Equilibrium in risk-sharing games

Although the agents’ set of strategic choices is infinite-dimensional, one of our main contributions is to show that a 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 us to prove the existence of Nash equilibrium for an arbitrary number of players. In the important case of two participating agents, we even show that a 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. Whereas a 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 a 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 Sect. 2 and consists of a two-period financial economy with uncertainty, containing possibly infinitely many 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 the simpler setup of a finite state space, there would not be any significant simplification in the mathematical treatment.

In our 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 to-be-shared 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 an Arrow–Debreu equilibrium. (Under CARA preferences, the optimal sharing rules have been extensively studied; see e.g. [10, 13, 7].) 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 an Arrow–Debreu equilibrium, treating reported beliefs as actual ones, since these sharing rules are 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 and hence also 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 [15], 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 [9], a 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 [14], it is entrepreneurs who strategically design securities that investors with non-securitised hedging need competitively trade. In [24], the role of security-designers is played by arbitrageurs who issue innovated securities in segmented markets. A mixture of strategic and competitive stages has also been used in models with asymmetric information. For instance, in [11], 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.

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 a 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 Sect. 3. Agents have motive to exploit other agents’ reported beliefs and hedging needs and drive the sharing transaction 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 the best probability response. Although this is a highly nontrivial 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 to-be-shared risky portfolios. According to the best probability response procedure, each institution tends to use a 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 3.8 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, then 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 Sect. 4, we characterise a Nash equilibrium as the solution of a certain finite-dimensional problem. Based on this characterisation, we establish the existence of a 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 equilibria also provides an algorithm that can be used to approximate the Nash equilibrium transaction by standard numerical procedures, such as Monte Carlo simulation.

Having characterised Nash equilibria, we are able to further perform a joint qualitative and quantitative analysis. Not only do we verify the expected fact that in any nontrivial 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 a strategy from all 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. [25]. When securities are endogenously designed, [14] highlights that imperfect competition among issuers results in risk-sharing inefficiency, even if securities are traded among perfectly competitive investors.)

One may wonder whether the revealed agents’ subjective beliefs in a 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 a 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 from 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 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 a 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 an Arrow–Debreu equilibrium.

Inspired by the involvement of the risk-tolerance coefficients in the agents’ utility gain or loss, in Sect. 5, we focus on induced Arrow–Debreu and Nash equilibria of two-agent games when one of the agents’ preferences approaches 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 a loss of aggregate utility when agents act strategically, there is always a utility gain in the Nash transaction compared to an 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 for 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 a 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 securities equal half of the limiting Arrow–Debreu equilibrium securities, hinting towards the fact that a Nash risk-sharing equilibrium results in loss of trading volume.

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

The symbols \(\mathbf{N}\) and \(\mathbf{R}\) are used to denote the sets of all natural and real numbers, respectively. We have chosen to use the symbol ℝ to denote (reported, or revealed) probabilities.

In all that follows, random variables are defined on a probability space \((\Omega , \, \mathcal{F}, \, \mathbb{P})\). We stress that no finiteness restriction is enforced on the state space \(\Omega \). We use \(\mathcal{P}\) for the class of all probabilities that are equivalent to the baseline probability ℙ. For \(\mathbb{Q}\in \mathcal{P}\), we use \(\mathbb{E}_{\mathbb{Q}}\) to denote the expectation under ℚ. The space \(\mathbb{L}^{0}\) consists of all (equivalence classes, modulo almost sure equality) finite-valued random variables with the topology of convergence in probability. This topology does not depend on the representative probability from \(\mathcal{P}\), and \(\mathbb{L}^{0}\) may be infinite-dimensional. For \(\mathbb{Q}\in \mathcal{P}\), \(\mathbb{L}^{1} (\mathbb{Q})\) consists of all \(X \in \mathbb{L}^{0}\) with \(\mathbb{E}_{\mathbb{Q}}\left[ |X|\right] < \infty \). We use \(\mathbb{L}^{\infty }\) for the subset of \(\mathbb{L} ^{0}\) consisting of essentially bounded random variables.

Whenever \(\mathbb{Q}_{1} \in \mathcal{P}\) and \(\mathbb{Q}_{2} \in \mathcal{P}\), \(\mathrm{d}\mathbb{Q}_{2} / \mathrm{d}\mathbb{Q}_{1}\) denotes the (strictly positive) density of \(\mathbb{Q}_{2}\) with respect to \(\mathbb{Q}_{1}\). The relative entropy of \(\mathbb{Q}_{2} \in \mathcal{P}\) with respect to \(\mathbb{Q}_{1} \in \mathcal{P}\) is defined as

For \(X \in \mathbb{L}^{0}\) and \(Y \in \mathbb{L}^{0}\), we write \(X \sim Y\) when there exists \(c \in \mathbf{R}\) such that \(Y = X + c\). In particular, this notion of equivalence will ease notation on probability densities: for \(\mathbb{Q}_{1} \in \mathcal{P}\) and \(\mathbb{Q}_{2} \in \mathcal{P}\), we write \(\log \left( \mathrm{d} \mathbb{Q}_{2} / \mathrm{d}\mathbb{Q}_{1}\right) \sim \Lambda \) to mean that \(\exp (\Lambda ) \in \mathbb{L}^{1} (\mathbb{Q}_{1})\) and \(\mathrm{d}\mathbb{Q}_{2} / \mathrm{d}\mathbb{Q}_{1} = (\mathbb{E} _{\mathbb{Q}_{1}} [\exp (\Lambda )])^{-1} \exp (\Lambda )\).

We consider a market with a single future period, at which point all uncertainty is resolved. In this market, there are \(n + 1\) economic agents, where \(n \in \mathbf{N}\); for concreteness, define the index set \(I = \left\{ 0, \ldots , n\right\} \). 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 \(i \in I\) over future random outcomes is numerically represented via the concave exponential utility functional where \(\delta_{i} \in (0, \infty )\) is the agent’s risk-tolerance, and \(\mathbb{P}_{i} \in \mathcal{P}\) represents the agent’s subjective beliefs. For any \(X \in \mathbb{L}^{0}\), agent \(i \in I\) is indifferent between the cash amount \(\mathbb{U} _{i} (X)\) and the corresponding risky position \(X\); in other words, \(\mathbb{U}_{i} (X)\) is the certainty equivalent of \(X \in \mathbb{L}^{0}\) for agent \(i \in I\). The functional \(- \mathbb{U}_{i}\) is an entropic risk measure in the terminology of convex risk measure literature; see, for example, [18, Chap. 4].

Define the aggregate risk-tolerance \(\delta \mathrel{\mathop{:}}= \sum_{i \in I}\delta_{i}\) and the relative risk-tolerance \(\lambda _{i} \mathrel{\mathop{:}}=\delta_{i} / \delta \) for all \(i \in I\). Note that \(\sum_{i \in I}\lambda_{i} = 1\). Finally, set \(\delta_{-i} \mathrel{ \mathop{:}}=\delta - \delta_{i}\) and \(\lambda_{-i} \mathrel{ \mathop{:}}=1 - \lambda_{i}\) for all \(i \in I\).

Preference structures that are numerically represented via (2.1) are rich enough to include the possibility of already existing portfolios of random positions for acting agents. To wit, suppose that \(\widetilde{\mathbb{P}}_{i}\in \mathcal{P}\) are the actual subjective beliefs of agent \(i \in I\), 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 \(E_{i} \in \mathbb{L}^{0}\). In this setup, adding on top of \(E_{i}\) a payoff \(X \in \mathbb{L}^{0}\) for agent \(i \in I\) results in a numerical utility equal to \(\widetilde{\mathbb{U}}_{i}(X) \mathrel{ \mathop{:}}=- \delta_{i} \log \mathbb{E}_{\widetilde{\mathbb{P}}_{i}} [\exp (- (X+E_{i}) / \delta_{i}])\). Assume that \(\widetilde{\mathbb{U}}_{i}(0) > - \infty \), that is, that \(\exp (- E _{i} / \delta_{i}) \in \mathbb{L}^{1} ( \widetilde{\mathbb{P}}_{i} )\). Defining \(\mathbb{P}_{i} \in \mathcal{P}\) via \(\log ( \mathrm{d} \mathbb{P}_{i} / \mathrm{d}\widetilde{\mathbb{P}}_{i} ) \sim - E_{i} / \delta_{i}\) and \(\mathbb{U}_{i}\) by (2.1), we have that \(\mathbb{U}_{i}(X) = \widetilde{\mathbb{U}}_{i}(X) - \widetilde{\mathbb{U}}_{i}(0)\) for all \(X \in \mathbb{L}^{0}\). Hence, hereafter, the probability \(\mathbb{P}_{i}\) is understood to incorporate any possible random endowment of agent \(i \in I\), and the utility is measured in relative terms as the difference from the baseline level \(\widetilde{\mathbb{U}}_{i}(0)\).

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

We introduce a method that produces a geometric mean of probabilities, which will play a central role in our discussion. Fix \((\mathbb{R} _{i})_{i \in I} \in \mathcal{P}^{I}\). In view of Hölder’s inequality, we have \(\prod_{i \in I} \left( \mathrm{d}\mathbb{R}_{i} / \mathrm{d}\mathbb{P}\right) ^{\lambda_{i}} \in \mathbb{L}^{1}( \mathbb{P})\). Therefore, we may define \(\mathbb{Q}\in \mathcal{P}\) via \(\log \left( \mathrm{d}\mathbb{Q}/ \mathrm{d}\mathbb{P}\right) \sim \sum_{i \in I} \lambda_{i} \log \left( \mathrm{d}\mathbb{R}_{i} / \mathrm{d}\mathbb{P}\right) \). Since \(\sum_{i \in I} \lambda_{i} \log \left( \mathrm{d}\mathbb{R}_{i} / \mathrm{d}\mathbb{Q}\right) \sim 0\), we are allowed to formally write The fact that \(\mathrm{d}\mathbb{R}_{i} / \mathrm{d}\mathbb{Q}\in \mathbb{L}^{1}(\mathbb{Q})\) implies \(\log^{+} \left( \mathrm{d} \mathbb{R}_{i} / \mathrm{d}\mathbb{Q}\right) \in \mathbb{L}^{1}( \mathbb{Q})\) and Jensen’s inequality give \(\mathbb{E}_{\mathbb{Q}}[ \log \left( \mathrm{d}\mathbb{R}_{i} / \mathrm{d}\mathbb{Q}\right) ] \leq 0\) for all \(i \in I\). Note that (2.2) implies the existence of \(c \in \mathbf{R}\) such that \(\sum_{i \in I} \lambda _{i} \log \left( \mathrm{d}\mathbb{R}_{i} / \mathrm{d}\mathbb{Q}\right) = c\); therefore, we in fact have \(\mathbb{E}_{\mathbb{Q}}[\log \left( \mathrm{d} \mathbb{R}_{i} / \mathrm{d}\mathbb{Q}\right) ] \in (- \infty , 0]\) for all \(i \in I\). In particular, \(\log \left( \mathrm{d}\mathbb{R}_{i} / \mathrm{d}\mathbb{Q}\right) \in \mathbb{L}^{1}(\mathbb{Q})\) for all \(i \in I\), and

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 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 \(\mathcal{P}\). Given a valuation measure, agents agree on a collection \((C_{i})_{i \in I} \in (\mathbb{L}^{0})^{I}\) of zero-value securities satisfying the market clearing condition \(\sum_{i \in I} C_{i} = 0\). The security that agent \(i \in I\) takes a long position in as part of the transaction is \(C_{i}\).

As mentioned in the introductory section, our model can find applications in OTC markets. For instance, the design of asset-backed securities involves only a small number of financial institutions; in this case, \(\mathbb{P}_{i}\) stands for the subjective beliefs of each institution \(i \in I\) and, in view of the discussion of Sect. 2.3, 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 exchanging a random payoff and a fixed payment.

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 maximises each agent’s respective utility.

Under risk preferences modelled by (2.1), a unique Arrow–Debreu equilibrium can be explicitly obtained. In other guises, Theorem 2.2 that follows has appeared in many works; see, for instance, [10, 13, 12]. Its proof is based on standard arguments; however, for reasons of completeness, we provide a short argument in Sect. A.1.

The securities that agents obtain at an Arrow–Debreu equilibrium described in (2.4) provide higher payoffs on events where their individual subjective probabilities are higher than the “geometric mean” probability \(\mathbb{Q}^{\ast }\) of (2.3). In other words, discrepancies in beliefs result in allocations where agents receive higher payoffs on their corresponding relatively more likely events.

Let us note an interesting decomposition for the securities traded at an Arrow–Debreu equilibrium. To wit, in view of the string of equalities agent \(i\) is indifferent between no trading and the first “random” part \(\delta_{i} \log (\mathrm{d}\mathbb{P}_{i} / \mathrm{d} \mathbb{Q}^{\ast })\) of the security \(C^{\ast }_{i}\). The second “cash” part \(\delta_{i} \mathcal{H}(\mathbb{Q}^{\ast }\, | \, \mathbb{P}_{i} )\) of \(C^{\ast }_{i}\) is always nonnegative and represents the monetary gain of agent \(i\) resulting from the Arrow–Debreu transaction. After this transaction, the position of agent \(i\) has certainty equivalent The aggregate monetary value resulting from the Arrow–Debreu transaction equals

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 depend not only on their existing risky portfolios, but also on the beliefs about the future outcomes they report for sharing. In general, agents have an incentive to report subjective beliefs that may differ from their true views about future uncertainty; in fact, these also depend on subjective beliefs reported by the other parties.

As discussed in Sect. 2.6, for a given set of agents’ subjective beliefs, the optimal sharing rules are governed by the mechanism resulting in an 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 the absence of a social planner, agents adapt this sharing mechanism for any collection \((\mathbb{R}_{i})_{i \in I} \in \mathcal{P}^{I}\) of subjective probabilities they choose to report; see also the related discussion in the introductory section. More precisely, in accordance to (2.3) and (2.4), the agreed-upon valuation measure \(\mathbb{Q}\in \mathcal{P}\) is such that \(\log \mathrm{d} \mathbb{Q}\sim \sum_{i \in I}\lambda_{i} \log \mathrm{d}\mathbb{R} _{i}\), and the collection of securities that agents will trade are \(C_{i} \mathrel{\mathop{:}}=\delta_{i} \log (\mathrm{d}\mathbb{R}_{i} / \mathrm{d}\mathbb{Q}) + \delta_{i} \mathcal{H}(\mathbb{Q}\, | \, \mathbb{R}_{i} )\), \(i \in I\).

Given the sharing rules consistent with an Arrow–Debreu equilibrium, 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 \(\mathcal{P}\) being the agents’ set of strategic choices. The subject of the present Sect. 3 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 Sect. 4.

We now describe how agents respond to the reported subjective probability assessments from their counterparties. For the purposes of Sect. 3.2, we fix an agent \(i \in I\) and a collection \(\mathbb{R}_{-i} \mathrel{\mathop{:}}=(\mathbb{R}_{j})_{j \in I \setminus \left\{ i\right\} } \in \mathcal{P}^{I \setminus \left\{ i\right\} }\) of reported probabilities of the remaining agents and seek the subjective probability that is going to be submitted by agent \(i \in I\). According to the rules described in Sect. 3.1, a reported probability \(\mathbb{R}_{i} \in \mathcal{P}\) from agent \(i \in I\) leads to entering a long position on the security with payoff where \(\mathbb{Q}^{(\mathbb{R}_{-i}, \mathbb{R}_{i})} \in \mathcal{P}\) is such that By reporting subjective beliefs \(\mathbb{R}_{i} \in \mathcal{P}\), agent \(i \in I\) also indirectly affects the geometric-mean valuation probability \(\mathbb{Q}^{(\mathbb{R}_{-i}, \mathbb{R}_{i})}\), resulting in a highly nonlinear overall effect on the security \(C_{i}\). With the above understanding, and given \(\mathbb{R}_{-i} \in \mathcal{P}^{I \setminus \left\{ i\right\} }\), the response function of agent \(i \in I\) is defined to be where \(\mathcal{H}(\mathbb{Q}^{(\mathbb{R}_{-i}, \mathbb{R}_{i})} \, | \,\mathbb{R}_{i} ) < \infty \) follows from the discussion of Sect. 2.4. The problem of agent \(i\) is to report the subjective probability that maximises the certainty equivalent of the resulting position after the transaction, that is, to identify \(\mathbb{R}^{\mathrm{r}}_{i} \in \mathcal{P}\) such that Any \(\mathbb{R}^{\mathrm{r}}_{i} \in \mathcal{P}\) satisfying (3.1) is called a 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 Sect. 2.3, it allows for random endowments with infinite support, like ones with a Gaussian distribution or arbitrarily fat tails, a substantial feature in the modelling of risk.

We show in the sequel (Theorem 3.7) that best responses in (3.1) exist and are unique. We start with a result that gives necessary and sufficient conditions for a best probability response.

The proof of Proposition 3.2 is given in Sect. A.2. The necessity of the stated conditions for a best response follows from the first-order optimality conditions. Establishing the sufficiency of the stated conditions is nontrivial due to the fact that it is far from clear (and in fact not known to us) whether the response function is concave.

Proposition 3.2 sets a roadmap for proving the existence and uniqueness in the best response problem via a one-dimensional parameterisation. Indeed, in accordance to (3.2), to find a best response, we consider for each \(z_{i} \in \mathbf{R}\) the unique random variable \(C_{i}(z_{i})\) that satisfies the equation Then, upon defining \(\mathbb{Q}_{i} (z_{i})\) via in accordance to (3.5), we seek \(\widehat{z}_{i} \in \mathbf{R}\) such that \(C_{i}(\widehat{z}_{i}) \in \mathbb{L}^{1}(\mathbb{Q}_{i}(\widehat{z}_{i}))\) and \(\mathbb{E} _{\mathbb{Q}_{i}(\widehat{z}_{i})} \left[ C_{i}(\widehat{z}_{i})\right] = 0\). It turns out that there is a unique such choice; once found, we simply define \(\mathbb{R}^{\mathrm{r}}_{i}\) via \(\log (\mathrm{d} \mathbb{R}^{\mathrm{r}}_{i} / \mathrm{d}\mathbb{P}_{i}) \sim - \log \left( 1 + C_{i}(\widehat{z}_{i}) / \delta_{-i} \right) \), in accordance to (3.4), to obtain the unique best response of agent \(i \in I\) given \(\mathbb{R}_{-i}\). The technical details of the proof of Theorem 3.7 below are given in Sect. A.3.

The increase in agents’ utility caused by following the best probability response procedure can be regarded as a measure for the value of the strategic behaviour induced by problem (3.1). Consider, for example, the case where only a single agent (say) \(0 \in I\) applies the best probability response strategy and the rest of the agents report their true beliefs, that is, \(\mathbb{R}_{j} = \mathbb{P}_{j}\) for \(j \in I \setminus \left\{ 0\right\} \). 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 Sect. 3.2, we may calculate the gains, relative to the Arrow–Debreu transaction, that agent 0 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.

Considering the model from a more pragmatic point of view, we may argue that agents do not actually report subjective beliefs, but rather agree on a valuation measure \(\mathbb{Q}\in \mathcal{P}\) and zero-price sharing securities \((C_{i})_{i \in I}\) that clear the market. However, there is a one-to-one correspondence between reporting subjective beliefs and proposing a valuation measure and securities, as we describe below.

From the discussion of Sect. 3.1, a collection of subjective probabilities \((\mathbb{R}_{i})_{i \in I}\) gives rise to a valuation measure \(\mathbb{Q}\in \mathcal{P}\) such that \(\log \mathrm{d}\mathbb{Q}\sim \sum_{i \in I}\lambda_{i} \log \mathrm{d} \mathbb{R}_{i}\) and a collection \((C_{i})_{i \in I}\) of securities such that \(C_{i} = \delta_{i} \log (\mathrm{d}\mathbb{R}_{i} / \mathrm{d} \mathbb{Q}) + \delta_{i} \mathcal{H}(\mathbb{Q}\, | \,\mathbb{R}_{i} )\) for all \(i \in I\). Of course, \(\sum_{i \in I}C_{i} = 0\) and \(\mathbb{E}_{\mathbb{Q}}\left[ C_{i}\right] = 0\) for all \(i \in I\). A further technical observation is that \(\exp (C_{i} / \delta_{i}) \in \mathbb{L}^{1}(\mathbb{Q})\) for all \(i \in I\), which is then a necessary condition that an arbitrary collection of market-clearing securities \((C_{i})_{i \in I}\) must satisfy with respect to an arbitrary valuation probability \(\mathbb{Q}\in \mathcal{P}\) in order to be consistent with the aforementioned risk-sharing mechanism. The previous observations lead to a definition: for \(\mathbb{Q}\in \mathcal{P}\), we define the class \(\mathcal{C}_{\mathbb{Q}}\) of securities that clear the market and are consistent with the valuation measure ℚ via Note that all expectations of \(C_{i}\) under ℚ in the definition of \(\mathcal{C}_{\mathbb{Q}}\) above are well defined. Indeed, the fact that \(\exp (C_{i} / \delta_{i}) \in \mathbb{L}^{1}( \mathbb{Q})\) in the definition of \(\mathcal{C}_{\mathbb{Q}}\) implies that \(C_{i}^{+} \in \mathbb{L}^{1}(\mathbb{Q})\) for all \(i \in I\). From \(\sum_{i \in I}C_{i} = 0\) we obtain \(\sum_{i \in I}\!|C_{i}| = 2 \sum_{i \in I}C_{i}^{+}\), and hence \(C_{i} \in \mathbb{L}^{1}(\mathbb{Q})\) for all \(i \in I\).

Given a valuation measure \(\mathbb{Q}\in \mathcal{P}\) and securities \((C_{i})_{i \in I} \in \mathcal{C}_{\mathbb{Q}}\), we may define a collection \((\mathbb{R}_{i})_{i \in I} \in \mathcal{P}^{I}\) via \(\log (\mathrm{d}\mathbb{R}_{i} / \mathrm{d}\mathbb{Q}) \sim C_{i} / \delta_{i}\) for \(i \in I\) and note that this is the unique collection in \(\mathcal{P}^{I}\) that results in the valuation probability \(\mathbb{Q}\in \mathcal{P}^{I}\) and securities \((C_{i})_{i \in I} \in \mathcal{C}_{\mathbb{Q}}\). In this way, the probabilities \((\mathbb{R}_{i})_{i \in I} \in \mathcal{P}^{I}\) can be considered as revealed by the valuation measure \(\mathbb{Q}\in \mathcal{P}\) and securities \((C_{i})_{i \in I} \in \mathcal{C}_{\mathbb{Q}}\). 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 4.1 that follows: the objects of a Nash equilibrium are the valuation measure and designed securities, in consistency with the definition of an Arrow–Debreu equilibrium.

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

A use of Proposition 3.2 results in the characterisation Theorem 4.2, the proof of which is given in Sect. A.4. For this, we need to introduce the \(n\)-dimensional Euclidean space

In the important case of two acting agents, since \(C^{\diamond }_{0} = - C^{\diamond }_{1}\), applying simple algebra in (4.2), we obtain that a Nash equilibrium risk-sharing security \(C^{\diamond } _{0}\) is such that \(-\delta_{1} < C^{\diamond }_{0} < \delta_{0}\) and satisfies In Theorem 4.7, the existence of a unique Nash equilibrium for the two-agent case will be shown. Furthermore, a one-dimensional root-finding algorithm presented in Sect. 4.4 allows us to calculate the Nash equilibrium and further calculate and compare the final position of each individual agent. Consider, for instance, Example 3.8 and its symmetric situation illustrated in Fig. 2, where the limited liability of the security \(C^{\mathrm{r}}_{0}\) implies less variability and a flatter right tail for the agent’s position. Under the Nash equilibrium, as argued in Sect. 4.3.1, the security \(C^{\diamond }_{0}\) is further bounded from above, which implies that the probability density function of the agent’s final position is shifted to the left. This fact, in the setting of Example 3.8, is illustrated in Fig. 3.

Despite the above symmetric case, it is not necessarily true that all agents suffer a loss of utility at the Nash equilibrium risk sharing. As we shall see in Sect. 5, for agents with sufficiently large risk-tolerance, the negotiation game results in higher utility compared to the one gained through an Arrow–Debreu equilibrium.

According to Theorem 4.7, Nash equilibria in the sense of Definition 4.1 always exist. Throughout Sect.  4.3 , we assume that \((\mathbb{Q}^{\diamond }, (C^{\diamond }_{i})_{i \in I})\) is a Nash equilibrium and provide a discussion on certain of its aspects, based on the characterisation in Theorem 4.2.

Theorem 4.2 is used as a guide in order to search for an equilibrium, parameterising candidates for optimal securities using the \(n\)-dimensional space \(\Delta^{I}\) introduced in (4.1). Proposition 4.5 that follows, and whose proof is the content of Sect. A.5, enables us to reduce the search of a Nash equilibrium, an inherently infinite-dimensional problem in our setting, to a finite-dimensional one. The latter problem gives the necessary tools for numerical approximations of Nash equilibria (see also Example 4.8).

In the notation of Proposition 4.5, for each \(z \in \Delta^{I}\), define the probability \(\mathbb{Q}(z)\) via The uniform bounds \(- \delta_{-i} < C_{i}(z) < (n-1) \delta + \delta _{i}\) follow as in Sect. 4.3.1 and imply \(\exp ( C_{i}(z) / \delta_{i}) \in \mathbb{L}^{1}(\mathbb{Q}(z))\) for all \(i \in I\) and \(z \in \Delta^{I}\). In particular, \(\left( C_{i}(z)\right) _{i \in I} \in \mathcal{C}_{\mathbb{Q}(z)}\) for all \(z \in \Delta^{I}\). In view of Theorem 4.2, Nash equilibria amount to finding \(z \in \Delta^{I}\) such that \(\mathbb{E}_{\mathbb{Q}(z)}\left[ C_{i}(z)\right] = 0\) for all \(i \in I\). We can in fact define a function \(\ell : \Delta^{I}\to \mathbf{R}_{+}\) that gives a “distance from equilibrium” by the formula Since \(C_{i}(z) > - \delta_{-i}\) for all \(z \in \Delta^{I}\), \(\ell \) is well defined. Furthermore, the inequality \(\log x \leq x-1\), valid for \(x \in (0, \infty )\), gives in view of the fact that \(\sum_{i \in I} C_{i}(z) = 0\) for all \(z \in \Delta^{I}\), which shows that \(\ell \) is indeed \(\mathbf{R} _{+}\)-valued. Furthermore, since \(\log x < x-1\) for \(x \in (0, \infty ) \setminus \left\{ 1\right\} \), it follows for any \(z \in \Delta^{I}\) that \(\ell (z) = 0\) is equivalent to \(\mathbb{E}_{\mathbb{Q}(z)}\left[ C _{i}(z)\right] = 0\) for all \(i \in I\).

The following result summarises the above discussion.

Proposition 4.6 provides a one-to-one correspondence between Nash equilibria and roots of \(\ell \). Recalling the discussion in Sect. 4.3.4, any root of \(\ell \) belongs to the compact subset of \(\Delta^{I}\) consisting of \((z_{i})_{i \in I} \in \Delta^{I}\) with \(z_{i} \geq - u^{\ast }_{i}\) for all \(i\). This allows numerical approximations of Nash equilibria, for example, by Monte Carlo simulation.

Its practical usefulness notwithstanding, Proposition 4.6 does not answer the question of actual existence of Nash equilibria nor, in case of existence, the uniqueness. These issues are settled in Theorem 4.7, whose proof is the subject of Sect. A.6.

The question of uniqueness for three or more agents remains open and is significantly more challenging from a mathematical perspective. In all cases of numerical simulation that were carried out, we observed the (existence and) uniqueness of a Nash equilibrium. The next example is representative.

We start with the two-agent case \(I = \left\{ 0,1\right\} \), where the risk-aversion of only one agent approaches zero. We keep the risk-tolerance \(\delta_{1}\) and subjective probability \(\mathbb{P} _{1}\) of agent 1 fixed. On the other hand, for agent 0, we consider a sequence of risk-tolerance coefficients \((\delta_{0}^{m})_{m \in \mathbf{N}}\) with \(\lim_{m \to \infty }\delta^{m}_{0} = \infty \) and a fixed subjective probability \(\mathbb{P}_{0}\). In this setup, Theorems 2.2 and 4.7 state that for each \(m \in \mathbf{N}\), there exist a unique Arrow–Debreu equilibrium \((\mathbb{Q}^{m, \ast }, (C^{m, \ast }_{i})_{i \in I})\) and a unique Nash equilibrium \((\mathbb{Q}^{m, \diamond }, (C^{m, \diamond }_{i})_{i \in I})\). We use agent 0 as the baseline and focus on the securities \(C^{m, \ast }_{0}\) and \(C^{m, \diamond }_{0}\) since \(C^{m, \ast }_{1} = -C^{m, \ast }_{0}\) and \(C^{m, \diamond }_{1} = -C^{m, \diamond } _{0}\).

We first examine the limiting behaviour of the valuation rule and the securities in the Arrow–Debreu equilibrium transaction. For each \(m \in \mathbf{N}\), we obtain from (2.3) that \(\mathbb{Q} ^{m, \ast }\in \mathcal{P}\) has \(\log (\mathrm{d}\mathbb{Q}^{m, \ast }/ \mathrm{d}\mathbb{P}_{0}) \sim \lambda_{1}^{m} \log ( \mathrm{d}\mathbb{P}_{1} / \mathrm{d}\mathbb{P}_{0})\). More precisely, we have Given that \(\lim_{m \to \infty }\lambda_{1}^{m} = 0\), \(\mathbb{L}^{0} \mbox{-} \lim_{m \to \infty }\left( \mathrm{d}\mathbb{Q}^{m, \ast }/ \mathrm{d}\mathbb{P}_{0}\right) = 1\) readily follows from the dominated convergence theorem—in fact, with \(\left| \,\cdot \,\right| _{ \mathrm{TV}}\) denoting the total-variation norm, Scheffé’s lemma implies that \(\lim_{m \to \infty }\left| \mathbb{Q}^{m, \ast }- \mathbb{P}_{0}\right| _{\mathrm{TV}} = 0\). Since for all \(m \in \mathbf{N}\) and \((\mathbb{Q}^{m, \ast })_{m \in \mathbf{N}}\) converges to \(\mathbb{P}_{0}\), we expect the limiting relationship \(\mathbb{L}^{0} \mbox{-} \lim_{m \to \infty }C^{m, \ast }_{0} = \delta_{1} \log (\mathrm{d}\mathbb{P}_{0} / \mathrm{d} \mathbb{P}_{1}) - \delta_{1} \mathcal{H}(\mathbb{P}_{0} \, | \, \mathbb{P}_{1})\). Clearly, for the previous limit to be valid, the following (technical) assumption is necessary.

In Sect. A.7, it is shown that the latter assumption is also sufficient for the validity of Proposition 5.2, giving the limiting valuation and security in an Arrow–Debreu equilibrium and the limiting gains of both agents.

It is indeed expected that the utility gain of a nearly risk-neutral agent is almost zero. To see this, compare the limiting valuation measure \(\mathbb{P}_{0}\) with the limiting utility of agent 0, which is the linear expectation with respect to \(\mathbb{P}_{0}\). On the other hand, the only case where there is no limiting utility gain for agent 1 is when the two agents’ subjective beliefs coincide.

We now turn to a Nash risk-sharing equilibrium. From (4.4), we obtain Accepting that the sequence \((z^{m, \diamond }_{0})_{m \in \mathbf{N}}\) converges in \(\mathbf{R}\) and \((C^{m, \diamond }_{0})_{m \in \mathbf{N}}\) converges in \(\mathbb{L}^{0}\) (these conjectures actually have to be proved as part of Theorem 5.4 below), and given that \(\lim_{m \to \infty }\delta^{m}_{0} = \infty \), \(\lim_{m \to \infty }\lambda_{0} ^{m} = 1\) and \(\mathbb{L}^{0} \mbox{-} \lim_{m \to \infty }C^{m, \ast }_{0} = C^{\infty , \ast }_{0}\), the limiting security \(C^{\infty , \diamond }_{0} \mathrel{\mathop{:}}=\mathbb{L}^{0} \mbox{-} \lim_{m \to \infty }C^{m, \diamond }_{0}\) should satisfy where we set \(z^{\infty , \diamond }_{0} \mathrel{\mathop{:}}= \lim_{m \to \infty }z^{m, \diamond }_{0}\). This heuristic discussion gives a method to compute the limit. For \(z \in \mathbf{R}\), define the random variable \(C^{\infty }_{0}(z)\) by the equation Since the function \((- 1, \infty ) \ni x \mapsto x + \log \left( 1 + x\right) \) is strictly increasing and continuous and maps \((-1, \infty )\) to \((- \infty , \infty )\), it follows that \(C^{\infty }_{0}(z)\) is a well-defined \((- \delta_{1}, \infty )\)-valued random variable for all \(z \in \mathbf{R}\). So we should have \(C^{\infty , \diamond }_{0} = C^{\infty }_{0} (z^{\infty , \diamond }_{0})\). Although \(z^{\infty , \diamond }_{0}\) is given as the limit of \((z^{m, \diamond }_{0})_{m \in \mathbf{N}}\), we may actually identify a priori what its value will be. To make headway, note that from (4.3), and the fact that \(\lim_{m \to \infty }\left| \mathbb{Q}^{m, \ast }- \mathbb{P}_{0}\right| _{\mathrm{TV}} = 0\), the limiting Nash valuation probability \(\mathbb{Q}^{\infty , \diamond }\) should have \(\log ( \mathrm{d}\mathbb{Q}^{\infty , \diamond }/ \mathrm{d}\mathbb{P}_{0}) \sim -\log (1 + C^{\infty , \diamond }_{0} / \delta_{1})\); since we expect \(\mathbb{E}_{\mathbb{Q}^{\infty , \diamond }} [C^{\infty , \diamond }_{0}] = 0\), we actually obtain that the equality \(\mathbb{E}_{\mathbb{P}_{0}} [ (1 + C^{\infty , \diamond }_{0} / \delta _{1})^{-1} ] = 1\) should be satisfied. The next result, whose proof is given in Sect. A.8, ensures that a unique such candidate \(z^{\infty , \diamond }_{0} \in \mathbf{R}\) exists.

Before we state our main result on the limiting behaviour of a Nash equilibrium, we make a final observation. Note that in view of (4.5), for all \(m \in \mathbf{N}\), we have \(\log ( \mathrm{d}\mathbb{R}^{m, \diamond }_{1} / \mathrm{d} \mathbb{P}_{1}) \sim - \log (1 - C^{m, \diamond }_{0} / \delta^{m} _{0})\). Since \(\lim_{m \to \infty }\delta_{0}^{m} = \infty \) and, as it turns out, \((C^{m, \diamond }_{0})_{m \in \mathbf{N}}\) is convergent, the revealed subjective probability \(\mathbb{R}^{m, \diamond }_{1}\) of agent 1 when \(m\) is large is close to the actual \(\mathbb{P}_{1}\). (There is an alternative way to obtain the same intuition. From (4.6), note that \(\mathrm{d}\mathbb{R}^{m, \diamond }_{1} / \mathrm{d}\mathbb{P}_{1} \geq \lambda_{0}^{m}\) for all \(m \in \mathbf{N}\). Since \(\lim_{m \to \infty }\lambda_{0}^{m} = 1\) and \((\mathrm{d}\mathbb{R}^{m, \diamond }_{1} / \mathrm{d}\mathbb{P}_{1})_{m \in \mathbf{N}}\) has constant expectation 1 under \(\mathbb{P}_{1}\), \((\mathbb{R}^{m, \diamond }_{1})_{m \in \mathbf{N}}\) must converge to \(\mathbb{P}_{1}\).) This suggests the same asymptotic behaviour as in the case discussed in Sect. 3.3, where only agent 0 acts strategically as indicated by the best probability response, whereas agent 1 reports true subjective beliefs \(\mathbb{P}_{1}\). Indeed, the following result, whose proof is given in Sect. A.9, implies that the limiting security structure is the same, regardless of whether the risk-averse agent 1 enters the game or simply reports true subjective beliefs (in which case only the approximately risk-neutral agent behaves strategically).

The equality of the limits of \((C^{m, \diamond }_{0})_{m \in \mathbf{N}}\) and \((C_{0}^{m,\mathrm{r}})_{m \in \mathbf{N}}\) implies that the strategic behaviour of a risk-neutral agent dominates the risk-sharing transaction. Intuitively, agents with high risk-tolerance are willing to undertake more risk at the sharing transaction in return for a higher cash compensation. Thus, at the limit, the risk-neutral agent satisfies the reported hedging needs of other agents but achieves better prices by applying the best response strategy. On the other hand, for the risk-averse agent, the risk reduction is more important than a higher price to be paid. As a result, at equilibrium, the risk-averse agent prefers to submit true beliefs even though this results in a higher price to be paid 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 Sect. 4.3 that in any risk-transfer situation, the Nash equilibrium incurs some loss of efficiency. Although the aggregate utility is reduced in a Nash equilibrium when compared with the Arrow–Debreu one, certain agents may obtain a higher utility gain in risk-sharing games. In particular, Proposition 5.5 (whose proof is given in Sect. A.10) demonstrates that an agent with sufficiently high risk-tolerance enjoys a higher utility at a Nash equilibrium transaction than the utility at the Arrow–Debreu equilibrium sharing.

The limiting loss for the risk-averse agent comes from two sides. The first is \((1 / \delta_{1}) \mathrm{Var}_{\mathbb{Q}^{\infty , \diamond }} (C^{\infty , \diamond }_{0})\), which is the limiting gain of agent 0. The remaining quantity \(\delta_{1} \mathcal{H}( \mathbb{Q}^{ \infty , \ast }| \mathbb{Q}^{\infty , \diamond })\) is in fact the loss from the applied strategic behaviour as opposed to sharing in a Pareto-optimal way. Both terms are strictly positive as long as \(C^{\infty , \diamond }_{0}\) is not identically equal to zero.

The message of Proposition 5.5 is clear. The introduction of strategic behaviour allows agents with high risk-tolerance to achieve better prices that the more risk-averse agents are willing to pay in order to achieve risk reduction. In contrast to the Arrow–Debreu equilibrium where prices are given by the optimal sharing measure, agents with sufficiently high risk-tolerance are willing to accept more risk in the Nash game since their strategy drives the market to better cash compensation for them. In fact, a more risk-averse agent not only tends to undertake all the efficiency loss caused by the game, but also fuels the utility gain of the (sufficiently) risk-tolerant counterparty.

Recalling the discussion and notation of Sect. 4.3.5, we may offer some more detailed comments. From (4.11) and Proposition 5.5, it follows that the marginal valuation measure of agent 0 approaches the limiting optimal valuation measure \(\mathbb{Q}^{\infty , \ast }\). In turn, this implies that for large enough \(m \in \mathbf{N}\), the security that agent 0 gets in a Nash equilibrium is undervalued—indeed, note that According to (4.15) and the discussion that follows, we readily get that the utility of agent 0 is increased. For the risk-averse agent, the situation is different. From (4.13), it follows that \(\mathbb{Q}^{m, \diamond }_{1}\) is close to \(\mathbb{Q}^{m, \diamond }\) for large \(m \in \mathbf{N}\), which in turn is close to \(\mathbb{Q}^{\infty , \diamond }\). Hence, for large enough \(m \in \mathbf{N}\), the security received by agent 1 in a Nash equilibrium is overvalued; on top of this, agent 1 also carries all the risk-sharing inefficiency of a Nash equilibrium.

We have seen before that the strategic behaviour of a highly risk-tolerant agent dominates the Nash game and drives the market to his preferred transaction, regardless of the actions of the other agent. Here, we examine what happens to the equilibria when both agents approach risk-neutrality at the same speed. More precisely, we fix \(\lambda_{0} \in (0,1)\) and \(\lambda_{1} \in (0,1)\) with \(\lambda_{0} + \lambda_{1} = 1\) and consider a nondecreasing sequence \(\left( \delta ^{m}\right) _{m \in \mathbf{N}}\) with \(\lim_{m \to \infty }\delta^{m} = \infty \). Define \(\delta^{m}_{i} \mathrel{\mathop{:}}=\lambda_{i} \delta^{m}\) for all \(m \in \mathbf{N}\) and \(i \in \left\{ 0,1\right\} \). In contrast to the setup of Sect. 5.1, here the subjective beliefs of the agents have to depend on \(m \in \mathbf{N}\). To obtain intuition on why and how the subjective probabilities must behave, note that according to Theorem 2.2, the security \(C^{m, \ast }_{0}\) is for all \(m \in \mathbf{N}\) given as a multiple by \(\delta_{0}^{m}\) of a random variable whose dependence on risk-tolerance comes only through \(\lambda_{0}\) and \(\lambda_{1}\). Since the latter weights are fixed for each \(m \in \mathbf{N}\), to guarantee that the securities in an Arrow–Debreu equilibrium have a well-behaved limit, we make the following assumption.

Note that the condition \(\mathbb{E}_{\mathbb{P}}\left[ \xi_{i}\right] = 0\) for \(i \in \left\{ 0,1\right\} \) appearing in Assumption 5.6 is just a normalisation and does not constitute any loss of generality.

The proof of Theorem 5.7 is given in Sect. A.11. Interestingly, the risk-neutrality of both agents drives the Nash equilibrium to half of the Arrow–Debreu securities, which is evidence of the market inefficiency caused by the strategic behaviour of risk-neutral agents. The result of Theorem 5.7 is another manifestation of the claim (initially made in Sect. 4.3.5) that the trading volume in a Nash equilibrium tends to be lower than the Pareto-optimal allocations.