Schelling's segregation model, playableA Third Is Enough

Give everyone on this grid the mildest preference imaginable — "I don't mind being a minority; I just don't want to be outnumbered worse than two-to-one" — and a perfectly mixed city sorts itself into hard, single-colour blocks that nobody asked for. Not because anyone is a bigot. Because everyone is a little bit choosy, and choices compound. This is a random-relocation variant of Thomas Schelling's 1971 model. His nearest-satisfactory-vacancy rule gave only slight segregation at about a third; the strong sorting here is measured under the rule below.¹ Here it is, running, with the gap between what people want and what they get measured live.

Two kinds of people share a grid — call them gold and teal — with some cells left empty. Each person looks at their up-to-eight neighbours and is content as long as at least a fraction τ of the neighbours they can see are their own kind. Anyone who isn't content picks up and moves to a random empty cell. That's the entire model. Nobody wants a segregated city; the sharpest preference on offer here is "please don't leave me almost alone." Watch what a whole grid of that mild wish produces.

Press play at the default setting — τ = 1/3 — and the salt-and-pepper mix unmixes. Within a dozen-odd rounds the churn stops: the grid has broken into broad territories of one colour, veined with the empty cells people fled through. Now read the two numbers. The grid started with each person having about 50% same-kind neighbours — exactly what a random mix gives. It ends at about 76%.

Everyone asked for a third. Everyone got three-quarters.

That is the whole paradox in one line, and it is worth sitting with. No individual on the grid wants a segregated world. Each would be perfectly content in a neighbourhood that is two-thirds other. Yet the arrangement these moves reach leaves them all content, with the average person surrounded overwhelmingly by their own kind. A mixed arrangement can also leave everyone content. The preference is mild and the outcome is extreme, and nothing bridges the two except the fact that one person's move makes a neighbour a little more outnumbered, who then moves, and nudges someone else. Segregation here is not anyone's goal. It is a fixed point these moves reach.

And here is the part that should unsettle you: everyone is happy

Look at the second number when the motion stops: zero people still want to move. The segregated grid is not a tragedy anyone is trapped in — it is an equilibrium everyone accepts. You cannot fix it by asking people to be content, because they already are. You cannot point to the culprit, because there isn't one. A social outcome that no member of the society wanted, and that no member is unhappy with, is the specific thing Schelling built this toy to make visible. The prize committee discussed the segregation model among his other contributions when he shared the 2005 Nobel Prize in economics.¹

What you ask for, versus what you get

Slide the preference up and down and the endpoint moves with it — but never to where you pointed. This curve is the summary: for each value of τ, the height of the gold line is where the grid ends, after settling or reaching the 400-round cap (averaged over seeds 1 to 24; every point recomputed offline). The pale diagonal is the honest expectation — if the world gave you exactly the mix you asked for, it would sit on that line. It never does. The whole region between the two is the paradox.

Two features of that curve are real discoveries, not decoration. First, the settled points sit above the diagonal: even a whisper of preference (τ = 0.15) leaves the grid measurably more sorted than chance. That holds for the mild settings plotted here; at τ = 1/8, the same experiment ends near the mixed baseline, at about 0.51. Second, follow the line to the right and it doesn't just keep rising — just past τ = 0.75 it falls off a cliff, crashing back down to the mixed baseline.

The cliff and the local threshold

Past that point the random moves fail to find an arrangement that contents everyone within the 400-round cap, and the grid remains roughly as mixed as it started. That is a measured outcome, not a proof that no contented arrangement exists or that the motion lasts forever. With eight occupied neighbours, wanting "≥ 75% the same" means 6 of 8. Above 75%, within the slider's range, the requirement becomes 7 of 8. That local threshold is arithmetic; it does not prove the global cliff. Two solid groups separated by empty space can content everyone even at τ = 1, with this grid's same 1,125 gold, 1,125 teal and 250 empty cells. The wish is possible. The sampled runs above 0.75 do not settle within 400 rounds.

What this does and doesn't say

It is tempting to walk away from this grid believing it has explained the segregated city outside your window. It hasn't, and pretending otherwise would betray the one thing the model is good for. Here is the honest reading, which is more interesting than the lazy one.

The model proves a sufficiency, not a cause. It shows that mild same-kind preference is enough, all by itself, to produce stark segregation — you don't need hatred, money, or law to get there. That is a genuine and counterintuitive result: it means observing a segregated city tells you less about people's inner attitudes than you'd think, because even tolerant people generate it. But "sufficient" is not "actual." Real-world residential segregation is over-determined — driven by income and housing cost, by explicit discrimination, by lending and zoning and school catchments, by history this toy contains none of. Schelling's grid removes all of that on purpose, to isolate one mechanism, and its lesson is precisely about that one mechanism, not the whole phenomenon.²

Two more caveats the honest version keeps in view. The grid is perfectly symmetric — the two groups are identical, equally numerous, with identical preferences and no difference in power or wealth. Real segregation is none of those things, and the symmetry is a limitation, not a neutrality. And the exact number — 76% at τ = 1/3 — depends on the rules stated here (eight neighbours, a bounded grid, moves to a random empty cell). Change the neighbourhood or the moving rule and both the number and the tendency to sort can change; some variants lack segregation patterns.³ Here, the strong sorting persists across the grid sizes and vacancy fractions tested below. The decimal is ours, and you can check it.