An analysis of one-dimensional schelling segregation

Abstract

We analyze the Schelling model of segregation in which a society of n individuals live in a ring. Each individual is one of two races and is only satisfied with his location so long as at least half his 2w nearest neighbors are of the same race as him. In the dynamics, randomly-chosen unhappy individuals successively swap locations. We consider the average size of monochromatic neighborhoods in the final stable state. Our analysis is the first rigorous analysis of the Schelling dynamics. We note that, in contrast to prior approximate analyses, the final state is nearly integrated: the average size of monochromatic neighborhoods is independent of n and polynomial in w.

Tipping Points in 1-Dimensional Schelling Models with Switching Agents

“…The breakthrough in [4] was built upon in [1] by the present authors, which also provided a thorough analysis of an unperturbed Schelling ring, but over a much larger range of parameters. The present model, which we shall describe shortly, continues in the same vein in again providing rigorous mathematical analyses of unperturbed 1-dimensional models, but represents a significant generalization again in terms of their parameters.…”

“…Here a 1-dimensional unperturbed variant of the model is studied, which is open in the sense that agents may enter and exit the model. Following the authors' previous work [1] and that of Brandt, Immorlica, Kamath, and Kleinberg in [4], rigorous asymptotic results are established.This model's openness allows either race to take over almost everywhere. Tipping points are identified between the regions of takeover and staticity.…”

“…More immediately, the current paper has its roots in the work of Brandt, Immorlica, Kamath, and Kleinberg [4], which represented a major departure from the numerous previous studies of Schelling segregation, for the first time providing a rigorous mathematical analysis of an unperturbed Schelling ring (which is to say a 1-dimensional spacial proximity model). Earlier work, such as that by Young [26] and Zhang ([27], [28], [29]), had concentrated on perturbed models, where agents have a small probability (ε) of acting against their own interests.…”

“…The introduction of this tiny amount of noise ensures that the resulting Markov process is reversible, and thus considerably easier to analyse. Yet it was established in [4] that the unperturbed model may give rise to dramatically different patterns of segregation from the limiting case (letting ε → 0) of the corresponding perturbed model.…”

“…Tipping points are identified between the regions of takeover and staticity. In a significant generalization of the models considered in [1] and [4], the model's parameters comprise the initial proportions of the two races, along with independent values of the tolerance for each race. …”