doi: 10.3934/jdg.2022006
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Schelling redux: An evolutionary dynamic model of residential segregation

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.

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M. Bøg, Is segregation robust?, Unpublished manuscript, Stockholm School of Economics, 2006.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