How the Model Works: Rules, Grid, and Dynamics

Thomas Schelling’s 1971 paper described a model that could be run with coins on a checkerboard. The mechanics are simple enough to execute by hand, which is how Schelling originally did it. The precision of the specification matters, because every parameter choice affects the outcome, and the gap between local rules and global behavior is where the model’s scientific content lives.

Agent Types and the Satisfaction Condition

The model places two types of agents on a grid. In Schelling’s original formulation, these were pennies and dimes on a checkerboard. In the standard computational version, they are colored cells on a two-dimensional lattice.

A fraction of cells --- typically 20 to 30 percent --- are left empty. The empty cells are not background; they are the resource that makes the dynamics possible. Without vacant cells, no agent can relocate, and the system is frozen at its initial condition regardless of how dissatisfied agents might be. The vacancy rate is a structural parameter, not a detail.

Each agent evaluates a single quantity: the fraction of its occupied neighboring cells that share its type. The agent has a tolerance threshold f. If the same-type fraction meets or exceeds f, the agent is satisfied and remains in place. If the fraction falls below f, the agent is dissatisfied.

The original Schelling threshold was approximately 1/3 --- meaning an agent is content even when outnumbered 2:1 by agents of the other type. This is far below what most people would consider an exclusionary preference. An agent with f = 1/3 would be satisfied in a neighborhood that is 67 percent different-type. The label “mild preference” is precise, not rhetorical.

Setting f = 0 means agents have no type preference at all; the system does not sort. Setting f = 1 means agents demand complete homogeneity; the system sorts immediately and completely. The interesting behavior occurs at intermediate values, particularly the range 0.25 to 0.50, where the mismatch between individual preference and collective outcome is largest.

The Update Rule

At each step, one dissatisfied agent is selected. That agent moves to a randomly chosen empty cell. After the move, the grid configuration updates: the agent’s old cell becomes empty, the destination cell becomes occupied, and the neighborhoods of all agents adjacent to both the old and new positions change.

This is asynchronous, random-sequential update. One agent moves per step, and the consequences of that move propagate immediately to the neighborhoods of adjacent agents, potentially changing their satisfaction status before the next step.

The alternative --- synchronous update, where all dissatisfied agents move simultaneously --- produces qualitatively similar long-run segregation but different transient dynamics. In synchronous update, multiple agents may target the same empty cell, requiring a tie-breaking rule. The segregation outcome is robust to the choice of update protocol, which is itself an informative property: the macro behavior is determined by the feedback structure, not by the scheduling details.

The simulation terminates when one of two conditions is met: either all agents are satisfied (a stable equilibrium) or no remaining dissatisfied agent can find an acceptable empty cell (a frustrated equilibrium). In most parameter regimes with sufficient vacancy, the system reaches a stable equilibrium in which every agent meets its threshold.

Grid Structures and Neighborhood Definitions

The standard grid is a two-dimensional square lattice with the Moore neighborhood: each interior cell has eight neighbors (horizontal, vertical, and diagonal adjacents). Edge cells have five neighbors; corner cells have three.

The von Neumann neighborhood --- four neighbors, horizontal and vertical only --- reduces the interaction range and generally produces less segregation at the same threshold, because each agent has fewer neighbors to evaluate and fewer opportunities for boundary erosion.

Wrapping the grid as a torus (connecting the top edge to the bottom and the left edge to the right) eliminates edge effects and ensures every cell has exactly eight neighbors. Most simulation studies use toroidal boundaries because they remove the confounding effect of boundary geometry on cluster formation.

The model has been extended beyond regular grids. On random Erdos-Renyi graphs, the segregation outcome is similar to the grid case when the mean degree is comparable (Fagiolo, Valente, and Vriend, 2007). On scale-free networks, segregation is amplified: high-degree nodes (hubs) influence many neighbors and act as anchors around which homogeneous clusters crystallize. On empirically calibrated city street networks, the results depend on the specific network structure, but the qualitative finding --- mild preferences producing strong segregation --- is robust across topologies.

Measuring Segregation

The claim “mild preferences produce strong segregation” requires quantitative measurement of segregation. Several standard indices are used.

The dissimilarity index (Duncan and Duncan, 1955) measures how evenly two groups are distributed across neighborhoods. It ranges from 0 (perfectly integrated) to 1 (perfectly segregated) and indicates the fraction of one group that would need to relocate for perfect integration. U.S. metropolitan areas typically score between 0.40 and 0.70 on this index.

The isolation index measures the probability that a randomly selected neighbor of a type-A agent is also type-A. At f = 1/3 on a standard grid, simulations consistently produce isolation indices of 0.70 to 0.90 --- meaning that despite each agent being willing to tolerate a 2:1 disadvantage, the average agent ends up surrounded by 70 to 90 percent same-type neighbors.

Spatial autocorrelation measures the degree to which same-type agents cluster spatially. Moran’s I and Geary’s C are standard measures. High spatial autocorrelation confirms that the clustering is genuine spatial sorting rather than random fluctuation.

The core puzzle the model raises is visible in these numbers. An agent with f = 1/3 would be satisfied at an isolation index of 0.33. The system equilibrium consistently produces an isolation index above 0.70. The collective outcome overshoots the individual preference by a factor of two or more. This amplification is not a bug in the simulation --- it is the model’s central result, and understanding its mechanism is the subject of the tipping points page.

Further Reading