Stable Risk Sharing and Its Monotonicity

We consider a risk sharing problem in which agents pool their random costs together and seek an allocation rule to redistribute the risk back to each agent. The problem is put into a cooperative game framework and we focus on two salient properties of an allocation rule: stability and monotonicity employing concepts of core and population monotonicity from cooperative game theory. When the risks of the agents are measured by coherent risk measures, we construct a risk allocation rule based on duality theory and establish its stability. When restricting the risk measures to the class of distortion risk measures, the duality-based risk allocation rule is population monotonic if the random costs are independent and log-concave. For the case with dependent normally distributed random costs, a simple condition on the dependence structure is identified to ensure the monotonicity property.