In the simplest game-theoretic formulation of Schelling's model of segregation on graphs, agents of two different types each select their own vertex in a given graph so as to maximize the fraction of agents of their type in their occupied neighborhood. Two ways of modeling agent movement here are either to allow two agents to swap their vertices or to allow an agent to jump to a free vertex. The contributions of this paper are twofold. First, we prove that deciding the existence of a swap-equilibrium and a jump-equilibrium in this simplest model of Schelling games is NP-hard, thereby answering questions left open by Agarwal et al. [AAAI '20] and Elkind et al. [IJCAI '19]. Second, we introduce two measures for the robustness of equilibria in Schelling games in terms of the minimum number of edges or the minimum number of vertices that need to be deleted to make an equilibrium unstable. We prove tight lower and upper bounds on the edge- and vertex-robustness of swap-equilibria in Schelling games on different graph classes.
翻译:在Schelling的图形隔离模型的最简单的游戏理论配方中,两种不同类型的代理人在某个图表中选择了自己的顶点,以便最大限度地增加其类型代理人在被占领社区中的分数。两种模拟剂移动方式是允许两个代理人交换他们的头顶,或允许一个代理人跳跃到一个自由的顶点。本文的贡献是双重的。首先,我们证明在Schelling游戏这一最简单模型中决定是否存在交换平衡和跳平衡是NP-硬的,从而回答Agarwal等人[AAI'20]和Elkind等人[IJCAI'19]留下的开放问题。第二,我们从最小边缘数或最小的顶点数量的角度为Schelling游戏的平衡不稳定性设定了两种衡量标准。在Schelling游戏中,我们发现在Scheling-stex-bustrucle的边缘和顶端-顶端-顶点游戏中,在不同的平面和顶端-平面的平流层中,我们证明是紧紧的界限。