Equilibrium in Risk-Sharing Games

Abstract

The large majority of risk-sharing transactions involve few agents, each of whom can heavily influence the structure and the prices of securities. In this paper, we propose 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 an arbitrary number of agents and to be unique in the two-agent game. In equilibrium, agents declare beliefs on future random outcomes different from 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 than compared to the Arrow-Debreu one. Keywords

Effective risk aversion in thin risk‐sharing markets

“…In fact, it follows from the analysis of the bilateral game that the noncompetitive equilibrium is beneficial in terms of utility gain for two types of traders: those with sufficiently high pre‐transaction beta, and those with sufficiently high risk tolerance. Such findings in noncompetitive markets are consistent with results in Anthropelos (2017) and Anthropelos and Kardaras (2017). (A result in that spirit also appears in Malamud and Rostek, 2017; namely it is shown that, when the market is centralized, less risk averse agents have greater price impact.…”

The Effect of Market Power on Risk-Sharing

“…Kardaras [4] (see also the discussion in subsection 6.3 below). below, the predatory trader's utility surplus decreases when the average agents' assessments give higher value to the securities she buys at equilibrium.…”

“…According to the proposed model, she should respond as her to-be-shared endowment the random quantity (called bestendowment response) that maximizes her own utility, when the agreed sharing rules are applied. 3,4 Under mean-variance preferences (hereafter M-V preferences), it is shown that it is never optimal for an agent to submit for sharing her true risk exposure. Instead, each agent has motive to declare only a fraction of her true endowment and also to report exposure to the aggregate endowment submitted by the other agents.…”

“…This set of choices is in fact equivalent to the set of securities and pricing kernel that are consistent with the optimal sharing rules. 4 The generality of the agent's random endowment in our model allows its application to a number of financial markets, such as the trading of innovated derivative products, the use of reinsurance contracts or the transactions among inter-dealers (see also the related discussion in Vayanos [51]). 5 Note that under M-V preferences, the demand schedules are linear and hence a demand function can be inferred by knowing only two of its values (orders).…”

“…This implies, in particular, that the linear sharing rules (as the ones of Proposition 2.5) could not give optimal sharing. This problem has recently been studied in Anthropelos and Kardaras[4] (see Theorem 1.2), where exponential utility is imposed and thus agents' utility functional coincides with M-V preferences under normal distributed…”