Thomas Schelling’s segregation model, discussed in class and in his book Micromotives and Macrobehavior, yields pessimism for those who believe in and strive for integrated neighborhoods. Even a mild preference on the part of all residents for neighbors of the same color (Schelling uses 33%, the example in class used 40%) eventually the neighborhood becomes segregated or at least divides into segregated sub-neighborhoods.  Despite confirming data and evidence of the model’s robustness, a variation on the model’s assumptions leads to a hopeful challenge.

One confirmation that this theory actually plays out in reality with respect to politics if not race is seen in the resent book, “The Big Sort” by Bill Bishop, reviewed in the April 22 edition of the Wall Street Journal. The review quotes Bishop as saying “As Americans have moved over the past three decades they have clustered in communities of sameness, among people with similar ways of life, beliefs and in the end, politics.” Bishop attributes this to ‘demographic resorting’ — when individuals are in a position, economically and otherwise, to make their own lifestyle choices they chose to cluster along cultural lines. The review dwells primarily on the political implications of this tendancy, which Bishop observes as particularly prevalent over the last eight years and attributes it to the political polarization, but suggests that the current presidential campaign involves candidates who (at least until recently) strive to be less polarizing, and that “The Big Sort” may “turn out to be a captivating account of recent history rather than an enduring explanation of American social life.”

Curiously, the review fails to mention that Schelling has shown such clustering to be a mathematical consequence rather than attributable to causes such as real estate agent ’steering’, bank policies, or ‘political polarization’.   Nor does the book itself refer to Schelling’s work, according to the index.  But we know from Schelling’s work that such clustering is a natural and mathematical consequence of mild preferences to live among people like oneself.

In fact the robustness of Schelling’s work has been affirmed in “Schelling’s spatial proximity model of segregation revisited” by Pancs and Vriend, Journal of Public Economics 91 (2007) 1-24. The authors apply Schelling’s approach to residents with a variety of utility functions u(x) representing attitudes for integration, all captured in a single parameterized form. Different attitudes towards segregation are represented by different values of the parameters. With one set of values this formulation gives Schelling’s original model, while other parameter values express a positive attitude towards integration.

However neither Schelling’s original model nor this generalization, nor any of the three Java-based online simulations linked from the class notes consider a case in which residents actually prefer to live among a number of people unlike themselves.   In fact in his book Schelling himself points out that “the only asymmetry is that we did not postulate a lower limit to the acceptable proprtion of opposite color, i.e. an upper limit to the proportion of like color in the neighborhood.”   

Suppose, instead of preferring (as in the in-class example) that at least 40% of immediate neighbors are like them, residents prefer that at most 60% of immediate neighbors are like them, i.e. at least 40% of neighbors are unlike them. The following illustrations show what would happen, starting with precisely the arrangement diagrammed in Schelling’s book but with this altered utility function.

Schelling’s initial state:

To indicate the nine residents who are ‘unhappy’ with respect to this revised criterion,  pennies have been placed under their Go tokens. 

 
One result of each ‘unhappy’ resident moving to a location that satisfies their criterion is shown below.  It should be confirmed by replication that this result is typical, but from the dynamics by which they arrive at this point it seems obvious that it is.

Whether Schelling’s criterion (as generalized by Pancs and Vriend and implemented in the inline Java models) or this revised criterion is more representative of actual human attitudes is not a mathematical matter, nor is it immutable.  (For one thing, it does not account for interracial families, who may well  prefer mixed neighborhoods.)    I know for a fact that some people actually employ the criterion proposed here, and Bishop’s book notwithstanding there are heterogeneous neighborhoods in this country, and stable ones at that.  

By ignoring the possibility that people may prefer to live in neighborhoods where there are at least some people unlike them (rather than preferring to live in neighborhoods where there are some people like them), Schelling’s work and other work that draws on it is biased against such an outcome.   To the extent that the work is widely known, it becomes a self-fulfilling prophecy.   Imagine a young couple planning to move into an integrated neighborhood because of a desire for diversity, only to be discouraged by their parents or friends on the grounds that the neighborhood is bound to tip one way or another.