Social distance games have been extensively studied as a coalition formation model where the utilities of agents in each coalition were captured using a utility function $u$ that took into account distances in a given social network. In this paper, we consider a non-normalized score-based definition of social distance games where the utility function $u^s$ depends on a generic scoring vector $s$, which may be customized to match the specifics of each individual application scenario. As our main technical contribution, we establish the tractability of computing a welfare-maximizing partitioning of the agents into coalitions on tree-like networks, for every score-based function $u^s$. We provide more efficient algorithms when dealing with specific choices of $u^s$ or simpler networks, and also extend all of these results to computing coalitions that are Nash stable or individually rational. We view these results as a further strong indication of the usefulness of the proposed score-based utility function: even on very simple networks, the problem of computing a welfare-maximizing partitioning into coalitions remains open for the originally considered canonical function $u$.

翻译：社交距离博弈作为一种联盟形成模型已被广泛研究，其中每个联盟中智能体的效用通过考虑给定社交网络中的距离的效用函数$u$来刻画。本文考虑一种非归一化的基于得分的社交距离博弈定义，其中效用函数$u^s$依赖于一个通用得分向量$s$，该向量可根据每个具体应用场景的特征进行定制。作为主要技术贡献，我们证明了对于任意基于得分的函数$u^s$，在树状网络上计算将智能体划分为联盟以实现社会福利最大化的可处理性。针对$u^s$的特定选择或更简单的网络，我们提供了更高效的算法，并将所有这些结果扩展到计算纳什稳定或个体理性的联盟。我们认为这些结果进一步有力证明了所提出的基于得分的效用函数的实用性：即使在非常简单的网络上，对于最初考虑的规范函数$u$，计算最大化社会福利的联盟划分问题仍然悬而未决。