Modified Schelling Games

Abstract

We introduce the class of modi ed Schelling games in which there are di erent types of agents who occupy the nodes of a location graph; agents of the same type are friends, and agents of di erent types are enemies. Every agent is strategic and jumps to empty nodes of the graph aiming to maximize her utility, de ned as the ratio of her friends in her neighborhood over the neighborhood size including herself.is 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)e ciency of equilibria that arise in such modi ed 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.

Welfare Guarantees in Schelling Segregation

“…Another question is whether we can obtain in polynomial time a better approximation of social welfare than the factor of 2 in Theorem 3.2, or whether there is in fact an inapproximability result. From a more conceptual perspective, one could try to extend our results to a model with more than two types of agents or more complex friendship relations (e.g., friendship relations defined by a social network, Elkind et al, 2019) or modified utility functions (Kanellopoulos et al, 2020). Questions concerning the convergence behavior in best-response dynamics also remain open: do such dynamics always converge to an optimal assignment?…”

“…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.…”

“…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.…”

“…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.…”

Not all Strangers are the Same: The Impact of Tolerance in Schelling Games

“…ey introduced a simpler utility function which the agents aim to maximize, and studied the existence, complexity and quality of equilibria in jump games with multiple types of agents and general topologies. ey also proposed many interesting variants, such as enemy aversion and social Schelling games, which have been partially studied by Kanellopoulos et al [2020] and Chan et al [2020], respectively. Agarwal et al [2020] studied similar questions for swap games, and Bilò et al [2020] considered a constrained se ing, in which the agents can only view a small part of the topology near their current location.…”