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Schelling redux: An evolutionary dynamic model of residential segregation

  • *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

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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  • 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.

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

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  • Figure 1.  Schelling's first example

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

    Figure 3.  Schelling's second example: stable integration

    Figure 4.  Lyapunov stable equilibria that are not asymptotically stable

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

    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

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

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

    Figure 9.  Taxation to sustain integration

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    [2] M. Bøg, Is segregation robust?, Unpublished manuscript, Stockholm School of Economics, 2006.
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    [10] M. M. Möbius, The Formation of Ghettos as a Local Interaction Phenomenon , Unpublished manuscript, MIT, 2000.
    [11] R. Pancs and N. J. Vriend, Schelling's spatial proximity model of segregation revisited, Journal of Public Economics, 91 (2007), 1-24. 
    [12] W. H. Sandholm, Evolution in Bayesian games II: Stability of purified equilibria, Journal of Economic Theory, 136 (2007), 641-667.  doi: 10.1016/j.jet.2006.10.003.
    [13] T. C. Schelling, Dynamic models of segregation, Journal of Mathematical Sociology, 1 (1971), 143-186. 
    [14] T. C. Schelling, Micromotives and Macrobehavior, Norton, New York, 1978.
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