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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

* 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

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

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.

Keywords: Residential segregation, Bayesian games, Evolutionary game theory, Bayesian best response dynamic, Stochastic stability.
Mathematics Subject Classification: Primary: 91A10, 91A07, 91A22, 91A80; Secondary: 37N40.
Citation: Emin Dokumacı, William H. Sandholm. Schelling redux: An evolutionary dynamic model of residential segregation. Journal of Dynamics and Games, doi: 10.3934/jdg.2022006
References:
[1]

L. D. Bobo and C. L. Zubrinsky, Attitudes on residential segregation: Perceived status difference, mere in-group difference, or racial prejudice, Social Forces, 74 (1996), 883-909. 

[2]

M. Bøg, Is segregation robust?, Unpublished manuscript, Stockholm School of Economics, 2006.

[3]

C. Z. Charles, The dynamics of racial residential segregation, Annual Review of Sociology, 29 (2003), 167-207.  doi: 10.1146/annurev.soc.29.010202.100002.

[4]

W. A. V. Clark, Residential preferences and neighborhood racial segregation: A test of the Schelling segregation model, Demography, 28 (1991), 1-19.  doi: 10.2307/2061333.

[5]

J. C. Ely and W. H. Sandholm, Evolution in Bayesian games I: Theory, Games and Economic Behavior, 53 (2005), 83-109.  doi: 10.1016/j.geb.2004.09.003.

[6]

R. Farley, E. L. Fielding and M. Krysan, The residential preferences of blacks and whites: A four-metropolis analysis, Housing Policy Debate, 8 (1997), 763-800.  doi: 10.1080/10511482.1997.9521278.

[7] M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, an Diego, 1974. 
[8] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9781139173179.
[9] D. Massey and N. Denton, American Apartheid: Segregation and the Making of the Underclass, Harvard University Press, Cambridge, 1993. 
[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.

[15]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995.

[16]

C. M. Tiebout, A pure theory of local public expenditures, Journal of Political Economy, 64 (1956), 416-424. 

[17] H. P. Young, Individual Strategy and Social Structure, Princeton University Press, Princeton, 1998. 
[18]

H. P. Young, The dynamics of conformity, in Social Dynamics (eds. S. N. Durlauf and H. P. Young), Brookings Institution Press/MIT Press, Washington/Cambridge, 2001,133–153.

[19]

J. Zhang, A dynamic model of residential segregation, Journal of Mathematical Sociology, 28 (2004), 147-170.  doi: 10.1080/00222500490480202.

[20]

J. Zhang, Residential segregation in an all-integrationist world, Journal of Economic Behavior and Organization, 24 (2004), 533-550.  doi: 10.1016/j.jebo.2003.03.005.

show all references

References:
[1]

L. D. Bobo and C. L. Zubrinsky, Attitudes on residential segregation: Perceived status difference, mere in-group difference, or racial prejudice, Social Forces, 74 (1996), 883-909. 

[2]

M. Bøg, Is segregation robust?, Unpublished manuscript, Stockholm School of Economics, 2006.

[3]

C. Z. Charles, The dynamics of racial residential segregation, Annual Review of Sociology, 29 (2003), 167-207.  doi: 10.1146/annurev.soc.29.010202.100002.

[4]

W. A. V. Clark, Residential preferences and neighborhood racial segregation: A test of the Schelling segregation model, Demography, 28 (1991), 1-19.  doi: 10.2307/2061333.

[5]

J. C. Ely and W. H. Sandholm, Evolution in Bayesian games I: Theory, Games and Economic Behavior, 53 (2005), 83-109.  doi: 10.1016/j.geb.2004.09.003.

[6]

R. Farley, E. L. Fielding and M. Krysan, The residential preferences of blacks and whites: A four-metropolis analysis, Housing Policy Debate, 8 (1997), 763-800.  doi: 10.1080/10511482.1997.9521278.

[7] M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, an Diego, 1974. 
[8] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9781139173179.
[9] D. Massey and N. Denton, American Apartheid: Segregation and the Making of the Underclass, Harvard University Press, Cambridge, 1993. 
[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.

[15]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995.

[16]

C. M. Tiebout, A pure theory of local public expenditures, Journal of Political Economy, 64 (1956), 416-424. 

[17] H. P. Young, Individual Strategy and Social Structure, Princeton University Press, Princeton, 1998. 
[18]

H. P. Young, The dynamics of conformity, in Social Dynamics (eds. S. N. Durlauf and H. P. Young), Brookings Institution Press/MIT Press, Washington/Cambridge, 2001,133–153.

[19]

J. Zhang, A dynamic model of residential segregation, Journal of Mathematical Sociology, 28 (2004), 147-170.  doi: 10.1080/00222500490480202.

[20]

J. Zhang, Residential segregation in an all-integrationist world, Journal of Economic Behavior and Organization, 24 (2004), 533-550.  doi: 10.1016/j.jebo.2003.03.005.

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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