App for running Schelling’s model

Schelling’s Segregation Model, I

In 1971, the economist Thomas Schelling (Nobel Prize 2005) formulated a simple model to help understand patterns of racial segregation. It can be thought of in terms of a game. There are two kinds of players — A and B, represented by light and dark blue in the case of the image on the left. The players are distributed on a grid, with certain percentage of the squares vacant. The players can be distributed in any way whatsoever, but we have have placed them at random. Below we describe their rules of play and look at some of the patterns that emerge from it.

Rules of the game

Each player has an emotional state, “satisfied” or “unsatisfied.” That state is determined by whether he feels surrounded by sufficiently many people like himself. To make this notion precise, consider a certain player, X. Unless he is on the edge of the grid or in a corner, he has eight neighbors — two to the North and South, two to the East and West, and also four at the adjacent corners. Player X also has a threshold. This is the minimum number of neighbors like him, A or B, that will make him feel satisfied. Thus X computes his emotional state as a function of just two things: the identity of his immediate neighbors and his threshold.

Playing the Game

Schelling ran the model he created using graph paper and coins — dimes and pennies. It took many months to play, though he had sketched out the initial idea on a plane ride writing X’s and O’s with pencil and paper. Nowadays, of course, we use computers, e.g., the web app pictured below. There are many implementations of the game, some given in the references at the end.

Experiments

We are going to run four experiments using the Schelling model, each with a different value of the threshold, running from low to high. The goal is to see what kind of macro patterns emerge, if any, from the micro behavior of the agents. At the end we will draw a few conclusions.

Schelling, initial state

Experiment: threshold 2

We set the threshold to 2, meaning that a player will be satisfied if he has at least two neighbors out of eight who are like him. Below is the result of running the game through 14 rounds of play After the 7th round, there was no change in the configuration. The system had reached a steady state, with a satisfaction level of 100% and not very much change from the initial configuration.

Schelling, threshold 2, round 14

Experiment: threshold 3

We set the threshold to 3, so that a player is satisfied if he has at least three out of eight neighbors who are like him. After 22 rounds of play, the grid looks like the one below. Satisfaction is 100%. This time there is a pronounced grouping into homogeneous neighborhoods.

Schelling, threshold 3, round 22

Experiment: threshold 4

We set the the threshold to 4, so that a player is satisfied if he has at least four neighbors who are like him. After 23 rounds of play, the grid looks like the one below. Large, homogeneous neighborhoods have formed. The satisfaction level is 96.2%. Further play results in small changes, but with the same general characteristics.

Schelling, threshold 4, round 24

Experiment: threshold 5

Let us now set the threshold to 5 This means that a player is satisfied if he has at least five out of eight neighbors who are like him — a majority. After 106 rounds of play, the grid looks like the one below. The population has almost completely separated into two homogeneous regions. The satisfaction level is 94.2%. Further play results in small changes, but with the same general characteristics — and a lot of churn in the border region.

Schelling-5–106

Conclusions

Very modest thresholds expressing the desire to have some neighbors like oneself. At threshold 2 there is very little change from a random configuration, but at threshold 3 there is already a quite marked grouping into homogeneous neighborhoods. At threshold 5, the first case where agents prefer a majority of neighbors like themselves, there is essentially completely segregation.

Appendix

Variants and descendants of the Schelling model have been used in other fields, including biology and generative art. The image below is an example of the latter.

See reference 8

References

  1. Thomas Schelling (Wikipedi
  2. Schelling’s original article
  3. Followup on Schelling’s article (2011)
  4. Ising, Schelling, and Self-Organizing Segregation (connections with the Ising model in physics)
  5. A physical analogue of the Schelling model
  6. Schelling’s Model of Segregation
  7. The Schelling Model of Residential Dynamics
  8. Parable of the Polygons
  9. Seeing Around Corners, Jonathan Rauch, Atlantic Magazine 2002
  10. Schelling model, generative art, and biology
  11. Elm-lang.org
  12. Code on GitHub
  13. Schelling Model App

Related Random References

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