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Economics 1971

Dynamic Models of Segregation

Thomas C. Schelling

Mild private preferences, multiplied across a crowd, can tip a mixed city into stark separation.

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In depth · the introduction

Nobody in the model is a bigot — and the city still splits cleanly in two. That gap is the discovery.

The big idea

Picture a crowd of two kinds of people scattered on a grid. Each person is easy-going: perfectly happy in a mixed area, uncomfortable only if they become a small minority right around their own home — say, if fewer than a third of their immediate neighbours are like them. When someone is uncomfortable, they move to the nearest spot where they aren't. That is the entire rule.

Run it, and the mixed grid slowly sorts itself into large blocks of one kind and large blocks of the other, with clean borders between. The astonishing part is the size of the effect: the final separation is far more extreme than anything any single person ever wanted.

A board game with coins

Thomas Schelling, an economist at Harvard, worked this out around 1969–1971 — famously, by hand. He laid pennies and dimes on a checkerboard, gave each coin a simple rule about how many like-neighbours it needed, and shuffled the unhappy ones to nearby empty squares, again and again. He watched a salt-and-pepper mixture collapse into stark territories. There were no computers in the original experiment; the surprise emerged on a tabletop grid, and it has been reproduced on every computer since.

Why it mattered

It cut a link people assume is tight — that a segregated society must be made of prejudiced individuals. Schelling showed that even a population of tolerant people, each willing to be a minority, can tumble into sharp separation purely from the mathematics of who moves when. The lesson runs both ways: segregation can persist with no one intending it, and gentle private wishes can compound into harsh public facts. And the same shape — small personal choices, surprising collective forms — turns up far beyond housing.

Like a slow stampede that no one is running

Think of a half-full theatre where each person just wants a couple of familiar faces nearby. One person shifts seats to sit near a few similar others; that empties one seat and crowds another, nudging the next person to move, and the next. No one is rushing for the exits, yet seat by seat the audience self-sorts into clumps. No single move looks dramatic — but the room ends up arranged in a way nobody chose.

A square grid of two coloured cells, indigo and amber, with a few empty cells, starting fully mixed. A slider sets how many like-coloured neighbours each cell demands; pressing run lets discontented cells hop to the nearest empty square that satisfies them, round after round, until the mixture reorganises into large single-colour regions with sharp borders. A readout reports the share of content cells and a segregation index that ends far higher than the demand each cell set.

Where it sits

Schelling's checkerboard is a founding example of what is now called emergence — how simple local rules build complex global patterns, the same theme that runs through Conway's Game of Life and through statistical physics. It helped launch agent-based modelling and the field of complexity economics, and won him a share of the 2005 Nobel Memorial Prize in Economics. When you next hear the phrase “tipping point,” it traces back to this work.

The original document
Original source text
Thomas C. Schelling · Journal of Mathematical Sociology, Vol. 1 (1971), pp. 143–186
The question
People get separated along many lines and in many ways.
Schelling opens by cataloguing the axes of separation — sex, age, income, language, religion, colour — and by distinguishing its causes. Some segregation is imposed by organizations; some is deliberately organized; and some, he notes, “results from the interplay of individual choices that discriminate.” It is this last, decentralized kind — no authority, no conspiracy, only many small personal moves — that the paper sets out to model.
Two models
The first is the spatial proximity model: agents of two kinds sit on a line or a checkerboard, each content only if a high-enough fraction of its near neighbours share its kind, and each discontented agent moving to the nearest spot that satisfies it. The second is the bounded neighbourhood model — the “tipping” model — in which agents share one common area and differ in how large a minority of the other kind they will tolerate; departures shift the ratio against those who stay, so a neighbourhood can flip wholesale from mixed to uniform.
The surprise
The systemic effects are overwhelming: there is no simple correspondence of individual incentive to collective results. Exaggerated separation and patterning result from the dynamics of movement.
Worked by hand with coins on a board, the mixed start collapses into broad single-colour regions even though every agent was willing to be a local minority. The aggregate segregation is far more extreme than any individual demanded — the macro-pattern is an amplification of the micro-motive, not a mirror of it.
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Journal of Mathematical Sociology · 1971