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The Math of Segregation

Brian Hayes

My childhood home was a stone’s throw away from a racial boundary line. I lived just outside of Philadelphia in an all-white suburb, but from my window I could look across a creek and a strip of parkland into a city neighborhood that was almost entirely black. As a boy I never thought to ask how my world came to be divided in this way, but others were puzzling over such questions. One of them was the economist Thomas C. Schelling, now of the University of Maryland.

In the 1960s Schelling devised a simple model in which a mixed group of people spontaneously segregates by race even though no one in the population desires that outcome. Initially, black and white families are randomly distributed. At each step in the modeling process the families examine their immediate neighborhood and either stay put or move elsewhere depending on whether the local racial composition suits their preferences. The procedure is repeated until everyone finds a satisfactory home (or until the simulator’s patience is exhausted).


The outcome of the process depends on the families’ preferences. If everyone is indifferent to race, no one has reason to move. If everyone is absolutely intolerant, refusing to live near anyone of the other race, then total segregation is the only stable solution. Between these extremes lies an interesting spectrum of nonobvious behavior. Schelling discovered that segregation can emerge among residents who have only a mild preference for living with their own kind: They choose not to be in the minority within their immediate environment. These families would be happy with a 50–50 milieu, but they wind up forming enclaves where about 80 percent of their neighbors are of the same race.

Over the years the Schelling model has intrigued not just social scientists but also mathematicians, physicists, and others. Dozens of variants have been explored through computer simulations. Nevertheless, not much about the model could be established with mathematical certainty. It was not clear how the degree of segregation varies as a function of individual intolerance, nor was it certain that the system would always settle into a stable final state. Now two groups of computer scientists, returning to a version of the model very similar to the one Schelling first described, supply some provable, analytic results. Their findings include a few surprises. For example, in some cases the segregation process is self-limiting: The monochromatic enclaves stop growing at a certain size, well before they reach the scale of metropolitan apartheid I knew in Philadelphia.

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