2. Core Maths: teaching and learning resource

3. Using Game Theory to Investigate Segregation in Society

4. 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

5. 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.

6. 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.

7. Parable of the Polygons Use the link http://ncase.me/polygons and work through the site.http://ncase.me/polygons Using a series of apps, you will start to build an understanding of how certain parameters may result in segregation.

8. 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 assumptions made in constructing the different apps. How might these assumptions limit any conclusions made?

9. 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?

10. 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.

11. Percentage of alike neighbours Explain your answer. Are all percentages possible?

12. 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.

13. Percentage of alike neighbours As you did in the previous example, work out all the possible values for the percentage of alike neighbours.

14. Parable of the Polygons Now try a square along one of the edges of the large grid, for example (B).

15. 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).

16. 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?

17. Percentage of alike neighbours The screenshot below suggest that there is 56% segregation:

18. Percentage of alike neighbours How is this figure calculated? Is this a good way to measure segregation?

19. 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.

20. Percentage of alike neighbours Is there a maximum level you can set for this figure beyond which a solution becomes impossible? Explain why.

21. 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?

22. Wrapping Up

23. Cartoon by Vi Hart

25. 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

26. 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.A Beautiful Mind Teacher notes: Parable of the Polygons

27. 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.

28. % (to 1%)No. of alike neighbours 12345678 No. of neighbours 1100 250100 33367100 4255075100 520406080100 61733506783100 7142943577186100 813253850637588100

29. 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.

30. 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.

31. Acknowledgements http://ncase.me/polygons https://www.idlethumbs.net/forums/topic/9815- parable-of-the-polygonshttps://www.idlethumbs.net/forums/topic/9815- parable-of-the-polygons https://gbark.wordpress.com/2014/12/12/reblog- parable-of-the-polygonshttps://gbark.wordpress.com/2014/12/12/reblog- parable-of-the-polygons https://twitter.com/vihartvihart Accessed 11/03/2015