Swap Stability in Schelling Games on Graphs

Abstract

We study a recently introduced class of strategic games that is motivated by and generalizes Schelling's well-known residential segregation model. These games are played on undirected graphs, with the set of agents partitioned into multiple types; each agent either occupies a node of the graph and never moves away or aims to maximize the fraction of her neighbors who are of her own type. We consider a variant of this model that we call swap Schelling games, where the number of agents is equal to the number of nodes of the graph, and agents may swap positions with other agents to increase their utility. We study the existence, computational complexity and quality of equilibrium assignments in these games, both from a social welfare perspective and from a diversity perspective.

Welfare Guarantees in Schelling Segregation

“…For this setting, the authors showed results similar to those of Elkind et al (2019). Bilò et al (2020) improved some of the price of anarchy bounds of Agarwal et al (2020), and also studied a variation of the model in which the agents have a restricted view of the topology and can only swap with their neighbors. Kanellopoulos et al (2020) investigated the price of anarchy and stability in jump Schelling games, but with a slightly different utility function according to which an agent considers herself as part of her set of neighbors.…”

“…Most related to our present work are the papers by Elkind et al (2019), Agarwal et al (2020), Bilò et al (2020), and Kanellopoulos et al (2020), which studied gametheoretic and complexity questions related to the social welfare in Schelling games. In particular, Elkind et al (2019) considered jump Schelling games in which there are k ≥ 2 types of agents, and the topology is a graph with more nodes than agents so that there are empty nodes to which unhappy agents can jump.…”

“…Given that the agents are willing to relocate to be close to other agents of the same type, another natural question is whether the resulting assignments satisfy some sort of efficiency. This has been considered in part by a recent array of papers (Chauhan et al, 2018;Echzell et al, 2019;Elkind et al, 2019;Agarwal et al, 2020;Bilò et al, 2020;Chan et al, 2020;Kanellopoulos et al, 2020), which have studied Schelling's model from a game-theoretic perspective. In particular, instead of randomly relocating, the agents are assumed to be strategic and each of them aims to select a location that maximizes her utility, defined as the fraction of same-type agents in her neighborhood.…”

“… Agarwal et al (2020). showed that the hardness holds when the numbers of agents and nodes are equal, but still required stubborn agents and moreover assumed at least three types of agents.…”

Equilibria in Schelling Games: Computational Hardness and Robustness

“…Happy agents do not change location, while, depending on the model, unhappy agents either randomly swap vertices with other unhappy agents or randomly jump to empty vertices. Schelling [25,26] observed that even for tolerant agents with τ ∼ 1 3 , segregation patterns (large connected areas where agents have only neighbors of their type) are likely to occur. Over the last 50 years, Schelling's model has been thoroughly studied both from an empirical (see, e.g., [9,12]) and a theoretical (see, e.g., [3,4,7,17]) perspective in various disciplines including computer science, economics, physics, and sociology.…”