Schelling redux: An evolutionary dynamic model of residential segregation

Emin Dokumacı; William H. Sandholm

doi:10.3934/jdg.2022006

Article Contents

2022, Volume 9, Issue 4: 373-403. Doi: 10.3934/jdg.2022006

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Schelling redux: An evolutionary dynamic model of residential segregation

Emin Dokumacı^1,*, , and
William H. Sandholm^2,

1.
Freddie Mac, 1551 Park Run Dr, McLean, VA 22102, USA
2.
Department of Economics, University of Wisconsin, 1180 Observatory Drive, Madison, WI 53706, USA

^*Corresponding author: Emin Dokumacı
^*Corresponding author: Emin Dokumacı

^* This work is written in 2007 with my advisor William H. Sandholm. I dedicate this research to his memory

Received: October 2020

Early access: March 2022

Published: October 2022

We thank Marzena Rostek and Larry Samuelson for helpful comments. Financial support from NSF Grants SES-0092145 and SES-0617753 is gratefully acknowledged.

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Abstract

Schelling [13] introduces a seminal model of the dynamics of residential segregation in an isolated neighborhood. His model combines agent heterogeneity with explicit behavior dynamics; as such it is presented informally, and with the use of "semi-equilibrium" restrictions on out-of-equilibrium play. In this paper, we use recent techniques from evolutionary game theory to introduce a formal version of Schelling's model, one that dispenses with equilibrium restrictions on the adjustment process. We show that key properties of the resulting infinite-dimensional dynamic can be derived using a simple finite-dimensional dynamic that captures aggregate behavior. We determine conditions for the stability of integrated equilibria, and we derive a strong restriction on out-of-equilibrium dynamics that implies global convergence to equilibrium: along any solution trajectory, one population's aggregate behavior adjusts monotonically, while the other's changes direction at most once. We present a variety of examples, and we show how extensions of the basic model can be used to study both alternative specifications of agents' preferences and policies to promote integration.

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Mathematics Subject Classification: Primary: 91A10, 91A07, 91A22, 91A80; Secondary: 37N40.

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Figure 1. Schelling's first example

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Figure 2. Schelling's first example revisited: no stable integration

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Figure 3. Schelling's second example: stable integration

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Figure 4. Lyapunov stable equilibria that are not asymptotically stable

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Figure 5. Making the white population less tolerant can create a stable integrated equilibrium

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Figure 6. Increasing tolerances in the black population creates a stable integrated equilibrium. After this, increasing the mass of the black population destroys the equilibrium

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Figure 7. Dual thresholds, low tolerances. Stable equilibria are segregated and sparsely populated

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Figure 8. Dual thresholds, high tolerances. A stable integrated equilibrium exists; with enough committed types it is the unique equilibrium

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Figure 9. Taxation to sustain integration

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References

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