Segregation without a bigot
Nobody in this model wants to live apart. Every household is content in a mixed street and only moves when it is left in a small minority. Run it and the city separates anyway, and it separates far more than anybody in it asked for.
This is the oldest agent based model in the social sciences and it is still the one that changes minds fastest. Thomas Schelling published it in 1971. It has two kinds of household, a board, and one rule.
The rule is this. Look at the eight squares around you. Count the neighbours who are like you as a share of the neighbours you have. If that share is at least what you asked for, stay. If it is less, move to a vacant square. That is the whole of the model, and the number you ask for is the only thing the reader sets.
Set it to a third and you have said something very mild: I am content anywhere I am not badly outnumbered, and two thirds of my neighbours can be the other kind. Run it, and watch what a city of people holding exactly that opinion turns into.
01The board
Each square is a household or a vacancy. The colour is the kind. Nothing else about a household exists: no income, no price, no landlord, no history, no opinion about anybody. The only thing a household can do is stay or move.
The city
Separation, round by round
02What is being measured
The number to watch is the separation index. It is the plainest thing you could measure: for every household, the share of its occupied neighbours that are its own kind, averaged over the whole board.
separation index = average over households of ( like neighbours / occupied neighbours ) A perfectly stirred board sits near the share of the majority kind, because that is what a random neighbour is. A completely separated board sits at one.
The starting board is random, so the index begins near the mixing share and the model has nowhere to go but up. What matters is not that it rises. What matters is how far above the demand it climbs. Ask for a third and the board will not stop at a third. It will run past it and settle somewhere near four fifths, and it will do that without a single household ever wanting a street that looks like the one it ends up in.
03Why the outcome overshoots the wish
Because a move is not a private act. A household that leaves a street changes the count for everyone it leaves behind, and some of them now fall under their own threshold and leave too. A household that arrives changes the count for everyone it joins. Every move is a small push on two neighbourhoods, and the pushes do not cancel: they favour the direction of separation, because a move is only ever made towards more of your own kind and never away from it.
So the board has a ratchet in it. There is no force that pulls a segregated street back apart, because nobody in this model is made unhappy by having too many neighbours like themselves. Turn the demand up to something genuinely intolerant and you get the same result faster and harder. Turn it down towards nothing and the ratchet stops, because nobody moves at all.
Somewhere between the two there is a threshold below which the board barely moves and above which it separates almost completely. Find it with the slider. It is lower than most readers expect.
04Why this belongs to SIMEA
The reason I keep this model at the front of the programme is that it breaks the habit of reading an outcome back into the people inside it. Look at the finished board and every instinct says that the households wanted this. They did not. Not one of them holds a preference that looks anything like the city it is living in.
That gap between what an agent wants and what the arrangement of agents produces is the gap SIMEA is built to measure. Here it is opened by the geometry of a board. In the simulation next door it is opened deliberately, by a system that chooses what each agent sees. The mechanism differs. The lesson does not: the individual preference and the collective outcome are separate quantities, and reading either one off the other is a mistake that survives because it is comfortable.
05Where the model stops
Almost everything that makes housing housing is missing here, and the omissions are not incidental. Here is what this board is not entitled to tell you.
Assumptions and limits
- No moneyThere is no rent, no price, no wage and no landlord. In a real city the constraint that decides who lives where is usually what they can pay, and discrimination often works through the price rather than through the neighbour count. None of that exists on this board.
- Two kindsHouseholds come in exactly two kinds and everyone of a kind is identical. That is a very strong simplification of anything a real division is made of.
- Movement is freeA dissatisfied household moves instantly to a vacant square at no cost, with no search, no queue, no refusal and no notice period. Real moving is slow and expensive, which by itself would slow this board down a great deal.
- Nobody is turned awayThe model has no landlord who refuses, no agent who steers, no covenant and no policy. It shows one channel by which separation can arise. It does not show that this is the channel that produced any actual separated city, and the historical record of deliberate exclusion is not in this picture at all.
- A vacant square is a random squareMy version sends an unhappy household to a vacancy chosen at random, where Schelling's own account moves it to the nearest satisfying spot. The qualitative result is the same and the details of the pattern are not.
- It is not evidenceThis is a demonstration that a mechanism is sufficient to produce an outcome. Sufficiency is not the same as having happened. No number on this page describes any real place.
06Source
The model is Schelling's. The board here follows the area version in that paper: a grid, two kinds, roughly a third of the squares left vacant, and a household that counts the eight squares around it.
- The paper that introduced the model, with both a one dimensional line of neighbours and the two dimensional board used here. In the board version a household counts the eight surrounding squares, demands that at least half of its neighbours be of its own kind, and moves to the nearest vacancy that satisfies it. Schelling reports that these mild demands produce clusters in which the great majority of neighbours are alike, and writes that there is no simple correspondence between individual incentive and collective result. Thomas C. Schelling, Dynamic Models of Segregation, The Journal of Mathematical Sociology 1(2), 1971, pages 143 to 186. tandfonline.com/doi/abs/10.1080/0022250X.1971.9989794 Full text read at suz.uzh.ch. Checked 2026 08 22.