A Coin and Paper Model of Segregation

For our recent talk at Data Science London, we wanted to introduce the audience to Agent-Based Modelling through the work of Nobel Prize-winning economist (and inspiration for Stanley Kubrik’s ‘Dr Strangelove’) Thomas Schelling.

Schelling was interested in how individual preferences and behaviours lead to interesting and sometimes surprising aggregate results. Or, to draw on the title of his 1978 book on the topic, how micromotives lead to macrobehaviour.

One of the phenomena Schelling looked at in his book was the emergence of segregation in populations – particularly racial segregation – from the discriminatory behaviour of individuals. What does it take in terms of the kind and degree of preference for being near to people of one’s own ‘type’ for segregation to take place?

Incidentally, if you thought that urban racial segregation was a thing of the 70s, check out Eric Fischer‘s dot density maps of US cities. For any fans of The Wire out there, here’s Baltimore (Red is White, Blue is Black, Green is Asian, Orange is Hispanic, Yellow is Other – and each dot represents 25 residents):

Baltimore Map

Eric Fischer’s Dot Density Map of Baltimore

Schelling explored segregation by building a simple ‘self-forming neighbourhood model’, using graph paper and coins. Here’s how he introduces the idea in his book:

“Some vivid dynamics can be generated by any reader with a half-hour to spare, a roll of pennies and a roll of dimes, a tabletop, a large sheet of paper, a spirit of scientific inquiry, or, lacking that spirit, a fondness for games.”

Get a roll of pennies, a roll of dimes, a ruled sheet of paper divided into one-inch squares, preferably at least the size of a checkerboard (sixty-four squares in eight rows and eight columns) and find some device for selecting squares at random.”

The dimes and pennies – or 10p pieces and 2p pieces, or whatever different-coloured coins of similar sizes you can find in your local currency – are used to represent the two groups in the neighbourhood you’re modelling.

As Schelling says, there are lots of different choices to be made in setting up the board in terms of the distribution of the coins (random vs ordered, equal numbers vs majority and minority). There are also choices to be made in terms of the rules that govern the individual behaviours. Individuals can have different degrees of preference to be amongst neighbours of their own kind. They can have different degrees of preference not to be amongst neighbours who are not their kind. And on and on.

Schelling’s set-up – an environment (board), agent population (coins) with attributes (dimes or pennies), and rules governing behaviour – is instantly recognisable as an agent-based model. Once a board size has been selected, a starting population has been distributed, and rules chosen, the simulation can be ‘run’ to see what macro-effects emerge.

It’s easy enough to set up an equivalent simulation on a computer today, and it would have been easy enough to do it in 1978 with the help of a brand new Apple II personal computer, but Schelling advised against it:

“I cannot too strongly urge you to get the dimes and pennies and do it yourself … there is nothing like tracing it through for yourself and seeing the thing work itself out.”

With Schelling’s words ringing in our ears, we took up our 10ps, 2ps and checkerboard, and set about re-enacting a key scene in the history of agent-based modelling.

We started with 18 2p coins and 18 10p coins, distributed at random over an 8×8 grid. For the purposes of determining neighbours, we decided that the grid wrapped both horizontally and vertically.

Individuals of both types, we decided, followed the same simple behavioural rule:

  • If more than 55% of my neighbours are of the same type as me, stay, otherwise move to a randomly selected empty square.

In other words, each type of coin has a preference to have a slight majority of its neighbours of the same type (spaces don’t count). So, a 2p surrounded by 4 2ps, 3 10ps and a space (57% similar) just meets the criterion to stay where it is, whereas if it were surrounded by 4 2ps and 4 10ps (50%), it would move.

We set the board up on the floor, mounted a camera above it, and got to work, going over the board and moving or leaving coins in place according to the rules. From time to time, between moves, we took a still of progress. Here’s the result.

We fairly quickly get to a stable configuration where no coin wants to move any longer, and where, as we see, the populations have become segregated.

Whilst it is perhaps obvious that where preferences to be amongst people of similar type are strong, segregation emerges quickly in communities. But, using this simple modelling approach, Schelling was able to show that even a moderate preference for being amongst one’s own kind can in time lead to segregation, or disrupt pre-existing patterns of integration.

Schelling also saw that, sometimes, the emergence of segregated patterns depends on chain reactions started by the movements of just a few individuals. In such situations, as he puts it, “if we could persuade them to stay, everybody else would be all right.”

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