.
Nash risk-sharing equilibrium occurs when all agents apply the best endowment response strategy. In Section 3, we characterise Nash equilibrium as the solution of a certain finite-dimensional problem, the solution of which can be approximated using standard numerical procedures, such as Monte Carlo simulation. Based on this characterisation, we show that, in any non-trivial case, the Nash risk-sharing securities are different from the Arrow-Debreu ones. In particular, and in sharp contrast to the Arrow-Debreu risk-sharing equilibrium, even when agents' risky exposures are unbounded (and possibly negatively correlated) the Nash game prevents agents from sharing the tails of their exposures, since all Nash securities have bounded liability. As a result of behaving strategically, agents lose the protection that Arrow-Debreu risk-sharing provides exactly where they most need it: at the tails of their random endowments. This departure from Arrow-Debreu risk-sharing allocation caused by the game yields risk-sharing inefficiency, which we measure via the difference between the aggregate monetary utilities at Arrow-Debreu and Nash equilibria.
The case of two agents is of special practical interest, since it models a multitude of risk-sharing situations such as reinsurance contract design and financial contract design between firms and financial institutions. When only two agents are involved in risksharing, we prove existence of Nash equilibrium and, in contrast to the majority of Nash games with infinite dimensional strategic sets, its uniqueness.
In Section 4, we focus on induced Arrow-Debreu and Nash equilibria when one (or both) of the agents' preferences approach risk neutrality, first establishing that both aforementioned equilibria converge to well-defined limits. Notably, it is shown that an extremely risk tolerant agent drives the market to the same equilibrium regardless of whether the other agent acts strategically or just submits his true risk exposure. In other words, extremely risk tolerant agents dominates the transaction just like the asymmetric information one. The study of limiting equilibria indicates that, although there is loss of aggregate utility when agents act strategically, there is always higher utility gain in Nash as compared to Arrow-Debreu equilibrium for the extremely risk-tolerant agent, regardless of the risk aversion and endowment of the other agent. Extremely risk-tolerant agents are willing to undertake more risk in exchange of better cash compensation; under the risksharing game, they respond to the risk-averse agent's hedging needs by driving the market to higher price for the security they short. This implies that agents with sufficiently high risk tolerance, although still risk averse, prefer to share risk in markets where participation is limited. The case where both acting agents uniformly approach risk-neutrality is also treated, where it is shown that the limiting Nash equilibrium sharing securities equal half of the limiting Arrow-Debreu equilibrium securities, pointing towards the fact that Nash risk-sharing equilibrium results in loss of trading volume.
For the reader's convenience all the proofs of the paper are placed in Appendix A.
Probabilistic notation. In all that follows, random variables are defined on a standard probability space (Ω, F, P). We stress that no finiteness restriction is enforced on the state space Ω. We use P for the class of all probabilities that are equivalent to P. For Q ∈ P, we use "E Q " to denote expectation under Q, while "E" will be reserved for expectation under P. The space L 0 consists of all (equivalence classes, modulo almost sure equality) finitely-valued random variables endowed with the topology of convergence in probability-note that this topology does not depend on the representative probability from P, and that L 0 may be infinite-dimensional. For Q ∈ P, L 1 (Q) consists of all X ∈ L 0 with E Q [|X|] < ∞. We also use L ∞ for the subset of L 0 consisting of essentially bounded random variables. Whenever Q 1 ∈ P and Q 2 ∈ P, we use dQ 2 /dQ 1 to denote the (strictly positive) density of Q 2 with respect to Q 1 .
1. Optimal Sharing of Risk 1.1. Agents and preferences. We consider a market with a single future period, where all uncertainty is resolved. In this market, there are n + 1 economic agents, where n ∈ N = {1, 2, . . .}; for concreteness, define the index set I = {0, . . . , n}. Agents derive utility from the future consumption of a numéraire, and all considered securities are expressed in units of this numéraire; in particular, a deterministic amount in the future has the same present value for the agents. The preference structure of agent i ∈ I over future random outcomes is numerically represented via the concave exponential utility functional where δ i ∈ (0, ∞) is the agent's risk tolerance. For each i ∈ I, U i (X) is the certainty equivalent of position X ∈ L 0 , defined as the unique deterministic amount for which agent i ∈ I is indifferent between the new (riskless) and the previous (risky) position.
Define the cumulative (aggregate) risk tolerance Furthermore, define λ i := δ i /δ for all i ∈ I, and note that i∈I λ i = 1.
1.2. Endowments. Each agent in our framework carries a certain risky future payoff; we let E i ∈ L 0 denote the (cumulative, up to the point of resolution of uncertainty) position of agent i ∈ I. Following standard terminology, E i is called the random endowment of agent i ∈ I. The existence of these random endowments gives agents the incentive to design securities, the trading of which reduces their risk exposure.
It should be stressed that each agent i ∈ I is completely characterised by his risk tolerance level and his random endowment, i.e., by the pair (δ i , E i ). In other aspects, and unless otherwise noted, agents are considered symmetric (regarding information, bargaining power, cost of risk-sharing participation, etc).
Mild regularity conditions will be imposed on agents' endowments. In order to make headway with the formal description, we introduce the convex domain 6 (1.2) where X − denotes the negative part of X ∈ L 0 . Asking that X ∈ X only imposes a requirement on the magnitude of negative values that X can take. If X bb consists of uniformly bounded from below random variables, then L ∞ ⊆ X bb ⊆ X and X + X bb = X .
Remark 1.1. Whenever X ∈ X and Y ∈ X , it is straightforward to check that X ∧ Y ∈ X .
We now define the class E of all "admissible" endowments to be Throughout the paper, we work under the following assumption Assumption 1.2 constitutes a very weak requirement: indeed, asking that U i (E i ) > −∞ for i ∈ I (which in effect means that the initial agents' position is not dramatically risky) Let E := i∈I E i be the aggregate endowment. Set also E −i := E − E i to be the endowment of all agents except i ∈ I; analogously, define δ −i := δ − δ i for all i ∈ I. 7 Remark 1.3. Suppose that (F i ) i∈I ∈ E, and set F := i∈I F i and F −i := F − F i for all i ∈ I. Convexity of X implies that F ∈ δX , as well as F −i ∈ δ −i X for all i ∈ I. Therefore, 6 As can be easily checked, for the validity of all results until Section 4 one may use the convex domain where κ := i∈I (1/λ i ), which is larger than X defined in (1.2). The slightly more restrictive domain X is however needed for the results in Section 4. 7 More generally, a subscript "−i" in a quantity is understood to aggregate every agent except i ∈ I.
i.e., F i ∈ L 1 (Q F ) for all i ∈ I. In particular, the previous remarks apply to (E i ) i∈I ∈ E.
Remark 1.4. As is well known, our exponential utility setting with random endowment can incorporate different subjective probabilities for the agents. Indeed, if P i ∈ P is the subjective probability of agent i ∈ I, and upon defining for the purpose of our analysis, one may consider agent i ∈ I to have endowment G i and subjective probability P.
1.3. Risk equivalence. Since deterministic endowment does not entail any risk, we identify the risk of future positions that differ by a constant. More precisely, we introduce the following equivalence class: For X ∈ L 0 and Y ∈ L 0 , we write X ∼ Y and call X and Y risk-equivalent if and only if there exists c ∈ R such that Y = X + c. The classes generated by the equivalence relation ∼ will be called risk-equivalence classes.
The risk-equivalence relation ∼ is also extended on (L 0 ) I in the following way: if (F i ) i∈I and (G i ) i∈I belong to (L 0 ) I , we write (F i ) i∈I ∼ (G i ) i∈I if and only if there exist (c i ) i∈I ∈ R I such that G i = F i + c i holds for all i ∈ I. It will be clear from the context whether ∼ refers to the risk-equivalence relation on L 0 or (L 0 ) I .
1.4. Securities and valuation. As has been already mentioned, discrepancies amongst agents' preferences and risky positions give rise to mutually beneficial trading opportunities. In principle, agents may improve their risk exposure by designing and trading securities in any desirable way, which essentially leads to a complete market. Such transactions are characterised by a pair of a valuation measure (that assigns prices to all possible securities), and the collection of the securities that are actually traded. In particular, since all future payoffs are measured under the same numéraire, (no-arbitrage) valuation corresponds to taking expectations with respect to probabilities in P. Given a valuation measure, agents agree in a collection (C i ) i∈I ∈ (L 0 ) I of zero-value securities, satisfying the market-clearing condition i∈I C i = 0; the position of each agent i ∈ I after the transaction is E i + C i . 8 Formally, for Q ∈ P we define the class C Q of securities that clear the market and are consistent with the valuation measure Q via 1.5. Arrow-Debreu equilibrium. When agents do not apply any kind of strategic behaviour in designing the securities, the agreed-upon transaction will actually form an Arrow-Debreu equilibrium. The valuation measure will determine indifference prices at which agents trade, and securities will be constructed in a way that maximise each agent's respective utility given the aforementioned valuation measure.
Definition 1.5. Q * , (C * i ) i∈I ∈ P × (L 0 ) I will be called an Arrow-Debreu equilibrium if: Under risk preferences modelled by (1.1) and the force of Assumption 1.2, one may explicitly obtain a unique Arrow-Debreu equilibrium. The proof of the following result is based on standard arguments (see for instance [Bor62], [BJ79] and [Buh84]).
Theorem 1.6. In the above setting, there exists a unique Arrow-Debreu equilibrium Q * , (C * i ) i∈I . In fact, the valuation measure Q * ∈ P is such that (1.4) log (dQ * /dP) ∼ −E/δ, and the equilibrium securities (C * i ) i∈I ∈ C Q * are given by 9 (1.5) Note that the valuation measure Q * and the optimal securities (C * i ) i∈I in Theorem 1.6 depend on (E i ) i∈I ∈ E only through its risk-equivalence class.
The securities described in (1.5) work as follows: agent i ∈ I exchanges the risky position E i for a fraction λ i = δ i /δ of the total aggregated endowment E, resulting in a where E i stands for the aggregate exposure of each one. In order to share their risk, the companies agree on the way they price future payoffs and on the actual security they are going to exchange. The generality of our model, however, allows the application of this scheme to a wider variety of real-world transactions. 9 See Remark 1.3 for the fact that Q * is well defined and E i ∈ L 1 (Q * ) holds for all i ∈ I.
For future reference, define u i := U i (E i ) for all i ∈ I, and note that u i ≤ u * i holds for all i ∈ I by definition of Arrow-Debreu equilibrium. Furthermore, set in words, u (resp., u * ) is the aggregate monetary value of all agents before (resp., after) the risk-sharing transaction. By (1.6), one readily obtains that which is the monetary utility of the representative agent with aggregate risk-tolerance δ.
In particular, an application of Jensen's inequality gives The last inequality shows that C * i is indeed the optimally-designed security for agent i ∈ I under the valuation measure Q * . Furthermore, for any collection (C i ) i∈I with i∈I C i = 0 A standard argument using the monotone convergence theorem extends the previous inequality to with equality if and only if C i ∼ C * i for all i ∈ I. Therefore, (C * i ) i∈I is a maximiser of the functional i∈I U i (E i + C i ) over all (C i ) i∈I with i∈I C i = 0; in fact, the collection of all such maximisers are exactly (C * i + p i ) i∈I where (p i ) i∈I ∈ R I is such that i∈I p i = 0. It can be shown that all Pareto optimal securities are exactly of this form; see e.g., [JST08, Theorem 3.1] for a more general result. Because of this Pareto optimality, the collection ((C * i ) i∈I , Q * ) usually comes under the appellation of (welfare) optimal securities and valuation measure, respectively.
Remark 1.8. If we ignore the transaction costs, the cases where an agent has no motive to enter in a risk sharing transaction are extremely rare. Indeed, agent i ∈ I will not take part in the Arrow-Debreu transaction if and only if C i = 0, which happens when E i ∼ λ i E, or, equivalently, when E i /δ i ∼ E/δ. In particular, agents are already in Arrow-Debreu equilibrium and no transaction takes place if and only if the collection (E i /δ i ) i∈I of endowments in units of risk tolerance are in the same risk-equivalence class.

Agents' Best Endowment Response
2.1. Agent's strategic behaviour in risk sharing. In the Arrow-Debreu setting, the resulting equilibrium is based on the assumption that agents do not apply any kind of strategic behaviour. However, in the majority of practical risk-sharing situations, agents' participation is limited and the modelling assumption of absence of agents' strategic behaviour is unreasonable, resulting (amongst other things) in overestimation of market efficiency. In our indicative example of two insurers negotiating the design of a reinsurance contracts in order to share their accumulated risk exposures, the contract to be designed and its valuation will depend on the portfolios they report for sharing; therefore, it is reasonable to expect that what is reported by one party depends on the risk exposure submitted by the other party.
We model strategic behaviour in risk-sharing using a two-stages procedure. The first stage fixes the sharing rules given reported endowments as input. As discussed in §1.5, for any given set of endowments the optimal sharing rules are governed by the mechanism resulting in Arrow-Debreu equilibrium, as these are the rules that efficiently share any reported endowments among agents. It is then reasonable to assume that agents follow the same sharing mechanism for any collection of the endowments agents choose to share (F i ) i∈I ∈ E. 10 More precisely, in accordance to (1.4) and (1.5), and with F := i∈I F i , the agreed-upon valuation measure Q F ∈ P is such that log dQ F /dP ∼ −F/δ and the collection of securities that agents will trade are At the second stage, agents respond to the endowments that other agents have reported, forming a Nash-type game. The set of strategic choices that agents use is the family of admissible endowments, i.e., E of (1.3). The subject of the present section is to analyse the problem of a single agent and establish the best endowment response. The definition and analysis of the Nash risk-sharing equilibrium is taken up in Section 3.

2.2.
Best endowment response. Let (F i ) i∈I ∈ E be a collection of reported endowments. Recall that F = i∈I F i is the aggregate reported endowment, and set F −i := 10 Agreement on the sharing mechanism also reduces negotiation time and hence the related transaction costs. It is also consistent to what we observe in practice, where the general structure of the securities signed by institutions are standardised and adjusted according to the required inputs (in this case the agents' reported endowments).
11 Note that both the valuation measure and the traded securities introduced above depend on (F i ) i∈I only through its risk-equivalence class.
F − F i to be the reported endowment of all agents except i ∈ I. The assumption (F i ) i∈I ∈ E, coupled with convexity of X , implies that F −i ∈ δ −i X , for all i ∈ I.
Throughout the remainder of §2.2, we fix an agent i ∈ I and the aggregate reported endowment F −i ∈ δ −i X of the remaining agents, and seek the endowment that is going to be submitted for sharing by agent i, i.e., how agent i is going to respond to the aggregate reported risk from his counterparties. According to the rules described in §2.1, a reported endowment F i ∈ δ i X from agent i ∈ I will lead to entering a long position on the security with payoff reporting endowment F i , agent i ∈ I affects the security he obtains both by the random variable λ i F − F −i and via the valuation measure Q F used for the cash transaction ; the latter effect is highly non-linear. With the above understanding, and given F −i ∈ δ −i X , the response function of agent i ∈ I is where the fact that (λ i F −i − λ −i F i ) ∈ L 1 (Q F −i +F i ) follows from Remark 1.3. Therefore, the problem of agent i ∈ I is to reported the endowment that maximises the certainty equivalent of his position after the transaction, i.e., to identify F r i ∈ δ i X such that 12 Any F r i ∈ δ i X satisfying (2.1) shall be called best endowment response. 13 In view of the equality , valid for all c ∈ R, it follows that best endowment responses for agent i ∈ I given F −i , if they exist, may only be unique up to risk-equivalence.
The following result gives necessary and sufficient conditions for best endowment response. 12 In contrast to the majority of the related literature, the agent's strategic set of choices in our model may be of infinite dimension. This is an important generalisation, since it allows consideration of endowments that have infinite support like ones with the Gaussian distribution or risky exposures of arbitrarily fat tails (an substantial feature in the modelling of risk). 13 The best response problem (2.1) imposes no constrains on the tails of the agent's reported endowment, as long as it belongs in δ i X . In principle, it is possible for an agent to report exposure to risks with extremely fat left or right tails. Severe departures of reported from actual endowments is undesirable from a modelling point of view, as it may be deemed unrealistic. However, it will be shown (see Theorem 3.2 and subsequent discussion in §3.3.1) that extreme responses are endogenously excluded.
where ζ i ∈ R is given by 14 The proof of Proposition 2.1 is given in §A.1. The necessity of the stated conditions for best endowment response basically follows from applying first-order optimality conditions.
Establishing the sufficiency of the stated conditions is far from being trivial, due to the fact that response functions are not concave in general-see Appendix B for a counterexample.
Equation (2.3) gives an economic interpretation for ζ i : it equals the difference of certainty equivalents in units of risk tolerance after the transaction between agent i ∈ I and an aggregate agent representing all others, given that the reported endowment F −i of the other agents is treated as the actual one. The interpretation becomes more transparent Proposition 2.1 sets a roadmap for proving existence and uniqueness of a best response. Indeed, (2.2) has to be satisfied with ζ i given in (2.3). In order to find a best endowment response, we consider for each z i ∈ R the unique random variable It turns out that there is a unique such choice; once found, one simply sets ) to obtain the unique (up to risk equivalence) best endowment response of agent i ∈ I given F −i . The technical details of the proof of Theorem 2.2 below are given in §A.2.
Theorem 2.2. For i ∈ I and F −i ∈ δ −i X there exists a unique, up to risk-equivalence, Hence, the best response of agent i ∈ I equals to his true endowment (up to risk equivalence) if and only if log (1 + C r i /δ −i ) ∼ 0, which holds if and only if C r i = 0. 15 In words, the risky endowment that an agent chooses to share is in the same risk equivalence class as 14 U −i is the exponential monetary utility of an agent with risk tolerance δ −i .
the true endowment if and only if the agent has no incentive to participate in the risksharing transaction, given the aggregate reported endowment of other agents. Hence, in any non-trivial cases, agents' strategic behaviour implies a departure from reporting their true endowment.
Remark 2.3. In sharp contrast to the securities (C * i ) i∈I formed in Arrow-Debreu equilibrium, which are linear combinations of the reported endowments, the security that agent i ∈ I is going to enter after declaring endowment F r i ∈ δ i X is bounded from below by the constant −δ −i . Taking into account the reported aggregate endowment of the rest of the agents, each agent submits an endowment that results in limited liability from his side in the risk-sharing transaction.
2.3. Exploiting asymmetric information. One case where the best endowment response problem arises is when the endowment of every but a single agent constitutes public information. In this scenario, the (only) agent with information on other agents' endowment will drive the market to his preferable equilibrium transaction, based on his best endowment response. We regard this situation as exploiting asymmetric information, where a single agent has market power to strategically choose a position to report, while the rest of the agents submit their true endowments. Example 2.4. Suppose that I = {0, 1}, δ 0 = 1 = δ 1 , and that E i for i = 0, 1 have Gaussian law with mean zero and common variance σ 2 > 0, while ρ ∈ [−1, 1] denotes the correlation coefficient of agents' endowments. It is straightforward to check that C * 0 = (E 1 − E 0 )/2; therefore, within the Arrow-Debreu equilibrium the position of agent 0 is E 0 + C * 0 = E/2. On the other hand, given that agent 1 has reported the true endowment E 1 , the security C r 0 corresponding to the best response of agent 0 satisfies 2C r 2) and (2.3). For σ 2 = 1 and ρ = −0.5, straightforward Monte-Carlo simulation allows for the numerical computation of the probability density of E 0 + C r 0 (position after exploiting the asymmetric information), which is depicted in Figure 1 together with the probability density functions of the agent's initial position and his position after the Arrow-Debreu transaction. As compared to the Arrow-Debreu case, note that the lower bound of the security C r 0 guarantees a heavier right tail of the agent's position after transaction when applying strategic behaviour.

Nash Risk Sharing Equilibrium
In this section, we consider the situation where every single agent follows the same strategic behaviour indicated by the best response problem. As previously mentioned, the first stage of the game is to agree on the sharing rules for any collection of reported endowments, per the Arrow-Debreu ones determined by Theorem 1.6. With the wellposedness of the best endowment response problem established, we are ready to examine the second stage of the risk sharing game. In view of the analysis of Section 2, each individual agent has motive to declare a risk exposure different than the actually possessed one. Each agent is going to perform a best endowment response as in (2.1), given what other agents have reported for sharing. In a way, best response functions indicate a scheme of negotiation which determines the acting valuation measure and actual shared securities.
The fixed point of this Nash-type game (if it exists) will produce the securities and the valuation measure that equilibrate the agents' risk sharing transaction.
3.1. Revealed endowments. Looking the model from a more practical point of view, one may comment that agents do not report actual endowments, but rather agree on a valuation measure Q ∈ P and sharing securities (C i ) i∈I ∈ C Q . In fact, there is a one-to-one correspondence (modulo risk equivalence) between reporting endowments and proposing a valuation measure and securities' payoffs. More precisely, from the discussion of §2.1 it follows that the collection (F i ) i∈I ∈ (L 0 ) I that is consistent with a given valuation measure Q ∈ P and security payoffs (C i ) i∈I ∈ C Q is the one that satisfies Hence, one may find a uniquely defined (modulo risk equivalence) and securities (C i ) i∈I ∈ C Q . This viewpoint underlies the definition of Nash equilibrium that follows.
An alternative, possibly more enlightening, way to perceive revealed endowments is through revealed subjective probabilities. As mentioned in Remark 1.4, allowing different subjective probabilities is equivalent to adjusting agents' random endowments. In this sense, revealed endowments become the way that individual agents report subjective views regarding the overall risk in the economy. Agents then negotiate through the individual reported subjective probabilities on the overall risky exposure.
3.2. Nash equilibrium and its characterisation. Following classic literature, we give the formal definition of a Nash risk-sharing equilibrium below.
the corresponding revealed endowments, we have (F i ) i∈I ∈ E and A use of Proposition 2.1 results in the characterisation Theorem 3.2 below, the proof of which is given in §A.3. For this, we introduce the n-dimensional Euclidean space Theorem 3.2. The collection (Q , (C i ) i∈I ) ∈ P × (L 0 ) I is a Nash equilibrium if and only if the following three conditions hold: (1) C i > −δ −i for all i ∈ I, and there exists z = (z i ) i∈I ∈ ∆ I such that (2) with Q * of (1.4), i.e., such that log(dQ * /dP) ∼ −E/δ, it holds that (3) E Q [C i ] = 0 holds for all i ∈ I.
Remark 3.3. Suppose that the agents' endowments are such that no trade occurs in Arrow-Debreu equilibrium, which happens when (E i /δ i ) i∈I belong in the same risk-equivalence class-see Remark 1.8. In this case, Q * = P and C * i = 0 for all i ∈ I. It is then straightforward from Theorem 3.2 to see that a Nash equilibrium is also given by Q = P and C i = 0 (as well as z i = 0) for all i ∈ I. In fact, as will be argued in §3.3.4, this is the unique Nash equilibrium in this case. Conversely, suppose that a Nash equilibrium is given by Q = P and C i = 0 for all i ∈ I. Then, (3.3) shows that Q * = Q = P and (3.2) 3.3. Within equilibrium. Throughout §3.3, we assume that (Q , (C i ) i∈I ) is a Nash risk-sharing equilibrium, and provide a discussion on certain aspects of it.
3.3.1. If trading, you never share your true position. It was argued after the statement of Theorem 2.2 that each agent's best response differs from the actual risky position that the agent carries in any case of risk transfer. This result becomes more pronounced when we consider the Nash risk-sharing equilibrium. To wit, if (F i ) i∈I are revealed endowments corresponding to a Nash equilibrium, it follows as a consequence of Theorem 3.2 that Note that F i ∼ E i holds if and only if C i = 0 for any fixed i ∈ I; therefore, whenever an agent takes part (by actually trading) in Nash equilibrium, the agent's reported endowment is never in the same risk-equivalence class as the actual one. 16 Note also that F : The quantity L in (3.5) indicates the additional risk added to the true aggregate endowment E that is traded as a result of the game among agents. In fact, not only are the securities in Nash equilibrium far from optimal, but typically this equilibrium transaction increases the risk traded in the market; indeed, as we will see below, L is always bounded from above, but not necessarily bounded from below. 16 According to the endogenous security bounds established in §3.3.2, the difference between the revealed endowment in Nash equilibrium and the actual endowment of an agent, which is the risk-equivalence class of log (1 + C i /δ −i ), is always bounded from above. In Nash equilibrium, agents may overstate only their downside exposure, revealing endowments with fatter left tail than the true ones. However, (3.2) (and

3.3.2.
Endogenous bounds on traded securities. As was pointed in the discussion following Theorem 2.2, the security that each agent is entering is bounded from below. In the case where all participating agents follow the same strategic behaviour, Nash equilibrium securities are bounded from above as well. Indeed, since C i > −δ −i is valid for all i ∈ I and i∈I C i = 0 holds, it also follows that C i = − j∈I\{i} C j < j∈I\{i} δ −j = (n − 1)δ + δ i , for all i ∈ I. Note that these endogenous bounds −δ −i < C i < (n−1)δ +δ i depend only on the risk tolerance profile of the agents, and not on their actual endowments. In addition, these bounds become stricter in games where quite risk-averse agents are playing, as they become more hesitant towards encountering risk.
There is clear economic interpretation of this sharp difference with the Arrow-Debreu equilibrium of (1.5). According to the best endowment response, each agent bounds the liability of the security that he takes a long position in. Since the market clears, the security that agents take a long position into is shorted by the rest of the agents, who similarly intend to bound their liabilities. Therefore, the potential gain from the agents' payoffs are also bounded. Naturally, the resulting endogenous bounds are an indication of how the negotiation among agents results in restricting the risk-sharing transaction, which in turn may be a source of large loss of efficiency. It should be also highlighted that boundedness of the securities' payoff reduces the risk improvement for the agents exactly where they mostly need it, that is on the (left) tails of their endowments-see Example 3.4 later on for an extreme situation.
3.3.3. Loss of efficiency. It is clear from the previous discussion that the agents' strategic behaviour results in risk-sharing inefficiency, which we may measure through the difference of the aggregate utility under the Arrow-Debreu transaction and the aggregate utility under the Nash equilibrium risk-sharing transaction. Indeed, since utilities (U i ) i∈I are numerically represented by certainty equivalents, aggregate utilities are measured in monetary units and hence can well be compared. 17 Mathematically, the loss of efficiency and u * are defined in (1.6) and (1.7), while 17 Similar measures of inefficiency have been used in the related to risk sharing literature-see e.g., [Vay99] or [AB05].
From (1.8), (3.2), (3.3) and (3.5), it follows that Recalling that E Q [C i ] = 0 holds for all i ∈ I, and noting that Adding up (3.6) over all i ∈ I and using the fact that i∈I z i = 0, one obtains holds for all i ∈ I, we indeed have u ≤ u * (which was anyway known from Remark 1.7); furthermore, the equality u = u * happens if and only if C i = 0 holds for all i ∈ I, which happens if and only if C * i = 0 holds for all i ∈ I-see Remark 3.3. In other words, the Nash risk-sharing equilibrium always implies a strict loss of efficiency, except for the case where there is no trading within Nash equilibrium (which is equivalent to the case where there is no trading within Arrow-Debreu equilibrium as well).
The next example demonstrates large inefficiency in a simple setting. Later on, in Figure 3, the loss of utility in another two-agent example is visualised.
Example 3.4. We illustrate this by use of the set-up of Example 2.4, for σ 2 > 0 and ρ = −1. To make the notation more concrete, let X ∈ L 0 have the standard Gaussian law under P, and set E 1 = σX = −E 0 . Note that C * 0 = −E 0 and C * 1 = −E 1 , which means that within Arrow-Debreu equilibrium the agents face no risk. Such risk-sharing in this particular example is ideal, since it ensures the full hedging of both agents' risk exposure. On the other hand, as will be established in Theorem 3.8, there exists a unique (up to risk-equivalence) Nash equilibrium. In fact, in this symmetric case we have that −1 < C 0 < 1, and it can be checked that Clearly, the loss of efficiency caused by the game is enormous, especially if σ 2 is large. In fact, if σ converges to infinity, it can be shown that C 0 converges to sign(X) = I {X<0} − I {X>0} . Hence, when the agent's risky position resolves to any negative value (and, given that σ is huge, this outcome will be quite negative), the hedge from risk-sharing will only provide a single unit of the numéraire.
3.3.4. A priori information on z . From (3.6) and (3.7), one obtains The last equality implies an economic interpretation for is the fraction λ i , corresponding to agent i ∈ I, of the aggregate loss of utility caused by forming a Nash equilibrium, instead of Arrow-Debreu one; on the other hand, u i − u * i is the difference between the actual utility that agent i ∈ I acquires in Nash equilibrium from the Arrow-Debreu one.
Note that, although the aggregate utility u in Nash equilibrium risk sharing can never be higher than the Arrow-Debreu aggregate utility u * , it may happen that some agents benefit from the game, in the sense that their individual utility after the negotiation game is higher when compared to the utility gain of the Arrow-Debreu equilibrium. We will address such cases in Section 4. Equation (3.8) is useful in obtaining tight bounds on z = (z i ) i∈I . Using the facts that u i ≤ u i and u ≤ u * , and the equality Combined with i∈I z i = 0, the previous a priori bounds imply that z has to live in a compact simplex on ∆ I . The bounds in (3.9) are indeed sharp: in the no-trade setting of Remark 3.3, it follows that u * i − u i = 0 for all i ∈ I, which implies that z i ≥ 0 should hold for all i ∈ I; since z ∈ ∆ I , it follows that z i = 0 should hold for all i ∈ I. This also shows that the trivial Nash equilibrium obtained in Remark 3.3 is unique.
3.3.5. Individual marginal indifference valuation. In view of (3.7) and the subsequent discussion, and recalling Remark 1.7, it follows that the allocation in Nash equilibrium fails to be Pareto optimal (except in the trivial no-trade case). Another way to demonstrate the inefficiency of Nash equilibrium is through the disagreement between the individual agent's marginal (utility) indifference valuation measures (Q i ) i∈I after the Nash risksharing transaction, for which log (dQ i /dP) ∼ − (E i + C i ) /δ i for all i ∈ I. 18 Indeed, in 18 By definition, for a fixed agent i ∈ I, given a position G i ∈ δ i X , the marginal indifference valuation view of (3.2) and (3.3), it follows that log (dQ i /dP) ∼ log (dQ /dP) + log (1 + C i /δ −i ); Pareto optimality would require all (Q i ) i∈I to agree, which is equivalent to all securities (C i ) i∈I belonging to the same risk-equivalence class. The latter is possible only if C i = 0, for all i ∈ I, i.e., exactly when no trades occurs.
All Nash securities (C i ) i∈I have zero value under Q . For each individual agent i ∈ I, we can measure the marginal indifference value of C i via In particular, note that This observation implies that (except in trivial situations of no trading) all agents would be better off if they would take a larger position in their individual securities; for all a ∈ R + the collection (aC i ) i∈I of securities clears the market, and for some a > 1 this collection of securities would result in higher utility for each agent than using securities (C i ) i∈I . Of course, what prevents agents from doing so is that they would find themselves in (Nash) disequilibrium. The fact that agents will not agree on market-clearing collections (aC i ) i∈I which for some a > 1 would be individually (and therefore, also collectively) preferable also indicates that trading volume within Nash equilibrium tends to be reduced.
The Nash valuation measure Q can be nicely expressed in terms of the individual marginal indifference valuation measures (Q i ) i ∈ I. To wit, define weights (α i ) i∈I via α i := δ −i /nδ = (1 − λ i )/n for all i ∈ I (noting that 0 < α i < 1/n holds for all i ∈ I, and that i∈I α i = 1); then, from (3.10) and the market clearing condition i∈I C i = 0, it follows that is maximised at q = 0 for all X ∈ L ∞ , which in words means that if prices given by Q i , then agent i ∈ I has no incentive to take any position other than the given G i ; using first-order conditions, it is straightforward to show that log (dQ i /dP) ∼ −G i /δ i holds. In Arrow-Debreu equilibrium, the collection In words, the Nash valuation measure Q is a convex combination of the individual agent's marginal indifference valuation measures, assigning weight α i to agent i ∈ I. Note also that risk-averse agents carry more weight; however, since max i∈I α i < 1/n, Q is almost equal to the equally-weighted average of (Q i ) i∈I for large numbers of agents. 19 The marginal indifference valuation measures (Q i ) i∈I of (3.10) can be used to provide a decomposition of the utility gain or loss of agents between the Nash and Arrow-Debreu transactions. Before providing the details below, recall that the relative entropy of Q 2 ∈ P with respect to Q 1 ∈ P is defined via Continuing, note first that L + (u * − u ) = −δ log (dQ /dQ * ), which combined with (3.8) . In turn, recalling (3.2) and (3.10), we obtain , for all i ∈ I; taking expectations with respect to Q * , it follows that (3.14) The difference of individual agents' utilities in the two equilibria comes from two distinct sources. The first stems from the discrepancy (measured via the relative entropy) of the Arrow-Debreu optimal valuation from the individual marginal indifference valuation of agent i ∈ I in Nash equilibrium. When the agents' marginal indifference valuation measure in Nash equilibrium is close to the Arrow-Debreu measure, his loss of utility caused by the Nash game is lower. In a sense, this is the part of aggregate loss of utility that is "paid" by agent i ∈ I (see also (3.15) below). The other term on the right-hand-side of (3.14) regards the price under the Arrow-Debreu valuation measure Q * of the actual security that agent i ∈ I buys at Nash equilibrium. Recall that Nash equilibrium prices of the Nash securities (C i ) i∈I are zero, positivity of E Q * [C i ] implies that the security C i is undervalued in Nash equilibrium transaction. Again, note that if Q i is close to Q * , the valuation E Q * [C i ] tends to be positive, since E Q i [C i ] is always nonnegative (see (3.11)).
19 It is also interesting to consider the situation of two interacting agents, with one of them being considerably more risk tolerant than the other. In this case, Q will be very close to the risk-averse agent's marginal utility-based valuation measure, resulting in him agreeing with the quoted prices; on the other hand, the possible discrepancy of Q from the risk tolerant agent's marginal utility-based valuation measure is beneficial from his side, as he is given the opportunity to purchase a security for zero price that has actually positive value for him. A limiting instructive case is treated in Section 4.
To recapitulate the previous discussion: agents whose marginal indifference valuation measure is close to the Arrow-Debreu one tend to benefit from the Nash game. 20 Due to the market-clearing condition i∈I C i = 0, the aggregate loss takes into account only the aggregate discrepancy of individual marginal measures from the Arrow-Debreu optimal one: under-valuation of certain securities balances off by over-valuation of others.
Indeed, adding up (3.14) over all i ∈ I, gives which measures Nash inefficiency as aggregate discrepancy from optimal valuation of the individual agents' marginal indifference valuation in Nash equilibrium.
3.4. Equilibrium via root finding. In order to search for equilibrium, we use Theorem 3.2 as a guide, parametrising candidates for optimal securities using the n-dimensional space ∆ I introduced in (3.1). Proposition 3.5 that follows enables to reduce the search of Nash equilibrium, an inherently infinite-dimensional problem in our setting, to a finitedimensional one. The latter problem gives the necessary tools for numerical approximations of Nash equilibria (see also Example 3.7 below).
Proposition 3.5. For all z ∈ ∆ I there exists a unique (C i (z)) i∈I ∈ (L 0 ) I such that (Note that, necessarily, i∈I C i (z) = 0 for all z ∈ ∆ I .) Furthermore, upon defining it follows that (F i (z)) i∈I ∈ E.
In the notation of Proposition 3.5, for each z ∈ ∆ I , define the probability Q(z) via In view of Theorem 3.2, establishing Nash equilibria amounts to finding z ∈ ∆ I such that E Q(z) [C i (z)] = 0 holds for all i ∈ I. 21 We can in fact define a function : ∆ I → R + that 20 As we will see in Section 4, this happens for example when agent i ∈ I is sufficiently risk tolerant. 21 Note that −δ −i < C i (z) < (n − 1)δ + δ i for all i ∈ I and z ∈ ∆ I follows exactly as in §3.3.2. The previous bounds imply that C i (z) ∈ L 1 (Q(z)) holds for all i ∈ I and z ∈ ∆ I .
gives a "distance from equilibrium" via the formula Since C i (z) > −δ −i holds for all z ∈ ∆ I , is well defined. Furthermore, the inequality in view of the fact that i∈I C i (z) = 0 for all z ∈ ∆ I , which shows that is indeed R +valued. Furthermore, since log(x) < x − 1 holds for all x ∈ (0, ∞) \ {1}, for any z ∈ ∆ I it follows that (z) = 0 is equivalent to E Q(z) [C i (z)] = 0 for all i ∈ I.
The following result summarises the discussion of this section.
Theorem 3.6. With the previous notation, the following are true: • Assume that (Q , (C i ) i∈I ) is a Nash equilibrium, and let z ≡ (z i ) i∈I ∈ ∆ I be as in (3.2). Then, (z ) = 0.
The previous result provides a one-to-one correspondence between Nash equilibria and roots of . Recalling the discussion in §3.3.4, any root of belongs to the compact subset of ∆ I consisting of (z i ) i∈I ∈ ∆ I with z i ≥ −(u * i − u i ) for all i ∈ I. This fact allows for numerical approximations of Nash equilibria via Monte-Carlo simulation.
Its practical usefulness notwithstanding, Theorem 3.6 does not answer the question of actual existence of Nash equilibria and, in case of existence, the uniqueness. As we shall show below in Theorem 3.8, both existence and uniqueness can be verified when two agents are involved; this case is rather special, since it corresponds to root-finding with a onedimensional parameter. The question of existence and uniqueness for three or more agents remains open, and seems significantly more challenging from a mathematical perspective.
In all cases of numerical simulation that were carried out, we observed existence and uniqueness of a Nash equilibrium. The next example is representative.
Since C 0 = −C 1 , applying simple algebra in (3.16), we obtain that the Nash equilibrium risk sharing security C 0 is such that −δ 1 < C 0 < δ 0 and satisfies Recall that aggregate utility at the Nash equilibrium is lower compared to the Arrow-Debreu one. We can further calculate and compare the final position of each individual agent at these equilibria. For instance, in the symmetric situation that is illustrated in Figure 1, the limited liability of the security C r 0 implies less variability and flatter right tail of the agent's position. Under the Nash equilibrium however, security C 0 is also bounded from above, which implies that the pdf of agent's final position is shifted to the left (as pictured in Figure 3). Despite the above symmetric case, it is not necessary true that all agents suffer a loss of utility at the Nash equilibrium risk sharing. As we will see in the Section 4 that follows, for agents with sufficiently large risk tolerance the negotiation game results in higher utility gain than the one gained through Arrow-Debreau equilibrium.

Extreme Risk Tolerance
In this section, we investigate the Arrow-Debreu and Nash risk-sharing equilibria when agents' risk preferences approach risk neutrality, in the sense that risk tolerance approaches infinity. In order to focus on the economic interpretation of the results, we consider the simplified (but representative) case of two agents.
The analysis that follows examines two cases: first, when only one agent becomes extremely risk tolerant and, second, when both agents' risk tolerance coefficients uniformly approaches infinity. Besides the interest of this analysis in its own right, it also offers the opportunity to compare the utility gain in Nash and Arrow-Debreu equilibria and substantiate the claim that highly risk tolerant agents benefit from the risk sharing game.
4.1. One extremely risk tolerant agent. In order to explore the asymptotic behaviour in Arrow-Debreu and Nash equilibrium when the risk aversion of an agent approaches zero, we consider the two-agent case I = {0, 1}. We keep the risk tolerance δ 1 and endowment E 1 of the agent 1 fixed, assuming that E 1 ∈ δ 1 X . On the other hand, for agent 0 we consider a sequence of risk tolerance coefficients (δ m 0 ) m∈N with the property that lim m→∞ δ m 0 = ∞ and a sequence of random endowments (E m 0 ) m∈N . In par with Assumption 1.2, E m 0 ∈ δ m 0 X holds for all m ∈ N. In order to ensure that all limiting formulas below are valid, we further assume that the sequence (E m 0 /δ m 0 ) m∈N of (normalised, in units of risk tolerance) endowments satisfies: Under the force of Assumption 4.1, we define G 0 ∈ L 0 + via Remark 4.2. When E m 0 has no dependence on m ∈ N, G 0 ≡ 0 and certain formulas in the sequel simplify; we shall comment accordingly. We prefer to consider a more general situation in order to include in the limiting analysis the interplay between the level of risk tolerance and the agent's actual risky position. 22 In this setup, Theorem 1.6 and Theorem 3.8 state that for each m ∈ N there ex- Note that the aggregate utility at optimal risk sharing for each m ∈ N is equal to . While this does not necessarily have a limit as m ∈ N, it is always the case that there is zero limiting excess certainty equivalent for the agent with preferences approaching risk neutrality. 22 It is in fact rather reasonable to assume that the risky position carried by an agent is of the same order as his risk tolerance. Indeed, suppose for example that this risky position came as a result of a previous optimal investment problem in a market with conic investment opportunities; then, it is straightforward to check that the solution of the aforementioned problem for an agent with risk tolerance δ 0 is actually equal to the corresponding solution of an agent with unit risk tolerance, multiplied with δ 0 .
It is instructive to consider the case G 0 = 0. Then, the limiting behaviour of the Arrow-Debreu risk sharing transaction is that agent 0 undertakes all the cumulative risk. The fact that agent 0 has no limiting utility gain is indeed expected, since a risk neutral agent's utility is linear and been matched with the price of the security that he buys under the limiting valuation measure, which when G 0 = 0 is actually equal to P. On the other hand, the only case where there is no limiting utility gain for agent 1 is when he actually carries no risk, i.e., when E 1 equals a constant.
We now turn to Nash risk-sharing equilibrium. From (3.19), we obtain Accepting that the sequence (z m, 0 ) m∈N converges in R and (C m, 0 ) m∈N converges in L 0 (these conjectures actually have to be proved as part of Theorem 4.5 below), and given that lim m→∞ δ m 0 = ∞, lim m→∞ λ m 0 = 1, and L 0 -lim m→∞ C m, * 0 = C ∞, * 0 , the limiting security This heuristic discussion gives a method to compute the limit. For any z ∈ R, define the random variable C ∞ 0 (z) satisfying the equation Since the function (−1, ∞) x → x + log (1 + x) is strictly increasing and continuous and maps (−1, ∞) to (−∞, ∞), it follows that C ∞ 0 (z) is a well defined (−δ 1 , ∞)-valued random variable for all z ∈ R. Then, we should have C ∞, given as the limit of (z m, 0 ) m∈N , we may actually identify a priori what its value will be. To make headway, note that from (3.3) the limiting Nash valuation probability Q ∞, should be such that log (dQ ∞, /dQ ∞, * ) ∼ − log(1 + C ∞, ] = 0 is expected to hold at the limit, we obtain actually that E Q ∞, * (1 + C ∞, 0 /δ 1 ) −1 = 1 would have to be satisfied. The next result, the proof of which is given in §A.8, ensures that a unique such candidate z ∞, ∈ R exists.
Lemma 4.4. There exists a unique z ∞, Before we state our main result on the limiting behaviour of Nash equilibrium, we make a final observation. Recall from (3.4) that F m, 1 ∼ E 1 + δ 1 log (1 − C m, 0 /δ m 0 ) holds for all m ∈ N. Since lim m→∞ δ m 0 = ∞ and, as it turns out, (C m, 0 ) m∈N is convergent, the revealed endowment F m, 1 of agent 1 when m is large is very close to the actual endowment E 1 . This suggests that the same asymptotic behaviour should also hold in the asymmetric information case discussed in Subsection 2.3, where agent 0 uses his best endowment response to construct a security C m,r 0 given that agent 1 actually reports his true endowment E 1 . Indeed, as part of the following result (whose proof is given in §A.9), we obtain that the limiting security structure is the same, regardless of whether the riskaverse agent 1 enters in the game or simply reports his true endowment (in which case, the almost risk neutral agent behaves strategically).
Theorem 4.5. With the previous notation (in particular, of Lemma 4.4), it holds that The equality of the limits of (C m, 0 ) m∈N and (C m,r 0 ) m∈N implies that the strategic behaviour of a risk neutral agent dominates the risk sharing transaction. We focus on the instructive case G 0 = 0. Agents with high risk tolerance are willing to undertake more risk at the sharing transaction in return of a higher cash compensation; at the limit, the risk neutral agent is willing to satisfy the reported hedging needs of other agents, but by applying the best response strategy he achieves better prices for doing so. On the other hand, for the risk averse agent the reduction of the risk exposure is more important than a higher price to be paid. Hence, at the equilibrium he prefers to submit his true risk exposure even though he has to pay a higher price to the risk neutral agent. The situation is totally different in an Arrow-Debreu equilibrium transaction, where agents act basically as price takers and the securities and prices are determined by the efficiency of the transaction.
We argued in Subsection 3.3 that in any risk-transfer situation the Nash equilibrium incurs some loss of efficiency. Although the aggregate utility is reduced in Nash equilibrium when compared with the Arrow-Debreu one, certain agents may obtain higher utility gain in risk-sharing games. In particular, Proposition 4.6 below (the proof of which is given in §A.10) demonstrates that the agent with sufficiently high risk tolerance enjoys higher utility at Nash equilibrium transaction than the utility at the Arrow-Debreu equilibrium sharing. Recall (3.13) for the definition of relative entropy.
Proposition 4.6. Define Q ∞, ∈ P such that dQ ∞, /dQ ∞, * = (1 + C ∞, 0 /δ 1 ) −1 . Then: Note that the loss for agent 1 comes from two sides. The first is (1/δ 1 )Var Q ∞, (C ∞, 0 ), which is the gain of agent 0. The remaining quantity δ 1 H Q ∞, * |Q ∞, is in fact the loss from playing the game as opposed to sharing in a Pareto optimal way. Both terms are strictly positive as long as C ∞, 0 is not identically equal to zero.
The message of Proposition 4.6 is clear. The introduction of strategic behaviour allows agents with high risk tolerance to achieve the highest price that the risk averse agents are willing to pay in order to hedge their risky exposures. In contrast to the Arrow-Debreu equilibrium where the prices are given by the welfare optimal sharing measure, agents with sufficiently high risk tolerance are more willing to accept even more risk in the Nash game, with their strategy driving the market to better cash compensation for them. In fact, a risk averse agent not only tends to undertake all the efficiency loss caused by the game, but also fuels the utility gain of the risk-tolerant counterparty.
Recalling the discussion and notation of §3.3.5, we may offer some more detailed comments. From (3.10) and Proposition 4.6, it follows that the marginal valuation measure of agent 0 approaches the limiting optimal valuation measure Q ∞, * . This implies that, for large enough m ∈ N, the security that agent 0 takes a long position in ); according to (3.14) and the discussion that follows, his utility is increased. Furthermore, Proposition 4.6 implies that the discrepancy between his marginal valuation measure and the Arrow-Debreu valuation measure tends to zero faster than his risk tolerance goes to infinity. Therefore, for large enough m ∈ N the total outcome is a gain of utility in Nash game. For risk averse agent, the situation is different. From (3.12), it follows that Q m, 1 will be close to Q m, for large m ∈ N, which in turn will be close to Q ∞, . Hence, for large enough m ∈ N, not only is the security that he gets at Nash equilibrium is overvalued, he also carries all the risk-sharing inefficiency of the Nash equilibrium. fix E i ∈ L 0 , for each i ∈ {0, 1}, as well as λ 0 ∈ (0, 1) and λ 1 ∈ (0, 1) with λ 0 +λ 1 = 1. Consider a non-decreasing sequences (δ m ) m∈N such that lim m→∞ δ m = ∞. Define δ m i := λ i δ m for all m ∈ N and i ∈ {0, 1}. Hence, we are looking at the case where both agents becomes risk neutral at the same speed.
The technical condition that we shall ask is The proof of the following Theorem is given in §A.11.
Theorem 4.7. In the above set-up, and under the force of (4.1), the sequences (C m, * 0 ) m∈N and (C m, 0 ) m∈N converge in L 0 to limiting securities C ∞, * 0 and C ∞, Interestingly enough, the risk neutrality of both agents drives Nash equilibrium to the half of the Arrow-Debreu securities, which is an evidence of the market inefficiency caused by the strategic behaviour of risk-neutral agents. The result of Theorem 4.7 is another manifestation of the claim (initially made in §3.3.5) that trading volume in Nash equilibrium tends to be lower than Pareto-optimal allocations.

Appendix A. Proofs
A.1. Proof of Proposition 2.1. In order to ease the reading, in the course of the proof we denote Q F −i +F r i by Q r i .
A.1.1. First-order conditions. We shall prove here the necessity of the stated conditions for best endowment response.
where note that (F r i + X) ∈ δ i X , dividing by λ −i and setting the resulting expression equal to zero (as it should, because of first-order conditions for optimality), we obtain which is exactly (2.3). Ranging X over L ∞ in (A.2) gives us that C r i should satisfy Necessarily, C r i > −δ −i should hold. Taking logarithms and rearranging (A.3) gives (2.2).
A.1.2. Optimality of candidates for best response. We now proceed to showing that the necessary conditions for best response are also sufficient. 23 Assuming the conditions stated, holds for all X ∈ L 0 such that (F r i − δX) ∈ δ i X , which will establish the optimality of F r i . Due to cash-invariance of V i (·; F −i ), we may assume without loss of generality that the utility from the response is equal to .
Note first that 23 As mentioned in the discussion following Theorem 2.2, V i (·; F −i ) is not necessarily concave (see counterexample in Appendix B); therefore, first order conditions do not necessarily imply optimality. 24 Since exp (X) X is bounded from below, E Q r i [exp (X) X] is always well-defined and may take the value ∞.
Since C r i > −δ −i , one may define the (0, ∞)-valued random variable D r i := 1 + C r i /δ −i , where note that E Q r i [D r i ] = 1. With this in mind and using (2.2), we obtain for some constant c which does not depend on X ∈ L ∞ . For X ≡ 0, the quantity

an application of Jensen's inequality under the probability which has density
On the other hand, upon defining the quantity χ := log E Q r i [exp (X)] ∈ R, note that where in the last equality we have used the facts C r Using the inequality exp(x) ≥ 1 + x, valid for all x ∈ R, we obtain that Putting everything together, and in view of Remark A.1, it follows that it suffices to show that Note that the function (0, ∞) z → φ(z) = z log z is convex, since φ (z) = 1/z for all z ∈ (0, ∞). Therefore, Jensen's inequality gives φ(E Q r i [exp(X)]) ≤ E Q r i [φ(exp(X))], which is exactly what was required.
A.2. Proof of Theorem 2.2. For z i ∈ R implicitly define C i (z i ) ∈ L 0 as the (−δ −i , ∞)- (Existence and uniqueness of the solution follows from the fact that the function (−1, ∞) y → (δ/δ i )y + log(1 + y) is strictly increasing from −∞ to ∞.) For z i ∈ R, define also the (0, ∞)-valued random variable D i (z i ) := 1 + C i (z i )/δ −i , and note that it is the unique solution to the equation Observe that C i (and, as a consequence D i ) is increasing as a function of z i . It is also Proof. Let ψ i : R → (0, ∞) denote the inverse of the strictly increasing function (0, ∞) Since ψ i is increasing and ψ i (y) ≤ 1+y holds for all y > 0, it follows that In order to establish Theorem 2.2, we claim that need to show that there exists a In view of Proposition 2.1, this is clearly necessary; conversely, if such a unique ζ i ∈ R exists, it is straightforward to see that F r i := F i (ζ i ) ∈ δ i X (the latter holding in view of Lemma A.2) is a best endowment response.
The "fixed point" equation written in an equivalent way, which will prove more useful. Note first that Lemma A.2 implies that the probability Q i (z i ) such that is well defined.
Proof. Calculate which shows that It also holds that . By the definition of Q i (z i ) and (A.5), it follows that from which the result immediately follows.
From Lemma A.3, it follows that we have to establish the existence of a unique ζ i ∈ R with the property that it follows that f i (z i ) < ∞ for all z i ∈ R. It is straightforward to check f i is continuous, in view of the monotone convergence theorem. Note that (A.5) implies that Let P i be the probability such that , for all z i ∈ R, with the above inequality holding in view of the fact that the covariance of exp(D i (z i )) and D i (z i ) is non-negative under any probability. Using the monotone convergence theorem and the above relationship, Furthermore, the monotone convergence theorem and lows that there exists at least one ζ i ∈ R such that f i (ζ i ) = 1. We also claim that f i is strictly increasing. In preparation, note that differentiating (A.4) with respect to z i and rearranging gives D i (z i ) = q i (D i (z i )), where (0, ∞) y → q i (y) := λ i y/ (λ i + y). In particular, since q i is a increasing function, the covariance between D i (z i ) and D i (z i ) will be non-negative for all z i ∈ R under any probability. Straightforward computations using .
≥ 0 for all z i ∈ R, Theorem 2.2 has been proved.
A.3. Proof of Theorem 3.2. Suppose that (Q , (C i ) i∈I ) is a Nash equilibrium and let (F i ) i∈I ∈ E be the revealed endowments. In view of Proposition 2.1, it follows that Adding up all the previous equations and using i∈I C i = 0, we obtain F ∼ E + L , where L := j∈I δ j log 1 + C j /δ −j . Since log(dQ /dP) ∼ −F /δ, we obtain (3.3). Furthermore, we i + λ i L for all i ∈ I. Therefore, (3.2) is valid for appropriate z ≡ (z i ) i∈I ∈ R I . The market clearing conditions i∈I C i = 0 = i∈I C * i show that i∈I z i = 0, i.e., z ∈ ∆ I . Finally, the fact that E Q [C i ] = 0 holds for all i ∈ I results directly from (C i ) i∈I ∈ C Q .
For the proof of the reverse implication, assume conditions (1), (2) and (3). Since F ∼ (1) and (2) combined give the existence of (ζ i ) i∈I ∈ R I such that (1/λ i ) ( where the next-to-last equality follows from the fact that Plugging back into the definition of L(z) in (A.6), we obtain that should be satisfied.
We now proceed backwards, by showing that (A.8) has a unique solution. With z ∈ ∆ I fixed, define w : Ω × R → R via w(y) = y − i∈I δ i log θ i (z i + C * i + λ i y) for y ∈ R, where the dependence of w in ω ∈ Ω is suppressed. The derivative of w with respect to the spatial coordinate is equal to Since θ i (y) behaves sub-linearly as y → ∞, lim y↑∞ w(y) = ∞ follows in a straightforward way. Furthermore, since x < δ i log θ i (x) holds for all x ∈ (−∞, 0) and i ∈ I, it follows that on the event y < ∧ j∈I −(z j + C * j )/λ j , one has w(y) < y − i∈I (z i + C * i + λ i y) = 0, which shows both that the equation w(L(z)) = 0 has a unique solution, and that where κ(z) ∈ R. Given the existence of a unique L(z) solving (A.8), the values of C i (z) are specified for all i ∈ I via (3.16). Furthermore, (3.16) and (3.17) combined imply that Given that (E j /δ j ) ∈ X for all j ∈ I, Remark 1.1 implies that F i (z) ∈ δ i X for all i ∈ I, which concludes the proof.
A.5. Coercivity of the distance function. We state and prove a result of independent interest, that will also be used in the proof of Theorem 3.8 later on.
Proof. If the result was not true, one would be able to find a ∆ I -valued sequence (z k ) k∈N such that lim k→∞ |z k | = ∞ and sup k∈N (z k ) < ∞ hold. Note first that since C i (z) < (n − 1)δ + δ i holds for all i ∈ I, there exists c ∈ R such that L(z k ) < c holds for all k ∈ N.
Furthermore, lim k→∞ |z k | = ∞ implies that lim k→∞ z k j = −∞ holds for at least one j ∈ I. (Indeed, this holds because i∈I z k i = 0 is true for all k ∈ N.) Recall from (A.7) and the notation set in §A.4 that 1 + C j (z k )/δ −j = θ j z k j + C * j + λ j L(z k ) holds for all k ∈ N; then, since lim y↓−∞ θ j (y) = 0, it follows that L 0 -lim k→∞ C j (z k ) = −δ −j . Continuing, note that log dQ(z k )/dP ∼ − E + L(z k ) /δ ∼ − E j + η j 1 + C j (z k )/δ −j /δ j . If P j denotes the probability with log (dP j /dP) ∼ −E j /δ j , it follows that the last inequality being true due to the fact that η j is an increasing function, implying that the covariance of exp −η j 1 + C j (z k )/δ −j and C j (z k ) is non-positive under any probability. Combined with L 0 -lim k→∞ C j (z k ) = −δ −j and the dominated convergence theorem, we obtain that lim k→∞ E Q(z k ) C j (z k ) = −δ −j , which shows that lim k→∞ (z k ) = ∞. We have reached a contradiction; therefore, is coercive.
A.6. Proof of Theorem 3.8. Note that (z 0 , z 1 ) ∈ ∆ I if and only if z 0 = −z 1 . In the course of the proof, we identify R and ∆ I via R z ↔ (z, −z) ∈ ∆ I , i.e., considering only the "zero" coordinate. Correspondingly, for z ∈ R we write C i (z) instead of C i ((z, −z)) for i ∈ {0, 1}; similarly, for z ∈ R we write L(z) instead of L(z, −z) of (A.6).
In view of Theorem 3.6, as well the equality C 1 (z) = −C 0 (z) for all z ∈ R, we need to prove the existence of a unique z ∈ R such that E Q(z ) [C 0 (z )] = 0 holds. Define the In the proof of Lemma A.4, it was it follows that −δ 1 = lim z↓−∞ f 0 (z) < lim z↑∞ f 0 (z) = δ 0 . Therefore, in order to prove Theorem 3.8, it suffices to show that f 0 is strictly increasing.
In other words, with q( L (z) = −δq(C 0 (z)) holds for all z ∈ R. Straightforward computations give that which implies that the covariance between C 0 (z) and −L (z) is non-negative under any probability, for all z ∈ R. Continuing, it is straightforward to compute that f 0 (z) = E Q(z) [C 0 (z)] − Cov Q(z) (C 0 (z), L (z)/δ) holds for all z ∈ R. Since C 0 (z) is (0, ∞)-valued and Cov Q(z) (C 0 (z), L (z)/δ) ≤ 0 for all z ∈ R, the claim is proved.
A.9. Proof of Theorem 4.5. We begin with a result on bounds of (C m, * 0 ) m∈N . Throughout, define p m, * Lemma A.5. There exists a constant c 0 ∈ (0, ∞) such that Proof. Let c 0 := sup m∈N |p m, * 0 | < ∞. Since C m, * , from which the second inequality follows.
Recall that Q m, ∈ P is such that log(dQ m, /dP) 3) and (3.5). The best response of agent 0 ∈ I for each m ∈ N is denoted by F m,r 0 . In order to ease the reading throughout the proof, for all m ∈ N define the (0, 1/λ m 1 )-valued random variable D m, and note by (2.2) and (3.19) that Applying (3.16) for i = 1 and using C m, It follows that log (dQ m, /dP) ∼ D m, . Therefore, if P 1 is the probability such that log (dP 1 /dP) ∼ −E 1 /δ 1 , then where the last inequality follows from Cov P 1 exp(D m, where the last inequality follows since Cov P m Continuing, since x ∈ (0, 1) implies φ m (x) ≤ λ m 0 δ 1 log(x), it follows that log (D m, Since the sequence ((z m, − p m, * ) /λ m 0 ) m∈N is bounded in R, the existence of a ∈ R such that (A.14) holds readily follows.
Lemma A.7. The sequence (z m,r 0 ) m∈N is bounded in R, and there exists c ∈ R such that Proof.
Recall that E Q m,r 0 D m,r 0 = 1, for all m ∈ N. Further, due to (A.5), it holds that log (dQ m,r 0 /dP) ∼ − (E 1 + F m,r 0 ) /δ m = − (E 1 − C m,r 0 ) /δ 1 ∼ −E 1 /δ 1 + D m,r 0 . Therefore, if P 1 is the probability such that log (dP 1 /dP) ∼ −E 1 /δ 1 , where the last inequality follows from the fact that Cov P 1 exp(D m,r 0 ), D m,r 0 ≥ 0 holds for all m ∈ N. It follows that E P 1 D m,r 0 ≤ 1 for all m ∈ N, which implies that (D m,r 0 ) m∈N is L 0 -bounded. Hence, the family {w m (D m,r 0 ) | m ∈ N} is also bounded from above in L 0 . Since (C m, * 0 ) converges in L 0 , (A.13) implies that (z m,r 0 ) m∈N is bounded from above (in R). Suppose that (z m,r 0 ) m∈N is not bounded from below. Passing to a subsequence if necessary, we may assume that (z m,r 0 ) m∈N is a sequence of negative numbers with lim m→∞ z m,r Since the sequence ((z m,r 0 − p m,r 0 )/δ 1 ) m∈N is bounded, the existence of c ∈ R such that (A.15) holds readily follows.
We now show that we have the expected limiting behaviour through subsequences. x → φ(x) = δ 1 (x − 1) + δ 1 log (x), and note that both (φ m ) m∈N of (A.12) and (w m ) m∈N of (A.11) converge uniformly to φ on compact subsets of (0, ∞). This fact, combined with Lemma A.6 and Lemma A.7 implies that (D m k , 0 ) k∈N has a (0, ∞)-valued L 0 -limit We first tackle the Nash equilibrium case. As in the proof of Lemma A.6, an application of (3.16) for i = 0 gives log (dQ m, /dP) ∼ − log D m, , ∀m ∈ N.
Lemma A.9. The sequence (z m, 0 ) m∈N is bounded in R.
Proof. We shall show that (z m, 0 ) m∈N is bounded above. A completely symmetric argument applied to agent 1 shows that (z m, 1 ) m∈N is bounded above; since z m, 0 = −z m, 1 holds for all m ∈ N, it will follow that (z m, 0 ) m∈N is also bounded below, which will complete the proof.