Schelling games model the wide-spread phenomenon of residential segregation in metropolitan areas from a game-theoretic point of view. In these games agents of different types each strategically select a node on a given graph that models the residential area to maximize their individual utility. The latter solely depends on the types of the agents on neighboring nodes and it has been a standard assumption to consider utility functions that are monotone in the number of same-type neighbors. This simplifying assumption has recently been challenged since sociological poll results suggest that real-world agents actually favor diverse neighborhoods. We contribute to the recent endeavor of investigating residential segregation models with realistic agent behavior by studying Jump Schelling Games with agents having a single-peaked utility function. In such games, there are empty nodes in the graph and agents can strategically jump to such nodes to improve their utility. We investigate the existence of equilibria and show that they exist under specific conditions. Contrasting this, we prove that even on simple topologies like paths or rings such stable states are not guaranteed to exist. Regarding the game dynamics, we show that improving response cycles exist independently of the position of the peak in the utility function. Moreover, we show high almost tight bounds on the Price of Anarchy and the Price of Stability with respect to the recently proposed degree of integration, which counts the number of agents with a diverse neighborhood and which serves as a proxy for measuring the segregation strength. Last but not least, we show that computing a beneficial state with high integration is NP-complete and, as a novel conceptual contribution, we also show that it is NP-hard to decide if an equilibrium state can be found via improving response dynamics starting from a given initial state.

翻译：谢林博弈从博弈论视角建模了大都市区域中普遍存在的居住隔离现象。在这类博弈中，不同类型的智能体各自策略性地选择给定图（模拟居住区域）上的节点，以最大化其个体效用。个体效用仅取决于邻居节点上的智能体类型，且传统标准假设认为效用函数与同类型邻居数量呈单调关系。这一简化假设近期受到挑战——社会学调查结果表明，现实中的智能体实际上偏好多元化社区。我们通过研究智能体具有单峰效用函数的跳跃型谢林博弈，为近期探索具有现实行为特征的居住隔离模型工作做出贡献。在此类博弈中，图中存在空节点，智能体可策略性地跳跃至这些节点以提升效用。我们研究了均衡的存在性，证明其在特定条件下存在。与此形成对比的是，我们证明即使在路径或环等简单拓扑结构上，此类稳定状态也无法保证存在。关于博弈动力学，我们证明改进响应循环的存在性独立于效用函数峰值位置。此外，我们针对近期提出的融合度指标（该指标统计拥有多元化邻居的智能体数量，可作为测量隔离强度的代理变量），给出了无政府代价和稳定代价的紧致上下界。最后但同等重要的是，我们证明计算具有高融合度的有益状态是NP完全的，并且作为新颖的概念贡献，我们还证明判定能否通过从给定初始状态出发的改进响应动力学找到均衡状态是NP难的。