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The Math of Segregation

Brian Hayes

Ground Truth

Can a game played with colored dots tell us anything useful about housing patterns and race relations in real cities? Racial segregation is not purely a result of individuals expressing preferences about their neighbors. There are legal, economic, and institutional forces at work as well. And segregation is generally not a symmetric process, in which two groups agree to live apart, but rather is a mechanism by which one group excludes the other. All of this is left out of the Schelling model.

But the absurd simplicity of the model is also its main fascination. Even mindless colored dots, entirely innocent of history and geography, can give rise to complex patterns no one expected or wanted. The dynamic at the heart of the model is more mathematical than sociological. It’s just this: If I move from Downtown to Uptown because I seek different neighbors, my presence in one place and my absence from the other alters both of those environments, which may induce others to move in turn. What could be more obvious? Yet the consequences are not easily foreseen.

My own qualms about the model focus not on oversimplification but on questions of robustness—of sensitivity to details of implementation. For example, in Schelling’s first version of the model, an unhappy agent moved to the nearest open location, not to a randomly chosen one; this deterministic choice seems to yield dramatically different results in some cases. Small changes in the rules governing favorable, neutral, and unfavorable moves also have major effects. Such sensitivity to minor variations is worrisome in a model meant to represent human behavior.

Markus M. Möbius of Microsoft Research and Tanya S. Rosenblat of Iowa State University have tried to test the model against real-world data. Working with census records for Chicago, they find evidence supporting the hypothesis that local interactions have the strongest influence on racial preferences, as in the Schelling model; the effective radius when people choose a neighborhood may be as small as 150 meters.

Meanwhile, back in my old neighborhood on the western edge of Philadelphia, the social fabric of segregation has begun to fray. According to the 2010 Census, the block group that includes my former home is now two-thirds African-American; on the other hand, the city neighborhood across the creek is still only 1 percent white.

Regrettably, the Schelling model can have nothing to say about this further evolution of the urban scene. The model describes only the genesis of segregation; once the colored dots have all sorted themselves into monochrome units, the map can never change. I would love to believe that the time has come for a model of racial reintegration.

©Brian Hayes


  • Barmpalias, G., R. Elwes, and A. Lewis-Pye. 2013 preprint. Digital morphogenesis via Schelling segregation.
  • Brandt, C., N. Immorlica, G. Kamath, and R. Kleinberg. 2012. An analysis of one-dimensional Schelling segregation. In Proceedings of the 44th Symposium on Theory of Computing, pp. 789–803.
  • Möbius, M. M., and T. S. Rosenblat. 2001. The process of ghetto formation: Evidence from Chicago. Unpublished manuscript.
  • Schelling, T. C. 1969. Models of segregation. American Economic Review 59:488–493.
  • Schelling, T. C. 1971. Dynamic models of segregation. Journal of Mathematical Sociology 1:143–186.
  • Schelling, T. C. 1978. Micromotives and Macrobehavior. New York: W. W. Norton.
  • Wilensky, U. 1997. NetLogo segregation model.

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