Equilibrium in risk-sharing games

Equilibrium in risk-sharing games

Michail Anthropelos Michail Anthropelos, Department of Banking and Financial Management, University of Piraeus anthropel@unipi.gr  and  Constantinos Kardaras Constantinos Kardaras, Statistics Department, London School of Economics k.kardaras@lse.ac.uk
July 13, 2019

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.

Key words and phrases:
Nash equilibrium, risk sharing, heterogeneous beliefs, reporting of beliefs
M. Anthropelos acknowledges support from the Research Center of the University of Piraeus. C. Kardaras acknowledges support from the MC grant FP7-PEOPLE-2012-CIG, 334540.


The structure of securities that optimally allocate risky positions under heterogeneous beliefs of agents has been a subject of ongoing research. Starting from the seminal works of [Bor62], [Arr63], [BJ79] and [Buh84], the existence and characterisation of welfare risk sharing of random positions in a variety of models has been extensively studied—see, among others, [BEK05], [JST08], [Acc07], [FS08]. On the other hand, discrepancies amongst agents regarding their assessments on the probability of future random outcomes reinforce the existence of mutually beneficial trading opportunities (see e.g. [Var85], [Var89], [BCGT00]). However, market imperfections—such as asymmetric information, transaction costs and oligopolies—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 has been addressed under both static and dynamic models (see, among others, [NN94], [MR00], [Par04], [Axe07], [Wil11]). The importance of frictions like transaction costs has be highlighted in [AG91]; see also [CRW12].

The present work aims to contribute to the risk-sharing literature by focusing on how over-the-counter (OTC) transactions with a small number of agents motivate strategic behaviour. The vast majority of real-world sharing instances involves only a few participants, each of whom may influence the way heterogeneous risks and beliefs are going to be allocated. (The seminal papers [Kyl89] and [Vay99] highlight such transactions.) As an example, two financial institutions with possibly different beliefs, and in possession of portfolios with random future payoffs, may negotiate and design innovative asset-backed securities that mutually share their defaultable assets. Broader discussion on risk-sharing innovative securities is given in the classic reference [AG94] and in [Tuf03]; a list of widely used such securities is provided in [Fin92].

As has been extensively pointed out in the literature (see, for example, [Var89] and [SX03]), it is reasonable, and perhaps even necessary, to assume that agents have heterogeneous beliefs, which we identify with subjective probability measures on the considered state space. In fact, differences in subjective beliefs do not necessarily stem from asymmetric information; agents usually apply different tools or models for the analysis and interpretation of common sets of information.

Formally, a risk-sharing transaction consists of security payoffs and their prices, and since only few institutions (typically, two) are involved, it is natural to assume that no social planner for the transaction exists, and that the equilibrium valuation and payoffs will result as the outcome of a symmetric game played among the participating institutions. Since institutions’ portfolios are (at least, approximately) known, the main ingredient of risk-sharing transactions leaving room for strategic behaviour is the beliefs that each institution reports for the sharing. We propose a novel way of modelling such strategic actions where the agents’ strategic set consists of the beliefs that each one chooses to declare (as opposed to their actual one) aiming to maximise individual utility, and the induced game leads to an equilibrium sharing. Our main insights are summarised below.

Main contributions

The payoff 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 work is the first instance that models the way agents choose the beliefs on future uncertain events that they are going to declare to their counterparties, and studies whether such strategic behaviour results in equilibrium. Our results demonstrate how the game leads to risk-sharing inefficiency and security mispricing, both of which are quantitatively characterised in analytic forms. More importantly, it is shown that equilibrium securities have endogenous limited liability, 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 contributions is to show that Nash equilibrium admits a finite-dimensional characterisation, with the dimensionality being one less than the number of participating agents. Not only does our characterisation provide a concrete algorithm for calculating the equilibrium transaction, it also allows to prove existence of Nash equilibrium for arbitrary number of players. In the important case of two participating agents, we even show that Nash equilibrium is unique. It has to be pointed out that the aforementioned results are obtained under complete generality on the probability space and the involved random payoffs—no extra assumption except from CARA preferences is imposed. While certain qualitative analysis could be potentially carried out without the latter assumption on the entropic form of agent utilities, the advantage of CARA preferences utilised in the present paper is that they also allow for substantial quantitative analysis, as workable expressions are obtained for Nash equilibrium.

Our notion of Nash risk-sharing equilibrium highlights the importance of agents’ risk tolerance level. More precisely, one of the main findings of this work is that agents with sufficiently low risk aversion will prefer the risk-sharing game rather than the outcome of an Arrow-Debreu equilibrium that would have resulted from absence of strategic behaviour. Interestingly, the result is valid irrespective of their actual risky position or their subjective beliefs. It follows that even risk-averse agents, as long as their risk-aversion is sufficiently low, will prefer risk-sharing markets that are thin (i.e., where participating agents are few and have the power to influence the transaction), resulting in aggregate loss of risk-sharing welfare.


Our model is introduced in Section 1, and consists of a two-period financial economy with uncertainty, containing possibly infinite states of the world. Such infinite-dimensionality is essential in our framework, since in general the risks that agents encounter do not have a-priori bounds, and we do not wish to enforce any restrictive assumption on the shape of the probability distribution or the support of agents’ positions. Let us also note that, even if the analysis was carried out in a simpler set-up of a finite state space, there would not be any significant simplification in the mathematical treatment.

In the economy we consider a finite number of agents, each of whom has subjective beliefs (probability measure) about the events at the time of uncertainty resolution. We also allow agents to be endowed with a (cumulative, up to the point of uncertainty resolution) random endowment.

Agents seek to increase their expected utilities through trading securities that allocate the discrepancies of their beliefs and risky exposures in an optimal way. The possible disagreement on agents’ beliefs is assumed on the whole probability space, and not only on the laws of the shared-to-be risky positions. Such potential disagreement is important: it alone can give rise to mutually beneficial trading opportunities, even if agents have no risky endowments to share, by actually designing securities with payoffs written on the events where probability assessments are different.

Each sharing rule consists of the security payoff that each agent is going to obtain and a valuation measure under which all imaginable securities are priced. The sharing rules that efficiently allocate any submitted discrepancy of beliefs and risky exposures are the ones stemming from Arrow-Debreu equilibrium. (Under CARA preferences, the optimal sharing rules have been extensively studied—see, for instance, [Bor62], [BJ79] and [BEK05].) In principle, participating agents would opt for the highest possible aggregate benefit from the risk-sharing transaction, as this would increase their chance for personal gain. However, in the absence of a social planner that could potentially impose a truth-telling mechanism, it is reasonable to assume that agents do not negotiate the rules that will allocate the submitted endowments and beliefs. In fact, we assume that agents adapt the specific sharing rules that are consistent with the ones resulting from Arrow-Debreu equilibrium, treating reported beliefs as actual ones, since we regard these sharing rules to be the most natural and universally regarded as efficient.

Agreement on the structure of risk-sharing securities is also consistent with what is observed in many OTC transactions involving security design, where the contracts signed by institutions are standardised and adjusted according to required inputs (in this case, the agents’ reported beliefs). Such pre-agreement on sharing rules reduces negotiation time, hence the related transaction costs. Examples are asset-backed securities, whose payoffs are backed by issuers’ random incomes, traded among banks and investors in a standardised form, as well as credit derivatives, where portfolios of defaultable assets are allocated among financial institutions and investors.

Combinations of strategic and competitive stages are widely used in the literature of financial innovation and risk-sharing, under a variety of different guises. The majority of this literature distinguishes participants among designers (or issuers) of securities and investors who trade them. In [DJ89], a security-design game is played among exchanges, each aiming to maximise internal transaction volume; while security design throughout exchanges is the outcome of non-competitive equilibrium, investors trade securities in a competitive manner. Similarly, in [Bis98], Nash equilibrium determines not only the designed securities among financial intermediaries, but also the bid-ask spread that price-taking investors have to face in the second (perfect competition) stage of market equilibrium. In [CRW12], it is entrepreneurs who strategically design securities that investors with non-securitised hedging needs competitively trade. In [RZ09], the role of security-designers is played by arbitrageurs who issue innovated securities in segmented markets. Mixture of strategic and competitive stages has also been used in models with asymmetric information. For instance, in [Bra05] a two-stage equilibrium game is used to model security design among agents with private information regarding their effort. In a first stage, agents strategically issue novel financial securities; in the second stage, equilibrium on the issued securities is formed competitively.

Our framework models oligopolistic OTC security design, where participants are not distinguished regarding their information or ability to influence market equilibrium. Agents mutually agree to apply Arrow-Debreu sharing rules, since these optimally allocate whatever is submitted for sharing, and also strategically choose the inputs of the sharing-rules (their beliefs, in particular).

Given the agreed-upon rules, agents propose accordingly consistent securities and valuation measures, aiming to maximise their own expected utility. As explicitly explained in the text, proposing risk-sharing securities and a valuation kernel is in fact equivalent to agents reporting beliefs to be considered for sharing. Knowledge of the probability assessments of the counterparties may result in a readjustment of the probability measure an agent is going to report for the transaction. In effect, agents form a game by responding to other agents’ submitted probability measures; the fixed point of this game (if it exists) is called Nash risk-sharing equilibrium.

The first step of analysing Nash risk-sharing equilibria is to address the well-posedness of an agent’s best response problem, which is the purpose of Section 2. Agents have motive to exploit other agents’ reported beliefs and hedging needs and drive the sharing transaction as to maximise their own utility. Each agent’s strategic choice set consists of all possible probability measures (equivalent to a baseline measure), and the optimal one is called best probability response. Although this is a highly non-trivial infinite-dimensional maximisation problem, we use a bare-hands approach to establish that it admits a unique solution. It is shown that the beliefs that an agent declares coincide with the actual ones only in the special case where the agent’s position cannot be improved by any transaction with other agents. By resorting to examples, one may gain more intuition on how future risk appears under the lens of agents’ reported beliefs. Consider, for instance, two financial institutions adapting distinct models for estimating the likelihood of the involved risks. The sharing contract designed by the institutions will result from individual estimation of the joint distribution of the shared-to-be risky portfolios. According to the best probability response procedure, each institution tends to use less favourable assessment for its own portfolio than the one based on its actual beliefs, and understates the downside risk of its counterparty’s portfolio. Example 2.8 contains an illustration of such a case.

An important consequence of applying the best probability response is that the corresponding security that the agent wishes to acquire has bounded liability. If only one agent applies the proposed strategic behaviour, the received security payoff is bounded below (but not necessarily bounded above). In fact, the arguments and results of the best response problem receive extra attention and discussion in the paper, since they demonstrate in particular the value of the proposed strategic behaviour in terms of utility increase. This situation applies to markets where one large institution trades with a number of small agents, each of whom has negligible market power.

A Nash-type game occurs when all agents apply the best probability response strategy. In Section 3, we characterise Nash equilibrium as the solution of a certain finite-dimensional problem. Based on this characterisation, we establish existence of Nash risk-sharing equilibrium for an arbitrary (finite) number of agents. In the special case of two-agent games, the Nash equilibrium is shown to be unique. The finite-dimensional characterisation of Nash equilibrium also provides an algorithm that can be used to approximate the Nash equilibrium transaction by standard numerical procedures, such as Monte Carlo simulation.

Having Nash equilibrium characterised, we are able to further perform a joint qualitative and quantitative analysis. Not only do we verify the expected fact that, in any non-trivial case, Nash risk-sharing securities are different from the Arrow-Debreu ones, but we also provide analytic formulas for their shapes. Since the securities that correspond to the best probability response are bounded from below, the application of such strategy from all the agents yields that the Nash risk-sharing market-clearing securities are also bounded from above. This comes in stark contrast to Arrow-Debreu equilibrium, and implies in particular an important loss of efficiency. We measure the risk-sharing inefficiency that is caused by the game via the difference between the aggregate monetary utilities at Arrow-Debreu and Nash equilibria, and provide an analytic expression for it. (Note that inefficient allocation of risk in symmetric-information thin market models may also occur when securities are exogenously given—see e.g. [RW15]. When securities are endogenously designed, [CRW12] highlights that imperfect competition among issuers results in risk-sharing inefficiency, even if securities is traded among perfectly competitive investors.)

One may wonder whether the revealed agents’ subjective beliefs in Nash equilibrium are far from their actual subjective probability measures, which would be unappealing from a modelling viewpoint. Extreme departures from actual beliefs are endogenously excluded in our model, as the distance of the truth from reported beliefs in Nash equilibrium admits a-priori bounds. Even though agents are free to choose any probability measure that supposedly represents their beliefs in a risk-sharing transaction, and they do indeed end up choosing probability measures different than their actual ones, this departure cannot be arbitrarily large if the market is to reach equilibrium.

Turning our attention to Nash-equilibrium valuation, we show that the pricing probability measure can be written as a certain convex combination of the individual agents’ marginal indifference valuation measures. The weights of this convex combination depend on agents’ relative risk tolerance coefficients, and, as it turns out, the Nash-equilibrium valuation measure is closer to the marginal valuation measure of the more risk-averse agents. This fact highlights the importance of risk tolerance coefficients in assessing the gain or loss of utility for individual agents in Nash risk-sharing equilibrium; in fact, it implies that more risk tolerant agents tend to get better cash compensation as a result of the Nash game than what they would get in Arrow-Debreu equilibrium.

Inspired by the involvement of the risk tolerance coefficients in the agents’ utility gain or loss, in Section 4 we focus on induced Arrow-Debreu and Nash equilibria of two-agent games, when one of the agents’ preferences approach risk neutrality. We first establish that both 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 plainly submits true subjective beliefs. In other words, extremely risk tolerant agents tend to dominate the risk-sharing transaction. The study of limiting equilibria indicates that, although there is loss of aggregate utility when agents act strategically, there is always utility gain in the Nash transaction as compared to Arrow-Debreu equilibrium for the extremely risk-tolerant agent, regardless of the risk tolerance level and subjective beliefs of the other agent. Extremely risk-tolerant agents are willing to undertake more risk in exchange of better cash compensation; under the risk-sharing game, they respond to the risk-averse agent’s hedging needs and beliefs by driving the market to higher price for the security they short. This implies that agents with sufficiently high risk tolerance—although still not risk-neutral—will prefer thin markets. 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, hinting towards the fact that Nash risk-sharing equilibrium results in loss of trading volume.

For convenience of reading, all the proofs of the paper are placed in Appendix A.

1. Optimal Sharing of Risk

1.1. Notation

The symbols “” and “” will be used to denote the set of all natural and real numbers, respectively. As will be evident subsequently in the paper, we have chosen to use the symbol “” to denote (reported, or revealed) probabilities.

In all that follows, random variables are defined on a standard probability space . We stress that no finiteness restriction is enforced on the state space . We use for the class of all probabilities that are equivalent to the baseline probability . For , we use “” to denote expectation under . The space consists of all (equivalence classes, modulo almost sure equality) finitely-valued random variables endowed with the topology of convergence in probability. This topology does not depend on the representative probability from , and may be infinite-dimensional. For , consists of all with . We use for the subset of consisting of essentially bounded random variables.

Whenever and , denotes the (strictly positive) density of with respect to . The relative entropy of with respect to is defined via

For and , we write if and only if there exists such that . In particular, we shall use this notion of equivalence to ease notation on probability densities: for and , we shall write to mean that and .

1.2. Agents and preferences

We consider a market with a single future period, where all uncertainty is resolved. In this market, there are economic agents, where ; for concreteness, define the index set . Agents derive utility only from the consumption of a numéraire in the future, and all considered security payoffs are expressed in units of this numéraire. In particular, future deterministic amounts have the same present value for the agents. The preference structure of agent over future random outcomes is numerically represented via the concave exponential utility functional


where is the agent’s risk tolerance and represents the agent’s subjective beliefs. For any , agent is indifferent between the cash amount and the corresponding risky position ; in other words, is the certainty equivalent of for agent . Note that the functional is an entropic risk measure in the terminology of convex risk measure literature—see, amongst others, [FS04, Chapter 4].

Define the aggregate risk tolerance , as well as the relative risk tolerance for all . Note that . Finally, set and , for all .

1.3. Subjective probabilities and endowments

Preference structures that are numerically represented via (1.1) are rich enough to include the possibility of already existing portfolios of random positions for acting agents. To wit, suppose that are the actual subjective beliefs of agent , who also carries a risky future payoff in units of the numéraire. Following standard terminology, we call this cumulative (up to the point of resolution of uncertainty) payoff random endowment, and denote it by . In this set-up, adding on top of a payoff for agent results in numerical utility equal to . Assume that , i.e., that . Defining via and via (1.1), holds for all . Hence, hereafter, the probability is understood to incorporate any possible random endowment of agent , and utility is measured in relative terms, as difference from the baseline level .

Taking the above discussion into account, we stress that agents are completely characterised by their risk tolerance level and (endowment-modified) subjective beliefs, i.e., by the collection of pairs . In other aspects, and unless otherwise noted, agents are considered symmetric (regarding information, bargaining power, cost of risk-sharing participation, etc).

1.4. Geometric-mean probability

We introduce a method that produces a geometric mean of probabilities which will play central role in our discussion. Fix . In view of Hölder’s inequality, holds. Therefore, one may define via . Since , one is allowed to formally write


The fact that implies , and Jensen’s inequality gives , for all . Note that (1.2) implies that the existence of such that holds; therefore, one actually has , for all . In particular, holds for all , and

1.5. Securities and valuation

Discrepancies amongst agents’ preferences provide incentive to design securities, the trading of which could be mutually beneficial in terms of risk reduction. In principle, the ability to design and trade securities in any desirable way essentially leads to a complete market. In such a market, transactions amongst agents are characterised by a valuation measure (that assigns prices to all imaginable securities), and a collection of the securities that will actually be traded. Since all future payoffs are measured under the same numéraire, (no-arbitrage) valuation corresponds to taking expectations with respect to probabilities in . Given a valuation measure, agents agree in a collection of zero-value securities, satisfying the market-clearing condition . The security that agent takes a long position as part of the transaction is .

As mentioned in the introductory section, our model could find applications in OTC markets. For instance, the design of asset-backed securities involves only a few number of financial institutions; in this case, stands for the subjective beliefs of each institution and, in view of the discussion of §1.3, further incorporates any existing portfolios that back the security payoffs. In order to share their risky positions, the institutions agree on prices of future random payoffs and on the securities they are going to exchange. Other examples are the market of innovated credit derivatives or the market of asset swaps that involve exchange of random payoff and a fixed payment.

1.6. Arrow-Debreu equilibrium

In the absence of any kind of strategic behaviour in designing securities, the agreed-upon transaction amongst agents will actually form an Arrow-Debreu equilibrium. The valuation measure will determine both trading and indifference prices, and securities will be constructed in a way that maximise each agent’s respective utility.

Definition 1.1.

will be called an Arrow-Debreu equilibrium if:

  1. , as well as and , for all , and

  2. for all with , holds for all .

Under risk preferences modelled by (1.1), a unique Arrow-Debreu equilibrium may be explicitly obtained. In other guises, Theorem 1.2 that follows has appeared in many works—see for instance [Bor62], [BJ79] and [Buh84]. Its proof is based on standard arguments; however, for reasons of completeness, we provide a short argument in §A.1.

Theorem 1.2.

In the above setting, there exists a unique Arrow-Debreu equilibrium . In fact, the valuation measure is such that


and the equilibrium market-clearing securities are given by


where the fact that holds for all follows from §1.4.

The securities that agents obtain at Arrow-Debreu equilibrium described in (1.4) provide higher payoff on events where their individual subjective probabilities are higher than the “geometric mean” probability of (1.3). In other words, discrepancies in beliefs result in allocations where agents receive higher payoff on their corresponding relatively more likely events.

Note also that the securities traded at Arrow-Debreu equilibrium have an interesting decomposition. Since , agent is indifferent between no trading and the first “random” part of the security . The second “cash” part of is always nonnegative, and represents the monetary gain of agent resulting from the Arrow-Debreu transaction. After this transaction, the position of agent has certainty equivalent


The aggregate agents’ monetary value resulting from the Arrow-Debreu transaction equals

Remark 1.3.

In the setting and notation of §1.3, let be the collection of agents’ random endowments. Furthermore, suppose that agents share common subjective beliefs; for concreteness, assume that , for all . In this case, and setting , the equilibrium valuation measure of (1.3) satisfies and equilibrium securities of (1.4) are given by , for all . In particular, note the well-known fact that the payoff of each shared security is a linear combination of the agents’ random endowments.

Remark 1.4.

Since , it is straightforward to compute


In particular, an application of Jensen’s inequality gives for , with equality if and only if . The last inequality shows that is indeed the optimally-designed security for agent under the valuation measure . Furthermore, for any collection with and for all , it follows that . A standard argument using the monotone convergence theorem extends the previous inequality to

with equality if and only if for all . Therefore, is a maximiser of the functional over all with . In fact, the collection of all such maximisers is where is such that . 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 usually comes under the appellation of (welfare) optimal securities and valuation measure, respectively.

Of course, not every Pareto optimal allocation , where is such that , is economically reasonable. A minimal “fairness” requirement that has to be imposed is that the position of each agent after the transaction is at least as good as the initial state. Since the utility comes only in the terminal time, we obtain the requirement , for all . While there may be many choices satisfying the latter requirement in general, the choice of Theorem 1.2 has the cleanest economic interpretation in terms of complete financial market equilibrium.

Remark 1.5.

If we ignore potential transaction costs, the cases where an agent has no motive to enter in a risk-sharing transaction are extremely rare. Indeed, agent will not take part in the Arrow-Debreu transaction if and only if , which happens when . In particular, agents will already be in Arrow-Debreu equilibrium and no transaction will take place if and only if they all share the same subjective beliefs.

2. Agents’ Best Probability Response

2.1. 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, the modelling assumption of absence of agents’ strategic behaviour is unreasonable, resulting, amongst other things, in overestimation of market efficiency. When securities are negotiated among agents, their design and valuation will depend not only on their existing risky portfolios, but also on the beliefs about the future outcomes they will report for sharing. In general, agents will have incentive to report subjective beliefs that may differ from their true views about future uncertainty; in fact, these will also depend on subjective beliefs reported by the other parties.

As discussed in §1.6, for a given set of agents’ subjective beliefs, the optimal sharing rules are governed by the mechanism resulting in Arrow-Debreu equilibrium, as these are the rules that efficiently allocate discrepancies of risks and beliefs among agents. It is then reasonable to assume that, in absence of a social planner, agents adapt this sharing mechanism for any collection of subjective probabilities they choose to report—see also the related discussion in the introductory section). More precisely, in accordance to (1.3) and (1.4), the agreed-upon valuation measure is such that , and the collection of securities that agents will trade are , .

Given the consistent with Arrow-Debreu equilibrium sharing rules, agents respond to subjective beliefs that other agents have reported, with the goal to maximise their individual utility. In this way, a game is formed, with the probability family being the agents’ set of strategic choices. The subject of the present Section 2 is to analyse the behaviour of individual agents, establish their best response problem and show its well-posedness. The definition and analysis of the Nash risk-sharing equilibrium is taken up in Section 3.

2.2. Best response

We shall now describe how agents respond to the reported subjective probability assessments from their counterparties. For the purposes of §2.2, we fix an agent and a collection of reported probabilities of the remaining agents, and seek the subjective probability that is going to be submitted by agent . According to the rules described in §2.1, a reported probability from agent will lead to entering a long position on the security with payoff

where is such that

By reporting subjective beliefs , agent also indirectly affects the geometric-mean valuation probability , resulting in a highly non-linear overall effect in the security . With the above understanding, and given , the response function of agent is defined to be

where the fact that follows from the discussion of §1.4. The problem of agent is to report the subjective probability that maximises the certainty equivalent of the resulting position after the transaction, i.e., to identify such that


Any satisfying (2.1) shall be called best probability response.

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 generalisation is important from a methodological viewpoint; for example, in the setting of §1.3 it allows for random endowments with infinite support, like ones with the Gaussian distribution or arbitrarily fat tails, a substantial feature in the modelling of risk.

Remark 2.1.

The best response problem (2.1) imposes no constrains on the shape of the agent’s reported subjective probability, as long as it belongs to . In principle, it is possible for agents to report subjective views that are considerably far from their actual ones. Such severe departures may be deemed unrealistic and are undesirable from a modelling point of view. However, as will be argued in §3.3.2, extreme responses are endogenously excluded in our set-up.

We shall show in the sequel (Theorem 2.7) that best responses in (2.1) exist and are unique. We start with a result which gives necessary and sufficient conditions for best probability response.

Proposition 2.2.

Fix and . Then, is best probability response for agent given if and only if the random variable is such that and


The proof of Proposition 2.2 is given in §A.2. The necessity of the stated conditions for best response follows from applying first-order optimality conditions. Establishing the sufficiency of the stated conditions is certainly non-trivial, due to the fact that it is far from clear (and, in fact, not known to us) whether the response function is concave.

Remark 2.3.

In the context of Proposition 2.2, rewriting (2.2) we obtain that


Using also the fact that , it follows that


Hence, holds if and only if , which holds if and only if . (Note that implies , since the expectation of under equals zero.) In words, the best probability response and actual subjective probabilities of an agent agree if and only if the agent has no incentive to participate in the risk-sharing transaction, given the reported subjective beliefs of other agents. Hence, in any non-trivial cases, agents’ strategic behaviour implies a departure from reporting their true beliefs.

Plugging (2.4) back to (2.3), and using also (2.2), we obtain


providing directly the valuation measure in terms of the security .

Remark 2.4.

A message from (2.4) is that, according to their best response process, agents will report beliefs that understate (resp., overstate) the probability of their payoff being high (resp., low) relatively to their true beliefs. Such behaviour is clearly driven by a desired post-transaction utility increase.

More importantly, and in sharp contrast to the securities formed in Arrow-Debreu equilibrium, the security that agent wishes to enter, after taking into account the aggregate reported beliefs of the rest and declaring subjective probability , has limited liability, as it is bounded from below by the constant .

Remark 2.5.

Additional insight regarding best probability responses may be obtained resorting to the discussion of §1.3, where incorporates the random endowment of agent , in the sense that , where denotes the subjective probability of agent . It follows from (2.4) that . It then becomes apparent that, when agents share their risky endowment, they tend to put more weight on the probability of the downside of their risky exposure, rather than the upside. For an illustrative situation, see Example 2.8 later on.

Remark 2.6.

In the course of the proof of Proposition 2.2, the constant in the equivalence (2.2) is explicitly computed; see (A.3). This constant has a particularly nice economic interpretation in the case of two agents. To wit, let , and suppose that is given. Then, from the vantage point of agent , (2.2) becomes

where the constant is such that

where denotes the utility functional of a “fictitious” agent with representative pair . In words, is the post-transaction difference, denominated in units of risk tolerance, of the utility of agent from the utility of agent (who obtains the security ), provided that the latter utility is measured with respect to the reported, as opposed to subjective, beliefs of agent . In particular, when agent does not behave strategically, in which case , it holds that .

Proposition 2.2 sets a roadmap for proving existence and uniqueness in the best response problem via a one-dimensional parametrisation. Indeed, in accordance to (2.2), in order to find a best response we consider for each the unique random variable that satisfies the equation ; then, upon defining via in accordance to (2.5), we seek such that and hold. It turns out that there is a unique such choice; once found, one simply defines via , in accordance to (2.4), to obtain the unique best response of agent given . The technical details of the proof of Theorem 2.7 below are given in §A.3.

Theorem 2.7.

For and , there exists a unique such that .

2.3. The value of strategic behaviour

The increase on agents’ utility that is caused by following the best probability response procedure can be regarded as a measure for the value of the strategic behaviour induced by problem (2.1). Consider for example the case where only a single agent (say) applies the best probability response strategy and the rest of the agents report their true beliefs, i.e., holds for . As mentioned in the introductory section, this is a potential model of a transaction where only agent 0 possesses meaningful market power. Based on the results of §2.2, we may calculate the gains, relative to the Arrow-Debreu transaction, that agent obtains by incorporating such strategic behaviour (which, among others, implies limited liability of the security the agent takes a long position in). The main insights are illustrated in the following two-agent example.

Example 2.8.

Suppose that and . We shall use the set-up of §1.3, where for simplicity it is assumed that agents have the same subjective probability measure. The agents are exposed to random endowments and that (under the common probability measure) have Gaussian law with mean zero and common variance , while denotes the correlation coefficient of and . In this case, it is straightforward to check that ; therefore, after the Arrow-Debreu transaction, the position of agent is . On the other hand, if agent 1 reports true beliefs, from (2.2) the security corresponding to the best probability response of agent should satisfy for appropriate that is coupled with . For and , straightforward Monte-Carlo simulation allows the numerical approximation of the probability density functions (pdf) of and under the best response probability , illustrated in Figure 1. As is apparent, the best probability response drives agent 0 in overstating the downside risk of and understating the downside risk of .

Figure 1. The solid black line is the pdf of endowments and under the agents’ common subjective probability measure, while the other curves illustrate the pdf of (dashed blue) and (dotted red) under the best probability response of agent 0. In this example, and .

The effect of following such strategic behaviour is depicted in Figure 2, where there is comparison between the probability density functions of the positions of agent 0 under (i) no trading; (ii) the Arrow-Debreu transaction; and (iii) the transaction following the application of best response strategic behaviour. As compared to the Arrow-Debreu position, the lower bound of the security guarantees a heavier right tail of the agent’s position after the best response transaction.

Figure 2. The solid black line is the pdf of the initial position , the dashed blue line illustrates the pdf of the position and the dotted red line is the pdf of the position , all under the common subjective probability measure. In this example, and .

3. Nash Risk-Sharing Equilibrium

We shall now consider the situation where every single agent follows the same strategic behaviour indicated by the best response problem of Section 2. As previously mentioned, sharing securities are designed following the sharing rules determined by Theorem 1.2 for any collection of reported subjective views. With the well-posedness of the best response problem established, we are now ready to examine whether the game among agents has an equilibrium point. In view of the analysis of Section 2, individual agents have motive to declare subjective beliefs different than the actual ones. (In particular, and in the setting of §1.3, agents will tend to overstate the probability of their random endowments taking low values.) Each agent will act according to the best response mechanism as in (2.1), given what other agents have reported as subjective beliefs. In a sense, the best response mechanism indicates a negotiation scheme, the fixed point (if such exists) of which will produce the Nash equilibrium valuation measure and risk-sharing securities.

Let us emphasise that the actual subjective beliefs of individual players are not necessarily assumed to be private knowledge; rather, what is assumed here is that agents have agreed upon the rules that associate any reported subjective beliefs to securities and prices, even if the reported beliefs are not the actual ones. In fact, even if subjective beliefs constitute private knowledge initially, certain information about them will necessarily be revealed in the negotiation process which will lead to Nash equilibrium.

There are two relevant points to consider here. Firstly, it is unreasonable for participants to attempt to invalidate the negotiation process based on the claim that other parties do not report their true beliefs, as the latter is, after all, a subjective matter. This particular point is reinforced from the a posteriori fact that reported subjective beliefs in Nash equilibrium do not deviate far from the true ones, as was pointed out in Remark 2.1 and is being further elaborated in §3.3.2. Secondly, it is exactly the limited number of participants, rather than private or asymmetric information, that gives rise to strategic behaviour: agents recognise their ability to influence the market, since securities and valuation become output of collective reported beliefs. Even under the appreciation that other agents will not report true beliefs and the negotiation will not produce an Arrow-Debreu equilibrium, agents will still want to reach a Nash equilibrium, as they will improve their initial position. In fact, transactions with limited number of participants typically equilibrate far from their competitive equivalents, as has been also highlighted in other models of thin financial markets with symmetric information structure, like the ones in [CRW12] and [RW15]—see also the related discussion in the introductory section.

3.1. Revealed subjective beliefs

Considering the model from a more pragmatic point of view, one may argue that agents do not actually report subjective beliefs, but rather agree on a valuation measure and zero-price sharing securities that clear the market. However, there is a one-to-one correspondence between reporting subjective beliefs and proposing a valuation measure and securities, as will be described below.

From the discussion of §2.1, a collection of subjective probabilities gives rise to valuation measure such that and collection of securities is such that , for all . Of course, and holds for all . A further technical observation is that holds for all , which is then a necessary condition that an arbitrary collection of market-clearing securities must satisfy with respect to an arbitrary valuation probability in order to be consistent with the aforementioned risk-sharing mechanism. The previous observations lead to a definition: for , we define the class of securities that clear the market and are consistent with the valuation measure via

Note that all expectations of under in the definition of above are well defined. Indeed, the fact that in the definition of implies that for all . From , we obtain and hence for all .

Starting from a given valuation measure and securities , one may define a collection via for , and note that this is the unique collection in that results in the valuation probability and securities . In this way, the probabilities can be considered as revealed by the valuation measure and securities . Hence, agents proposing risk-sharing securities and a valuation measure is equivalent to them reporting probability beliefs in the transaction. This viewpoint justifies and underlies Definition 3.1 that follows: the objects of Nash equilibrium are the valuation measure and designed securities, in consistency with the definition of Arrow-Debreu equilibrium.

3.2. Nash equilibrium and its characterisation

Following classic literature, we give the formal definition of a Nash risk-sharing equilibrium.

Definition 3.1.

The collection will be called a Nash equilibrium if and, with for all denoting the corresponding revealed subjective beliefs, and for , it holds that

A use of Proposition 2.2 results in the characterisation Theorem 3.2 below, the proof of which is given in §A.4. For this, we need to introduce the -dimensional Euclidean space

Theorem 3.2.

The collection is a Nash equilibrium if and only if the following three conditions hold:

  1. for all , and there exists such that

  2. with as in (1.3), i.e., such that , it holds that

  3. holds for all .

Remark 3.3.

Suppose that the agents’ preferences and risk exposures are such that no trade occurs in Arrow-Debreu equilibrium, which happens when all are the same (and equal to, say, ) for all —see Remark 1.5. In this case, and for all . It is then straightforward from Theorem 3.2 to see that a Nash equilibrium is also given by and (as well as ) for all . 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 and for all . Then, (3.3) shows that and (3.2) implies that , which means that for all . In words, the Nash risk-sharing equilibrium involves no risk transfer if and only if the agents are already in a Pareto optimal situation.

In the important case of two acting agents, since , applying simple algebra in (3.2), we obtain that a Nash equilibrium risk sharing security is such that and satisfies


In Theorem 3.7, existence of a unique Nash equilibrium for the two-agent case will be shown. Furthermore, a one-dimensional root-finding algorithm presented in §3.4 allows to calculate the Nash equilibrium, and further calculate and compare the final position of each individual agent. Consider for instance Example 2.8 and its symmetric situation that is illustrated in Figure 2, where the limited liability of the security implies less variability and flatter right tail of the agent’s position. Under the Nash equilibrium, as will be argued in §3.3.1, security is further bounded from above, which implies that the probability density function of agent’s final position is shifted to the left. This fact is illustrated in Figure 3.

Figure 3. The solid green line is the pdf of the position , the dashed blue line illustrates the pdf of the position and the dotted red line is the pdf of the position , all under the common subjective probability measure. In this example, , and .

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, for agents with sufficiently large risk tolerance the negotiation game results in higher utility compared to the one gained through Arrow-Debreu equilibrium.

3.3. Within equilibrium

According to Theorem 3.7, Nash equilibria in the sense of Definition 3.1 always exist. Throughout §3.3, we assume that is a Nash equilibrium and provide a discussion on certain aspects of it, based on the characterisation Theorem 3.2.

3.3.1. Endogenous bounds on traded securities

As was pointed in Remark 2.4, the security that each agent enters resulting from the best response procedure is bounded below. When all participating agents follow the same strategic behaviour, Nash equilibrium securities are bounded from above as well. Indeed, 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. Mathematically, since is valid for all and holds, it also follows that , for all . Therefore, a consequence of the agents’ strategic behaviour is that Nash risk-sharing securities are endogenously bounded. This fact is in sharp contrast with the Arrow-Debreu equilibrium of (1.4), where the risk transfer may involve securities with unbounded payoffs. An immediate consequence of the bounds on the securities is that the potential gain from the Nash risk-sharing transaction is also endogenously bounded. Naturally, the resulting endogenous bounds are an indication of how the game among agents restricts the risk-sharing transaction, which in turn may be a source of large loss of efficiency. The next example is an illustration of the such inefficiency in a simple symmetric setting. Later on, in Figure 3, the loss of utility in another two-agent example is visualised.

Example 3.4.

Let have the standard (zero mean, unit standard deviation) Gaussian law under the baseline probability . For , define via ; under , has the Gaussian law with mean and unit standard deviation. Fix , and set and . In this case, it is straightforward to compute that . It also follows that . If is large, the discrepancy between the agents’ beliefs results in large monetary profits to both after the Arrow-Debreu transaction. On the other hand, as will be established in Theorem 3.7, in case of two agents there exists a unique Nash equilibrium. In fact, in this symmetric case we have that , and it can be checked that (see also (3.4) later)

The loss of efficiency caused by the game becomes greater with increasing values of . In fact, if converges to infinity, it can be shown that converges to ; furthermore, both and will converge to , which demonstrates the tremendous inefficiency of the Nash equilibrium transaction as compared to the Arrow-Debreu one.

Note that the endogenous bounds depend only on the risk tolerance profile of the agents, and not on their actual beliefs (or risk exposures). In addition, these bounds become stricter in games where quite risk-averse agents are playing, as they become increasingly hesitant towards undertaking risk.

3.3.2. If trading, you never reveal your true beliefs

As discussed in Remark 2.3, agents’ best probability response differ from their actual subjective beliefs in any situation where risk transfer is involved. This result becomes more pronounced when we consider the Nash risk-sharing equilibrium. To wit, if are revealed subjective beliefs corresponding to a Nash equilibrium, it is as a consequence of Theorem 3.2 (see also (2.4)) that


Note that holds if and only if for any fixed ; therefore, whenever agents take part (by actually trading) in Nash equilibrium, their reported subjective beliefs are never the same as their actual ones.

Even though in any non-trivial trading situation agents will report different subjective beliefs from their actual ones, we shall argue below that (3.5) imposes endogenous constraints on the magnitude of the possible discrepancy; the discussion the follows expands on Remark 2.1. Start by writing (3.5) as