# Parable of the Polygons

```Core Maths:
teaching and
learning
resource
Using Game Theory to
Investigate Segregation in
Society
Parable of the Polygons
“Parable of the Polygons” is a website which uses
apps to explore how areas of cities can become
segregated.
Parable of the Polygons
• The site uses a mathematical model to show
that small individual biases can lead to a large
collective bias.
• This activity encourages you to think about the
mathematical model underlying the apps.
Parable of the Polygons
• The starting assumption is that there are two
types of individuals (squares and triangles).
• Each prefers being in a mixed neighbourhood
but is unhappy if too many neighbours are
different.
Parable of the Polygons
• Use the link http://ncase.me/polygons and work
through the site.
• Using a series of apps, you will start to build an
understanding of how certain parameters may
result in segregation.
The underlying message
• Take some time to experience the model in
action and ponder the underlying message from
the site.
• Give some thought to the
the different apps.
How might these
assumptions limit
any conclusions
Percentage of alike neighbours
• By filling the surrounding
spaces with either
squares, triangles, or
empty spaces, we arrive at
the percentage of alike
neighbours.
How does the
app measure
the percentage
of alike
neighbours?
Percentage of alike neighbours
• For example consider a square near the centre
of the large grid, see the diagram below.
• By filling the surrounding spaces with either
squares, triangles, or empty spaces, we arrive at
the percentage of alike neighbours.
Percentage of alike neighbours
Are all
percentages
possible?
Percentage of alike neighbours
• Consider how the possible values for the
percentage of alike neighbours may change
depending on the shape’s position.
• You could start by examining a square in one of
the corners of the large grid, for example (A) in
the diagram below.
Percentage of alike neighbours
• As you did in the previous example, work out all
the possible values for the percentage of alike
neighbours.
Parable of the Polygons
• Now try a square along one of the edges of the
large grid, for example (B).
Percentage of alike neighbours
• Compare the possible percentages you have
just obtained for (A) and (B), to the percentages
you previously obtained for a square in the
central section (C).
Percentage of alike neighbours
• Assume the simulation
was run many times, with
the same setting.
• Which of the three
squares considered
above (A, B, or C) ,
would you expect to
move more frequently on
average?
Percentage of alike neighbours
• The screenshot below suggest that there is 56%
segregation:
Percentage of alike neighbours
• How is this figure calculated?
• Is this a good way to measure segregation?
Percentage of alike neighbours
• The initial problem starts with both shapes
having the same requirements, that is, that at
least 1/3 of their neighbours need to be like
them for them to be happy.
• On the second app, shown in the next screen
shot, you are allowed to change this figure of
1/3 ≈ 33% to other values to investigate its
effect on the level of segregation.
Percentage of alike neighbours
• Is there a maximum level you can set for this
figure beyond which a solution becomes
impossible? Explain why.
Percentage of alike neighbours
• How about if the shapes
showed different levels of
preference?
• For example would it be
possible for a solution where
squares required 75% to be
alike, whilst triangles would be
happy with 1/3?
What constraint
does this place
on the settings?
Wrapping Up
Cartoon by Vi Hart
Teacher notes: Parable of the Polygons
Before using this resource with your students you
will want to check to ensure that you are
comfortable with them viewing the content on the
website and the underlying messages expressed
at the end, under ‘Wrapping up’.
You may find the apps work better in some
browsers than others – they work in Google
Chrome.
Teacher notes: Parable of the Polygons
Game theory is a branch of applied mathematics
which is used in economics, sociology, psychology
and biology. It deals with mathematical modelling
of strategy and behaviour in situations where the
decisions which individuals make are affected by
what others might do.
The 2001 film A Beautiful Mind was based on the
life of John Nash who won the Nobel prize for
economics for his work on game theory.
Mathematical Detail
Slides 10-15
Are all percentages possible?
No. Looking at a position in the middle of the
‘town’ each shape has 8 neighbours. Therefore
there can be 0, 1, 2, 3, 4, 5, 6, 7 or 8 neighbours,
of whom various numbers can be alike. The table
on the following slide gives the possibilities and the
percentages.
% (to 1%)
No. of alike neighbours
1
2
3
4
5
6
7
No. of neighbours
1
100
2
50
100
3
33
67
100
4
25
50
75
100
5
20
40
60
80
100
6
17
33
50
67
83
100
7
14
29
43
57
71
86
100
8
13
25
38
50
63
75
88
8
100
Mathematical Detail
Slides 10-15
Are all percentages possible?
On an edge there are up to 5 neighbours.
In a corner there are up to 3 neighbours.
Mathematical Detail
Slide 18
How is segregation calculated by this App?
It looks as if it is the average of the ‘like
neighbours’ for all of the shapes.
The advantage of this is that ‘partial segregation’ is
included.
An alternative would be to look at how many
shapes had no like neighbours at all. In this case
there are 126 out of 322, which is 39% of shapes
completely segregated.
Acknowledgements
• http://ncase.me/polygons
• https://www.idlethumbs.net/forums/topic/9815parable-of-the-polygons
• https://gbark.wordpress.com/2014/12/12/reblogparable-of-the-polygons