Agents' Strategic Behavior and Risk-Sharing Inefficiency

We consider the market of n financial agents who negotiate the sharing of their random incomes. Assuming that agents' risk measurements are induced by concave utility functionals, we write the optimal risk sharing as a solution of an equilibrium price-allocation problem. Given the optimal sharing rule, we address the situation where agents do not share their true random endowments but instead they report as endowments the random quantities that maximize their expected utility after the sharing rule is applied. It is shown that this strategic behavior results in a Nash-equilibrium type of agreement among the agents, which implies an ineffcient risk sharing. Under quadric utility functionals, we give closed form solutions and discuss the associated findings. Furthermore, the ineffciency of risk sharing is even more intense if the number of the agents becomes an endogenous problem. In particular, we give conditions under which the participation of a new agent is bene cial to all the existed agents. This discussion naturally leads to the problem of group formation in the market, which is addressed for the first time in a financial risk sharing setting. A related example under quadratic utility functionals is extensively analyzed.