We introduce the class of modified Schelling games in which there are different types of agents who occupy the nodes of a location graph; agents of the same type are friends, and agents of different types are enemies. Every agent is strategic and jumps to empty nodes of the graph aiming to maximize her utility, defined as the ratio of her friends in her neighborhood over the neighborhood size including herself. This is in contrast to the related literature on Schelling games which typically assumes that an agent is excluded from her neighborhood whilst computing its size. Our model enables the utility function to capture likely cases where agents would rather be around a lot of friends instead of just a few, an aspect that was partially ignored in previous work. We provide a thorough analysis of the (in)efficiency of equilibria that arise in such modified Schelling games, by bounding the price of anarchy and price of stability for both general graphs and interesting special cases. Most of our results are tight and exploit the structure of equilibria as well as sophisticated constructions.

我们介绍了一类改进的 Schelling博弈，其中有不同类型的代理占据位置图的节点；同类型的代理是朋友，不同类型的代理是敌人。每个智能体都是有策略的，并跳转到图的空节点，旨在最大化她的效用，定义为她邻里的朋友与邻域大小的比率，包括她自己。这与 Schelling 博弈的相关文献形成对比，后者通常假设代理在计算其大小时被排除在她的邻居之外。我们的模型使效用函数能够捕捉可能的情况，即代理更愿意与很多朋友而不是几个朋友在一起，这一方面在以前的工作中被部分忽略。我们通过限制一般图和有趣的特殊情况的无政府状态价格和稳定性价格，对此类修改后的谢林博弈中出现的均衡（无效）效率进行了全面分析。我们的大多数结果都很紧凑，并且利用了平衡结构和复杂的结构。