Why swaps with random empty cells instead of “best” locations?

Schelling’s story is about satisficing under friction: agents leave bad spots and take any vacancy they can find. Random swaps keep the rule set small while still producing strong sorting; more elaborate search rules change kinetics but not the qualitative lesson.

Why does τ ≈ 0.35–0.45 already look “extreme” with ~10% vacancies?

With two equal groups and random mixing, a typical interior agent sees about half of its occupied neighbors as “same” on average. Raising τ only a little above that baseline makes many agents unhappy simultaneously; repeated swaps amplify correlations into large clusters even without explicit attraction to identical types.

Does the torus change the story compared to an open boundary?

Wrap-around removes edge effects so interior-like statistics persist everywhere on a small grid. Open boundaries can pin clusters against walls; the torus keeps the focus on spontaneous pattern formation from local rules.

Schelling Segregation Model

Two agent types on a toroidal lattice with vacancies: unhappy agents swap into random empty cells when fewer than a fraction τ of occupied Moore neighbors share their type—mild local rules, global clustering.

About this model

Thomas Schelling’s classic spatial model shows how mildly coordinated preferences can produce macroscopic segregation even when nobody demands a fully homogeneous neighborhood. On this page, two agent types (red and blue) occupy a toroidal square lattice with a tunable fraction of empty sites. Each agent inspects its eight Moore neighbors that are occupied; it is unhappy if the fraction of neighbors of its own type is strictly below a similarity threshold τ. Asynchronous dynamics repeatedly picks a random unhappy agent (if one exists) and swaps it with a random empty cell—mimicking relocation with vacancy-limited housing. The emergent patchiness is a standard teaching example in complex systems and agent-based modeling: the global pattern is not encoded in any individual’s goal, yet it appears quickly on finite grids.

Who it's for: Students in statistical physics, computational social science, or complex systems after percolation or simple cellular automata.

Key terms

Schelling model
Segregation
Agent-based model
Moore neighborhood
Similarity threshold
Emergence

How it works

Thomas Schelling’s checkerboard model: two groups on a toroidal square lattice with vacancies. Each timestep unhappy agents swap into random empty cells when fewer than a fraction τ of their occupied Moore neighbors share their type—individually mild rules can yield strong spatial sorting.

Key equations

Let n_same and n_diff count occupied 8-neighbors of each type. An agent is happy if n_same + n_diff = 0 or n_same / (n_same + n_diff) ≥ τ. One move picks a random unhappy agent (if any) and exchanges it with a random empty site; asynchronous swaps approximate the original relocation story.

Frequently asked questions

Why swaps with random empty cells instead of “best” locations?: Schelling’s story is about satisficing under friction: agents leave bad spots and take any vacancy they can find. Random swaps keep the rule set small while still producing strong sorting; more elaborate search rules change kinetics but not the qualitative lesson.
Why does τ ≈ 0.35–0.45 already look “extreme” with ~10% vacancies?: With two equal groups and random mixing, a typical interior agent sees about half of its occupied neighbors as “same” on average. Raising τ only a little above that baseline makes many agents unhappy simultaneously; repeated swaps amplify correlations into large clusters even without explicit attraction to identical types.
Does the torus change the story compared to an open boundary?: Wrap-around removes edge effects so interior-like statistics persist everywhere on a small grid. Open boundaries can pin clusters against walls; the torus keeps the focus on spontaneous pattern formation from local rules.

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Schelling Segregation Model

About this model

How it works

Key equations

Frequently asked questions

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Schelling Segregation Model

About this model

How it works

Key equations

Frequently asked questions