Welfare Guarantees in Schelling Segregation

Abstract

Schelling’s model is an influential model that reveals how individual perceptions and incentives can lead to residential segregation. Inspired by a recent stream of work, we study welfare guarantees and complexity in this model with respect to several welfare measures. First, we show that while maximizing the social welfare is NP-hard, computing an assignment of agents to the nodes of any topology graph with approximately half of the maximum welfare can be done in polynomial time. We then consider Pareto optimality, introduce two new optimality notions based on it, and establish mostly tight bounds on the worst-case welfare loss for assignments satisfying these notions as well as the complexity of computing such assignments. In addition, we show that for tree topologies, it is possible to decide whether there exists an assignment that gives every agent a positive utility in polynomial time; moreover, when every node in the topology has degree at least 2, such an assignment always exists and can be found efficiently.

Schelling games on graphs

“…Next, we establish the hardness of maximizing social welfare for the case k = 2 and |V | > n; very recently, Bullinger et al [2021] complemented our result by showing that the problem is hard, even when k = 2 and |V | = n. eorem 4.2. Given a Schelling game with two types and a rational value ξ, it is NP-complete to decide whether the game admits an assignment with social welfare at least ξ. e hardness result holds even if all strategic agents belong to one type and the other type consists of a single stubborn agent.…”

“…Chan et al [2020] introduced an alternative model wherein multiple agents can occupy the same location and, similarly to our social Schelling games (Section 7.1), there is a friendship network. Bullinger et al [2021] presented results on the complexity of computing assignments ful lling welfare guarantees or other e ciency notions. Very recently, Kreisel et al [2021] built on our work to establish that nding equilibria is hard even if all agents are strategic; their result holds both for jump and for swap games.…”

Network Creation with Homophilic Agents

“…Residential segregation has recently received a lot of attention by a stream of research developing a game-theoretic framework based on Schelling's checkerboard model (Chauhan et al, 2018;Agarwal et al, 2021;Echzell et al, 2019;Kanellopoulos et al, 2021a;Bullinger et al, 2021;Kanellopoulos et al, 2021b;Bilò et al, 2022). There, agents of several types strategically select positions on a given fixed network and try to optimize the number of same-type agents in their neighborhood.…”

Tolerance is Necessary for Stability: Single-Peaked Swap Schelling Games

“…The modified version yields that agents are aware of their own contribution to the diversity of their neighborhood. Bullinger et al [2021] consider the number of agents with non-zero utility as social welfare function. They prove hardness results for computing the social optimal state and they discuss other stability notations such as Pareto optimality.…”

“…But only in the last decade progress has been made to understand the involved random process from a theoretical point of view. Even more recently, the Algorithmic Game Theory and the AI communities became interested in residential segregation and game-theoretic variants of Schelling's model were studied [Chauhan et al, 2018;Echzell et al, 2019;Bilò et al, 2020;Agarwal et al, 2021; Kanellopoulos et al, 2021a;Bullinger et al, 2021]. In these strategic games the agents do not perform random moves but rather jump or swap to positions that maximize their utility.…”