# An N-Race Schelling Segregation Model

21 Jan 2016

It is a common theme in non-linear modeling that small perturbations in initial conditions can result in massive deviations in the outcome of a simulation. In 1969, Nobel laureate Thomas Schelling explored how even small biases can have large sociological effects. In his paper "Models of segregation", Schelling described how a model in which a preference that one's neighbor's be of a specific mixture can lead to total segregation regardless of intent.

Figure 1: A 500x500 Schelling segregation model with 3 races. This simulation was conducted with a maximum minority threshold of 0.2 for 1000 ticks.

Using this concept as a starting point, I developed a Java-based application to simulate a Schelling segregation model for an arbitrary number of races (n>=1).

Schelling models are relatively simplistic. The world consists of a lattice of agents living on a torus. Each agent has a race. Every tick, we evaluate if the proportion of neighbors for that agent with a different race is above that agent's minority threshold. If the proportion is greater than the threshold, then the agent is considered unhappy. After each agent is checked, the unhappy agents are shuffled to new locations on the lattice.

The most important parameter for this type of simulation is the minority threshold. If the threshold is too high, then segregation occurs in small noisy clusters. Alternatively, if the threshold is very low, then segregation will form in large continuous bands. An example of the difference the threshold makes can be seen in Figure 2, below. On the left is a simulation with a minority threshold of 0.2, versus the simulation on the right with a threshold of 0.5. Notice that the patches of segregation on the left are larger than the patches on the right. Additionally, notice that the simulation on the left contains more noise around the segregated clusters than the simulation on the right.

Figure 2: A comparison of two simulations of a Schelling segregation model thresholds of 0.2 and 0.5 respectively. Each simulation was performed in a 500x500 torus world for 200 steps. The simulation on the left involves agents with a minority threshold of 0.2. This means that they will be unhappy and move if there their neighborhood is 20% minority or more. The simulation on the right involves agents with a minority threshold of 0.5.

One interesting implication with the Schelling model of segregation is that improving race relations does not fix segregation. Once a population has become segregated and all of the agents are content, then they have no reason to move even if they become more tolerant. This could be used to explain why it has been so difficult for historically segregated areas to desegregate. According to Schelling's model, the two ways for an area to desegregate are:

1. A disturbance in the area that displaces otherwise content people.
2. A system in which people are discontent living in an area without diversity.

While there are many external factors that drive people to move, the Schelling model suggests that embracing and actively pursuing diversity can go a long way to dissolve segregated communities.