Welfare Guarantees in Schelling Segregation

TL;DR

This paper investigates welfare guarantees and computational complexity in Schelling's segregation model, analyzing optimality notions, bounds on welfare loss, and efficient algorithms for specific topologies.

Contribution

It introduces new bounds and algorithms for welfare optimization and Pareto optimality in Schelling's model, including polynomial-time solutions for certain graph structures.

Findings

Maximizing social welfare is NP-hard, but near-optimal solutions are computable in polynomial time.

Bounds on welfare loss for Pareto optimal assignments are established.

Existence and efficient computation of positive utility assignments are shown for specific topologies.

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.