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: |
[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M. Bøg, Is segregation robust?, Unpublished manuscript, Stockholm School of Economics, 2006.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.![]() ![]() |
Schelling's first example
Schelling's first example revisited: no stable integration
Schelling's second example: stable integration
Lyapunov stable equilibria that are not asymptotically stable
Making the white population less tolerant can create a stable integrated equilibrium
Increasing tolerances in the black population creates a stable integrated equilibrium. After this, increasing the mass of the black population destroys the equilibrium
Dual thresholds, low tolerances. Stable equilibria are segregated and sparsely populated
Dual thresholds, high tolerances. A stable integrated equilibrium exists; with enough committed types it is the unique equilibrium
Taxation to sustain integration