We study a fair resource sharing problem, where a set of resources are to be shared among a group of agents. Each agent demands one resource and each resource can serve a limited number of agents. An agent cares about what resource they get as well as the externalities imposed by their mates, who share the same resource with them. Clearly, the strong notion of envy-freeness, where no agent envies another for their resource or mates, cannot always be achieved and we show that even deciding the existence of such a strongly envy-free assignment is an intractable problem. Hence, a more interesting question is whether (and in what situations) a relaxed notion of envy-freeness, the Pareto envy-freeness, can be achieved. Under this relaxed notion, an agent envies another only when they envy both the resource and the mates of the other agent. In particular, we are interested in a dorm assignment problem, where students are to be assigned to dorms with the same capacity and they have dichotomous preference over their dormmates. We show that when the capacity of each dorm is 2, a Pareto envy-free assignment always exists and we present a polynomial-time algorithm to compute such an assignment. Nevertheless, the result breaks immediately when the capacity increases to 3, in which case even Pareto envy-freeness cannot be guaranteed. In addition to the existential results, we also investigate the utility guarantees of (Pareto) envy-free assignments in our model.

翻译：我们研究了一个公平资源共享问题，其中一组资源需在一组智能体之间共享。每个智能体需求一个资源，且每个资源可服务于有限数量的智能体。智能体不仅关心自己获得的资源，还关心与其共享同一资源的同伴所施加的外部性。显然，强无嫉妒性（即没有智能体因他人获得的资源或同伴而嫉妒）并非总能实现，我们证明，甚至判断这种强无嫉妒分配的存在性都是一个棘手问题。因此，更有趣的问题是：在何种情况下，一种放宽的无嫉妒性概念——帕累托无嫉妒性——能够实现。在这一放宽概念下，智能体仅在既嫉妒另一智能体的资源又嫉妒其同伴时才会产生嫉妒。我们特别关注一个宿舍分配问题：学生需被分配到容量相同的宿舍，且他们对宿舍同伴具有二分偏好。我们证明，当每个宿舍容量为2时，帕累托无嫉妒分配总是存在，并给出一个多项式时间算法来构造这种分配。然而，当容量增至3时，结论立即失效，此时连帕累托无嫉妒性也无法保证。除了存在性结果，我们还研究了模型中（帕累托）无嫉妒分配的效用保证。