Residential segregation in metropolitan areas is a phenomenon that can be observed all over the world. It is characterized by the emergence of large regions populated by residents that are homogeneous in terms of ethnicity or other traits. In a recent research trend in the AI community this phenomenon was investigated via game-theoretic models. There, selfish agents of two types are equipped with a monotone utility function that ensures higher utility if an agent has more same-type neighbors. The agents strategically choose their location on a given graph that serves as residential area to maximize their utility. However, sociological polls suggest that real-world agents are actually favoring mixed-type neighborhoods, and hence should be modeled via non-monotone utility functions. We study Swap Schelling Games with non-monotone utility functions that are single-peaked. In these games pairs of agents may improve their utility by swapping their locations. Our main finding is that tolerance, i.e., that the agents favor fifty-fifty neighborhoods or even being in the minority, is necessary for equilibrium existence on almost regular or bipartite graphs. We show equilibrium existence on almost regular graphs via a potential function argument and we prove that this approach is impossible on arbitrary graphs even with tolerant agents. Regarding the quality of equilibria, we consider the recently introduced degree of integration, that counts the number of agents that live in a heterogeneous neighborhood, as social welfare function. We derive (almost) tight bounds on the Price of Anarchy and the Price of Stability. In particular, we show that the latter is constant on bipartite and almost regular graphs. Moreover, we prove that computing approximations of the social optimum placement and the equilibrium with maximum social welfare is NP-hard even on cubic graphs.
翻译:都市地区的住宅隔离是一种可以在世界各地观察到的现象,其特点是出现了由居民组成的大区域,其居民在种族或其他特征方面是同质的。在AI社区最近的一项研究趋势中,这一现象是通过游戏理论模型来调查的。在这种模式中,两种自私的代理商配备了单调功能,如果一个代理商拥有更相似的邻居,可以确保更高的效用。代理商在战略上选择他们的位置,该图表作为住宅区,以尽量扩大它们的效用。然而,社会民意测验表明,真实世界代理商实际上偏爱混合类型的社区,因此应该通过非monoton的公用事业功能来模拟。我们用非monotoon的公用事业功能来研究Swap Schell游戏。在这些游戏中,两种自私的代理商配有单调的单一的公用事业功能,如果一个代理商拥有更相似的邻里邻居,那么在某个特定的居住区里区里(比如说,代理商更喜欢50个邻里,甚至是少数的居住区),这是我们几乎定期或双向的图表上的平衡所必须的。我们甚至通过近mononononodal int 来在固定的图表中都存在一种平衡,我们通过一个固定的固定的货币价格的基 。我们通过一个固定的固定的货币价格和固定的汇率的基 展示的基 展示的 展示,我们可以证明,我们用这个固定的货币的货币的货币的货币的货币的货币的汇率的汇率的固定的固定的汇率 。我们用法 。这个固定的汇率的汇率的汇率的汇率的汇率的汇率的汇率的汇率, 。我们通过一个潜在的货币的汇率的汇率的汇率的汇率的汇率, 。我们通过一个潜在的货币的汇率的汇率的汇率的汇率的汇率的汇率的汇率的汇率, 证明, 。