| Schelling's Neighborhood
 In the turtle's segregation simulation model in chapter 8 of Flirting With Disaster, there are two very unrealistic characteristics. First, the model assumes that turtles have uniform like-me preferences. Second, the model assumes that each turtle's neighborhood is defined narrowly. What might happen if these assumptions are relaxed?Schelling provides an interesting approach to dealing with this situation that uses hardly any mathematics, although the logic is more complex than it appears.Nobel laureate Thomas SchellingSchelling's ApproachIn terms of like-me preferences, Schelling specifies that this factor ranges widely in the model population, but follows a known distribution. For example, imagine that the most tolerant community member will accept an "adverse" ratio of 2:1 (i.e., being in a one-third minority) while the least tolerant member will not accept any members of the opposite type.A straight-line relationship between these two extremes will produce a median tolerance of 50/50, a fact that you can confirm by drawing a quick graph. If we assume that both groups (Blacks/Whites, Israelis/Palestineans, Catholics/Protestants, Earthers/Klingons, you name it) have similar tolerance profiles but represent different percentages in the overall population---both reasonable assumptions---then it is clear that there are some of each group that can coexist, but not all of them.
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Gentrified Haight Ashbury, San Francisco. Personal picture by Urban, 2004 (CC2). | If we add to the model free entry and exit to and from such a "neighborhood," it becomes possible to see that there are some conditions under which the neighborhood will be stable and others under which it will not. For instance, if the ratio within the neighborhood is attractive to some outsiders, they will move in. Their actions might then push the least tolerant members of the opposite group out, thus changing the ratio again and, as a result, making it more attractive to some and less attractive to others. This gets very difficult to work out in your head, although it is easy to see that such scenarios fit many real world situations, such as neighborhood "gentrification."We are all familiar with the impact of increasing neighborhood affluence on local services, which are often driven out by rising rents and chain stores that move in to take advantage of a greater number of affluent shoppers. In some places, like much of France, for example, regulations require that certain community services, like bakeries, exist in certain numbers. Without such rules, there would be far fewer boulangeries in affluent neighborhoods, since they could not afford to pay local rents.To provide an illustration of this phenomenon, I modified Schelling's identical straight line tolerance profiles from his book Micromotives and Macrobehavior (1978) to produce two different curvilinear shapes. The majority population's tolerance is shown on the left curve while the minority population is shown on the right curve. Note that both the y-axis values and the curvatures are different.
Majority population tolerance Minority population tolerance
The tolerance graph at the left, for the majority population, shows that a few members will live with a 5:1 adverse ratio, but the overall tolerance quickly falls, and the median tolerance ratio is about 0.25. This means that the average citizen will accept only a quarter of his neighbors that are not like him. (Obviously, my numbers are selected for demonstration purposes, not realism.)The minority population's tolerance curve, at the right, is similarly shaped, though not as extreme. It also has higher numerical tolerance values as you might expect from the minority population, and thus the median tolerance is approximately 5.0. As a result, the average minority member will live with an adverse ratio of 5:1. You will see in a moment why I have chosen curves with these shapes.Following Schelling, the next stage of the modeling process is to translate from the generic tolerance curves to specific numerical tolerance figures. To do so, it is necessary to specify the relative sizes of the populations. Let's assume 100 members of the majority population and 25 members of the minority. For each point on the first tolerance curve, we multiply the number of citizens by their respective tolerance level. In the case of the second group, we must also divide the percent numbers on the x-axis by 4 to obtain the proper scaling, since the minority population is only 25. This produces the graphs in the figure below with the number of tolerated citizens of the opposite type on the y-axis and the number of citizens who will tolerate each of those numbers on the x-axis.
Number of "contrasting type" tolerated appears on the y-axis, number of own type appears
on the x-axis for the majority population (left) and the minority population (right).
Putting It All Together
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Combined tolerance chart "phase diagram"
The x-axis represents the number of members of the majority population in the mixed neighborhood and the y-axis represents the number of members of the minority population.
| The final preparatory step is to combine the two charts into one. To do so, it is necessary to rotate the minority group's chart 90 degrees. Since the tolerated number on one chart refers to the number of citizens of the opposite type on the other chart, rotating one chart and overlaying them connects the two charts logically. When we do the combination, the new chart looks as shown in the figure at the right. The number and letter annotations and "phase diagram" arrows are explained below.The x-axis represents the number of members of the majority population in the mixed neighborhood and the y-axis represents the number of members of the minority population. The rounded curve that opens to the left represents the combinations of numbers of the two groups that are acceptable to the minority population while the humped-shaped curve that opens downward represents the acceptable mixtures for the larger population. The darkest shaded overlap area represents the mixtures that are acceptable to both, while the partly-shaded areas represent combinations that are acceptable to one group but not the other. The white area toward the upper right comprises those population combinations (such as 85 members of the majority and 20 members of the minority, near the letter A) that will make some members of both groups unhappy, thus encouraging them to leave, and causing the population numbers of both groups to decrease along both axes.The final population of the mixed neighborhood depends on the shape of these curves, the starting populations in the mixed neighborhood, and the rates of movement of the two populations both in and out. For instance, if the populations were to start off in region A, at the upper right, members of both groups would leave, presumably the least tolerant members first. However, what happens over time depends on the rates of departure. Numerically equal departure rates would create one type of movement, while departure rates that are proportional to the starting populations would create another type of movement. These differences would affect the specific part of the lower combined curve that is intersected as the populations fall.If the falling population mix enters area C, the population mix will be drawn toward point 1, where it will remain. Point 1 is one of the three stable equilibrium points in the diagram, and the only one that is not an extreme point representing either 100 percent of one group or the other. Almost all starting points within region F---which corresponds to population mixtures acceptable to both groups---are also likely to end up at point 1. While all of region F represents mixtures that will attract new outside members from both groups---causing the population mix to move up and to the right---when the boundary with region C is reached, members of the majority population will stop entering because outside members find further increases in the minority population unattractive. However, region C is an area that is still attractive to outside members of the minority population, so they will continue to enter. In dynamic terms, the majority population is likely to rise and then fall, potentially falling to a fraction of its maximum value, since point 1 is far to the left of some parts of region F.Point 2 is unusual. It sits at the maximum number of the larger population that falls within the minority population's tolerance zone, and also on the boundary of region A. From this point, neither population has any incentive to increase their numbers from outside the neighborhood since doing so would push the population mix into region A, thus making it unattractive to the newcomers.(An equally valid view--although not necessarily a realistic one--is that newcomers would not realize the impact of their presence until after they entered, but would immediately leave. Assuming that entry and exit were symmetrical, the same sized group would repeatedly enter and then leave.) Similarly, from point 2, there is no reason for incumbents from either group to leave, since they are satisfied with the current ratio. However, this equilibrium is unstable. One can see that a decrease in either population (movement down, to the left, or both) will set into motion steps that either lead either to point 1 (through regions F and C) or to the complete dominance of the majority population following a path through region B to the extreme point at the lower right.
The idea that the behavior of the entire system may be sensitive to small changes at certain equilibrium points is a very important learning. (You may remember from calculus that such instabilities occur often in complex mathematical functions.) You might also recall from Flirting With Disaster, my message that rational decisions in complex dynamic systems do not always produce beneficial outcomes, even if they appear to do so when they are initiated.Finally, region D might be termed a ghetto. In this region, almost any minority population with only a handful of the majority will lead to the minority's exclusive dominance. As a consequence of their tolerance curve, very few majority population members will accept being in an extreme minority, so these initial conditions defined by region D lead inexorably to a 100 percent minority community. Minority members continue to enter while majority population members do not. Eventually, no majority population members find the relative number of minority population acceptable.With all these complex dynamics, you might be wondering why I did not simply build a somewhat more complex agent-based simulation, such as the one described in the book, to explore these relationships. Actually, there is no reason not to do so. The appropriate number of turtles could be randomly assigned a tolerance level from a suitable function and the turtles could be programmed to jump into and out of a designated neighborhood based upon both their tolerance level and the characteristics of the neighborhood.However, it would be necessary to run many experiments to understand the full nature of the dynamics, and there would always be a risk of missing certain important relationships. This diagram---which is a form of so-called "phase diagram" that is often used to study dynamical systems---provides an overview of all of the dynamics, albeit one that many people initially find somewhat difficult to interpret. Generally, phase diagrams and dynamic simulations work well together in exploring the nature of systems with complex dynamic properties. I probably need not remind you too strenuously about how difficult it would be to figure out dynamic situations like Shelling's neighborhood (not to mention even more complex real world systems) without such conceptual models and computational aids.
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