Counter-monotonic Risk Sharing with Heterogeneous Distortion Risk Measures

We study risk sharing among agents with preferences modeled by heterogeneous distortion risk measures, who are not necessarily risk averse. Pareto optimality for agents using risk measures is often studied through the lens of inf-convolutions, because allocations that attain the inf-convolution are Pareto optimal, and the converse holds true under translation invariance. Our main focus is on groups of agents who exhibit varying levels of risk seeking. Under mild assumptions, we derive explicit solutions for the unconstrained inf-convolution and the counter-monotonic inf-convolution, which can be represented by a generalization of distortion risk measures. Furthermore, for a group of agents with different levels of risk aversion or risk seeking, we consider a portfolio manager's problem and explicitly determine the optimal investment strategies. Interestingly, we observe a counterintuitive phenomenon of comparative statics: even if all agents in the group become more risk seeking, the portfolio manager acting on behalf of the group may not necessarily allocate a larger proportion of investments to risky assets, which is in sharp contrast to the case of risk-averse agents.