Hedonic games are fundamental models for investigating the formation of coalitions among a set of strategic agents, where every agent has a certain utility for every possible coalition of agents it can be part of. To avoid the intractability of defining exponentially many utilities for all possible coalitions, many variants with succinct representations of the agents' utility functions have been devised and analyzed, e.g., modified fractional hedonic games by Monaco et al. [JAAMAS 2020]. We extend this by studying a novel succinct variant that is related to modified fractional hedonic games. In our model, each agent has a fixed type-value and an agent's cost for some given coalition is based on the differences between its value and those of the other members of its coalition. This allows to model natural situations like athletes forming training groups with similar performance levels or voters that partition themselves along a political spectrum. In particular, we investigate natural variants where an agent's cost is defined by distance thresholds, or by the maximum or average value difference to the other agents in its coalition. For these settings, we study the existence of stable coalition structures, their properties, and their quality in terms of the price of anarchy and the price of stability. Further, we investigate the impact of limiting the maximum number of coalitions. Despite the simple setting with metric distances on a line, we uncover a rich landscape of models, partially with counter-intuitive behavior. Also, our focus on both swap stability and jump stability allows us to study the influence of fixing the number and the size of the coalitions. Overall, we find that stable coalition structures always exist but that their properties and quality can vary widely.

翻译：享乐博弈是研究策略性个体间联盟形成的基础模型，其中每个个体对其可能加入的每个潜在联盟都具有特定效用。为避免为所有可能联盟定义指数级数量效用函数带来的计算困难，学界已设计并分析了多种具有简洁效用函数表示的变体，例如Monaco等人[JAAMAS 2020]提出的修正分数享乐博弈。本文通过研究一种与修正分数享乐博弈相关的新型简洁变体对此进行拓展。在我们的模型中，每个个体具有固定的类型值，且个体对特定联盟的成本基于其自身值与其联盟其他成员值之间的差异。这可以模拟运动员按相近表现水平组建训练小组或选民沿政治光谱自我划分等自然场景。我们特别研究了以下自然变体：个体的成本通过距离阈值定义，或通过其与联盟内其他个体的最大/平均数值差异定义。针对这些设定，我们研究了稳定联盟结构的存在性、性质及其在无政府代价与稳定代价方面的质量。此外，我们探究了限制最大联盟数量的影响。尽管采用简单的线型度量距离设定，我们揭示了具有部分反直觉行为的丰富模型图景。同时，我们对交换稳定性和跳跃稳定性的双重关注使我们能够研究固定联盟数量与规模的影响。总体而言，我们发现稳定联盟结构始终存在，但其性质与质量可能存在显著差异。