Schelling's Threshold Cascades

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What is Schelling’s Threshold Cascades?

Schelling’s Threshold Cascades are the process in which locally mild moving rules, passed from neighbor to neighbor by the act of moving, sort a mixed area into clusters much sharper than any one person’s line.

The systemic effects are found to be overwhelming: there is no simple correspondence of individual incentive to collective results.

That sentence is from Schelling’s 1971 abstract. The incentive is a local share. The result is a pattern. The two need not match. Picture a cafeteria table. Each student stays if about half the nearby seats are friends. One friend stands up. The share falls, so the next student leaves. The table ends as two clusters. Nobody wrote down a wish for an empty middle. The same gap shows up between neighboring ideas. In-Group Bias is a tilt toward one’s own side. Group Polarization is a discussion that pushes the average opinion further out. Schelling’s Threshold Cascades can empty the middle even when nobody’s opinion moved. The people who would have sat in the middle have left, and their leaving is what the next person sees.

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Schelling’s Threshold Cascades in 3 Depths

• Beginner: When a mixed table becomes two tables, ask who left after the ratio changed. Do not start from the claim that everyone wanted a pure group.
• Practitioner: Write the leave-rule as a share before you interpret the map. Then remove one person and recompute the share for those who remain. The second departure is often a reply to the new share.
• Advanced: A stable mix is an overlap of tolerable shares, not the absence of rules. Schelling’s accounting says that, in general, the minority must be the more tolerant group, or a large part of the majority must stay out. Crossing from one stable pattern to another often takes a large shock or action in concert.

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Origin

Schelling’s Threshold Cascades is this page’s name for the sorting-and-tipping account in Thomas Schelling’s 1971 paper. He did not print this English title. His words were dynamic models of segregation, and neighborhood tipping. The paper grew from a RAND memorandum, RM-6014-RC, issued in May 1969. Schelling was at Harvard. He set aside two other causes on purpose: organized exclusion, and the economic sorting of rich from poor. He wanted the part that can come from unorganized individual choices alone. He warned that real cities mix all three, so a segregated map does not identify the cause. The demonstration that readers remember is a board. The runs he compares use 13 rows and 16 columns. Blank squares are usually about 30 percent of the cells, so a person who moves has an empty square to enter. A neighborhood is the eight surrounding squares. Two colors start mixed. A person who is short of a chosen share of like neighbors moves to a vacant square that meets the rule. Schelling wrote that his samples were too small for serious generalizations, and that the counts belong to an artificial board. They are unlikely, he said, to have any quantitative analogue in the living world. The shape of the result is what he asked readers to keep. He then turned the same arithmetic toward tipping. Tipping, in his definition, is what people say has happened when a recognizable new minority enters a neighborhood in sufficient numbers to cause the earlier residents to begin leaving.

Furthermore, it is interesting that in none of the cases shown does any important discontinuity necessarily occur at the modal or typical tolerance value.

He reports Morton Grodzins’s 1957 claim that, for the vast majority of white Americans, a tipping point exists, and that 20 percent was a commonly estimated upper limit in some Eastern cities. Schelling’s figures do not put the break at that named tolerance. If people can tolerate about 20 percent, he writes, any tipping point in the model will tend to sit at a quite different percentage. He also reports two observations from other authors, and he does not treat them as a proof. Duncan and Duncan, in a 1957 study of Chicago, found no mixed neighborhood, defined as 25 to 75 percent white, in which succession from white to Black occupancy was arrested or reversed between 1940 and 1950. Schelling notes that this could mean tipping is nearly universal and irreversible, or that tipping speeds up a change that was coming anyway. Albert J. Mayer’s account of Russell Woods, which Schelling summarizes, is a neighborhood of about 700 single-family homes. A few houses were sold to Black buyers in 1955. The third sale convinced residents that the neighborhood was destined to become mixed. A year later, 40 houses had been sold. Within another two years the share had passed 50 percent, and the end was no longer in doubt. Schelling’s footnote says residents expected large numbers from the first three sales or so, and that at 40 percent nearly everybody took an eventual complete turnover for granted. One result in the analytic section surprises people who think the model requires hatred. The schedules do not depend on a preference for living separately. They do not even depend on a preference for being in the majority. The same curves can be read as the point where a wish for a mixed neighborhood is outweighed by being too small a minority. Schelling did not add a floor on the other group. He did not assume that people want purity. The 1978 book Micromotives and Macrobehavior retells the argument for a general reader. It was not opened for this page. In 2005 Schelling shared the economics prize with Robert Aumann for work on conflict and cooperation through game theory. That prize was not awarded for the checkerboard. A Schelling Point is a different idea from the same author: a place people coordinate on without talking. Mark Granovetter’s 1978 threshold model gives each person a number of others who must already have joined before they join. That is a cascade of joining. This page is a cascade of moving.

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Key Points

Four findings carry Schelling’s Threshold Cascades. All four are from the 1971 paper. None of them is a city percentage you can copy.

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Applications

Four tactics follow from the mechanism. They are uses of the 1971 accounting. They are not results from an office trial or a feed experiment.

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Case Study

The counted case for Schelling’s Threshold Cascades is David Card, Alexandre Mas, and Jesse Rothstein’s 2008 paper in the Quarterly Journal of Economics. They asked whether white population flows in American neighborhoods show the break Schelling described. They used census tracts from 1970 through 2000 and a regression-discontinuity design. The published abstract reports tipping-like behavior in those flows in most cities. The tipping points are spread from 5 percent to 20 percent minority share. The pattern appears both in the suburbs and near existing minority enclaves. The authors report little evidence of a matching break in rents or housing prices at the tipping point. Cities where white residents held more tolerant racial attitudes had higher tipping points. The lesson is narrow and useful. A flow of moves can change sharply around a local share, and that share is not one number for the country. The boundary is just as narrow. This is racial composition of U.S. tracts across those decades, summarized from the abstract. The article PDF was not opened, so no table beyond the abstract is used here. The study does not measure political opinions. It does not show that prices collapse. It does not show that every resident who remains wanted the final map. It does not license a 10 percent rule for a chat, a campus, or another year.

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Boundaries and Failure Modes

Schelling’s Threshold Cascades stops being evidence when a board percentage, or a census range, is treated as a universal law of hatred. The first boundary is the board itself. Schelling wrote that the counts refer only to an artificial checkerboard and are unlikely to have a quantitative analogue in the living world. His samples were too small for serious generalizations. Blank squares, usually about 30 percent, are what make movement possible. If people cannot move, the cascade has no second step. He also notes that it takes more pressure to make a family leave than to make a family decide not to move in. Residents and outsiders need not share one line. The second boundary is the causes he set aside. Organized exclusion and income sorting produce separation with no mild similarity rule at all. A segregated city is not a measurement of the leave-share. Herding Effect is a further neighbor: people copy an action because others took it. Here the person answers a composition, and the answer changes the composition. Collapsing the two names hides which fact you would need to collect. The common misuse is the slogan that any tiny preference, for example leaving when 10 percent of neighbors differ in politics, must end in absolute opposition. That example is a strong similarity demand. The 1971 paper does not run it, does not use political opinions, and does not find that every mild rule produces total separation. At about one-third, separation on his board was slight.

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Common Misconceptions

Three readings flatten Schelling’s Threshold Cascades into a slogan. Each one drops a distinction the 1971 paper states.

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Related Concepts

These pages sit next to Schelling’s Threshold Cascades. Each one names a different object.

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One-Line Takeaway