How mild individual preferences for similar neighbors can lead to starkly segregated collective outcomes.
Schelling (1971): in a "checkerboard city" with two groups, each agent wants a minimum fraction of similar neighbors. Not a majority — just a minimum.
If unmet, the agent moves to an empty cell. No hostility, no coordination, no zoning. The model shows how reasonable individual choices can aggregate into unexpected collective outcomes.
On a 2,500-cell grid with two colors and 10% empty, each agent evaluates its 8 neighbors. If the fraction of similar neighbors falls below tolerance , the agent moves to a random empty cell.
Figure 1. Emergent segregation. Even with a tolerance threshold as low as 33%, the individual preference for similarity leads to the rapid formation of homogeneous clusters.
At 50% tolerance — happy if half of neighbors match — the grid still divides into segregated blocks.
The relationship is nonlinear: 20% gives mild clustering, 33% visible segregation, 50%+ near-total separation.
Figure 2. Measuring the transition. The segregation index tracks the average homogeneity of each agent's local neighborhood. A sharp rise in this metric signifies the point where random distribution collapses into a sorted state.
Schelling's model applies wherever agents sort on any visible trait: language, income, or even seating at a cafeteria. The math does not care what the trait is.