Schelling’s Segregation

Model

By Ron Aldad

In life, people move to places that they would rather live. They usually pick

places where people are like them, in terms of their race, economic status, and/or

religion. Thomas C. Schelling, a famous philosopher and professor, calls this his

Segregation Model. This model explains why this phenomenon occurs and how this

could be mathematically proven.

Thomas C. Schelling was born on April 14, 1921, in Oakland California. He

graduated from the University of California with a degree in economics in 1944. Mr.

Schelling later went on to Harvard University to obtain his Ph.D., and completed his

exams in June of 1948. In November of 1950, he joined the White House Staff of

foreign policy advisor to President Harry S. Truman. Later, in the fall of 1953, Mr.

Schelling joined the faculty of Yale University. At Yale, Thomas Schelling started

publishing his works including, an “Essay on Bargaining” in 1956 in the American

Economic Review, and “Bargaining, Communication, and Limited War” in the

inaugural issue of the Journal of Conflict Resolution, 1957. These works were made

before Mr. Schelling read the book, Game and Decisions by Howard Raiffa and R.

Duncan Luce, which inspired Mr. Schelling to continue to work with his game

theories. After spending hundreds of hours trying to understand this book, Thomas

Schelling took his family to London in the summer of 1958 to make a manuscript of

his game theory. He then submitted it to the Journal of Conflict Resolution. He called

his article, “Prospectus for a Reorientation of Game Theory”. He spent thirty-one

years at Harvard University; first in the Department of Economics and the Center for

International Affairs and, in 1969, in the John F. Kennedy School of Government.

Subsequently, Mr. Schelling continued his travels around the world and went to

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many places to learn more and find out more facts to prove his theories. Most of his

works and theories were described in the Harvard University Press in 1960 as “The

Strategy of Conflict”. One of these theories is Mr. Shelling’s segregation model.

Thomas C. Schelling’s segregation model is a model of life; it is an agent-

based model. This is a model that has three variables: the agents, or the people in

this case, their behaviors, and the results. It shows how humans, based on their likes

and dislikes of their neighbors, choose to move to areas with similar neighbors. He

shows this movement based on preferences through the use of mathematical

models. Mr. Schelling started by first making a checkerboard, with each square in

the checkerboard representing a home. The checkerboard represents a

neighborhood while each person has to decide whether or not to stay or to move

based on who is his or her neighbor. The criteria the agent or person uses to move

or not is based on a percentage of how much each person surroundings is similar to

his own. As this preference percentage increases the segregation becomes more

intense in the neighborhood, but as the preference percentage decreases, there is

less segregation. Current models use sophisticated computer programs to calculate

the movement of large populations such as cities based on neighbor preference. The

models use statistical models to demonstrate how people move to preferred

locations.

This model uses many graphs, pictures, and charts and is a very visual model.

An example of this model is shown in figure 1. The people are represented with plus

signs and zeros and the dots on top of them show if they are not satisfied with their

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neighbors. Schelling devised on a set of rules for this model such that the way a zero

or plus signs is able to move only if it is dissatisfied, and the order of which the plus

signs or zero is allowed to move is based on its order from left to right. For example,

in figure 1, the first plus sign from left to right would move first because it is the first

dissatisfied plus sign or zero in the model. The second plus sign would move next

for the same reason. They will move to the nearest spot on the model where they

are going to be satisfied. In this case, the people want 50% of his or her neighbors to

be like them and one’s neighbor is justified as the four people to the left and right of

him or her. As I said before, the first person to move would be the first plus sign.

This person would move to go in between the zero that was eighth from the left and

the plus sign that was ninth from the left. The next to move is the plus sign that was

fifth from the left. This person moves to the right of the peron that moved first. After

that, the zero that was tenth from the left will move further left, going over four plus

signs on the way. This zero moved to the left because if it moved to the right, it

would have to move a greater distance to achieve it’s satisfaction. After all of this

you would get a scenario like figure 2 where the plus signs and zeros are clustered

based on preference similarities and therefore all are satisfied. As you continue

doing this, you will get closer and closer to segregation and to figure 2.

If you were to add eight zeros and plus signs but keep the amounts of the

zeros and plus signs equal, the new people would have to accommodate to the

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addition. Most likely they would start out in a random spot where he or she is

dissatified; therefore, they would have to move. After they move they will look like

figure 2. This also shows that not only do the people become segregated, but they

also form groups of 8, 15,10,15,16, and 6 with an average of 12 members per group.

If you were to remove some of the zeros or stars form figure 1, you would get

similar results. If you take away, for example, 17 zeros out of the original in figure 1

you would get an uneven amount of plus signs and zeros. You would receive

something like figure 3. In this case, all of the zeros and three of the plus signs are

unhappy, if a neighbor is defined as the four people to the right and left of him or

her. If you follow the same rules as we used before, you will receive a segregated

group of zeros in the middle of the model, as shown in figure 4. In this figure, there

ends up to be three groups of 15,18, and 20 with and average of 18. This shows that

this always works and ends up segregated clusters.

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So far we have only looked at the linear model. To do this on a two-

dimensional model is a little harder. In a linear model, the plus signs or zeros just

moved to the left or right to make more room for the other plus signs or zeros. In a

two-dimensional model, there must be more rules because you can’t keep making

more room in a rectangular model. This being said, a new rule must be enforced.

First, you must make a rectangle with a fixed number of spaces, leaving some empty.

In this two-dimensional model, when a person moves, he always leaves a vacant

space behind and he can only move to a vacant space. Instead of making your own, it

is possible to use a checkerboard but just don’t pay attention to the colors on the

board. There must be a few empty squares on the board; therefore, to have a good

amount of vacant squares and to be accurate and fair, you

should have 25% to 30% of the board filled with empty squares. Figure 5 is a perfect

example of how this should look. It has thirteen rows, sixteen columns, and two

hundred eight squares. Each of the squares are filled in with number signs, filled in

with zeros, or not filled in at all. This is all done randomly.

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A person’s neighbor, in this case, is referring to a number sign’s or zero’s

eight squares around him. The way a star or zero moves in this model is by moving

to the nearest empty square that satisfies its needs. The word “nearest” is defined as

the closest square to the number sign or zero and is based on how few the squares

are in between the place the number sign or zero started at and where it ends.

If you use figure 5, there are 25 number signs and 18 zeros that start out

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dissatisfied. If you follow all of the rules stated before, move the dissatisfied number

signs and zeros to empty spaces where they are happy, and work from the upper left

hand side downward and to the right, you would receive something like figure 6, but

if you work from the center outwards, you get something like figure 7. They are both

segregated, but figure 6 has its number signs and zeros much more separated and

figure 7 has its stars and zeros much more fragmented. Due to the fact that it’s a

little hard to see the segregation in figure 7, figure 8 is the same thing just with

borderlines on it. This makes it easier to see the pattern of segregation in the model.

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If you think about it, this is quite unbelievable. This model allows you to see

how a neighborhood would be like and where everyone would live by just knowing

how many people living there, their tolerance level to the criteria you are testing,

and the category they belong to. These things are extraordinary. This model also

teaches us how even when our tolerance is high, we still segregate a lot. Even if you

wanted to have just thirty percent of the people living around you to be the same

race as you, lets say, there still would be major segregation, according to the model.

Lastly, this teaches how humans live and about sociology. People usually live where

they feel comfortable. Because of this, humans naturally segregate. Schelling’s model

also teaches us that whatever happens on the micro level does not always equal

what happens on the macro level. Lets say that a person wants 30 percent of his or

her neighbors to be like him or her. If you put this information into the model, you

will see that around 72 percent of the neighbors would end up like him or her. Even

though the person is pretty tolerant, there will still be segregation when looked at

on the macro level and when looking at things on a bigger scope. This model shows

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us all of these things and is a very important model because it helps us understand

sociology through the use of mathematics.

This work can relate to our lives today, it can show us how we pick and

choose where we want to live or be. Although this model does not include some

social phenomenon, this can still be used. If we are intolerant about the people we

live near or our friends, we wont have anywhere to live anywhere or we wont have

any friends. This model teaches us that we have to be tolerant in order to live a

relatively normal life. This is important and it is amazing that this can be proven

using a mathematical model. This model can still help us today and is an outstanding

presentation of what our lives might come to if people become intolerant of other

people.

This model can help us as people to see that we need to become more

tolerant. If one were to further extend the research, he or she would continue to find

more ways to prove this concept and would go into more complicated mathematics

and statistical modeling. This will show you that it is important to be tolerant of

other people because if we don’t, then none of us will be happy or satisfied and we

will just keep moving. Even if we think we are tolerant, that might just really be on

the micro level. On the macro level, we are most of the time intolerant and are

segregated by race, religion, and ways of life.

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Citations:

1) http://www.nobelprize.org/nobel_prizes/economics/laureates/2005/

schelling-autobio.html

2) http://www.youtube.com/watch?v=HWP3K2BLDiQ

3) http://statistics.berkeley.edu/~aldous/157/Papers/

Schelling_Seg_Models.pdf

4) http://meteo.lcd.lu/globalwarming/Schelling/

thomas_schelling_nobelprice_eco_2005.png

5) http://www.utc.edu/Research/ProbascoChair/images/Schelling.jpg

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