We study a weighted comonotonic risk-sharing problem among multiple agents with distortion risk measures under heterogeneous beliefs. The explicit forms of optimal allocations are obtained, which are Pareto-optimal. A necessary and sufficient condition is given to ensure the uniqueness of the optimal allocation, and sufficient conditions are given to obtain an optimal allocation of the form of excess of loss or full insurance. The optimal allocation may satisfy individual rationality depending on the choice of the weight. When the distortion risk measure is value at risk or tail value at risk, an optimal allocation is generally of the excess-of-loss form. The numerical examples suggest that a risk is more likely to be shared among agents with heterogeneous beliefs, and the introduction of the weight enables us to prioritize some agents as part of a group sharing a risk.

中文翻译：

---

异质信念下的加权单调风险分担

我们研究了在异质信念下具有失真风险度量的多个代理之间的加权共单调风险分担问题。得到了最优分配的显式形式，它们是帕累托最优的。给出了保证最优分配唯一性的充要条件，给出了获得超额损失或全额保险形式最优分配的充分条件。根据权重的选择，最优分配可以满足个体的合理性。当扭曲风险度量为风险价值或风险尾值时，最优分配通常为超额损失形式。数值例子表明，风险更有可能在具有不同信念的代理人之间分担，