Maths HL IA- Markov Chains

[Rahul Daswani]

Hey All,

 

I'm struggling a bit with my IA topic. So I was stuck for quite a whole and somebody directed me along the route of Markov Chains. For those of you who don't know what that is, I found a really good explanation video here: https://www.youtube.com/watch?v=uvYTGEZQTEs (I just learnt it myself). This was recommended was Matrices were just taken out of the core syllabus (so it is one of those better topics to achieve a high grade in the actual Maths part) and the theory itself is not that hard to learn. 

 

Anyways, I found this whole idea quite interesting and I wanted to apply it to board games, more specifically, Monopoly. So I have been doing a ton of research recently and I've come to the conclusion that this is going to be way to complicated. Basically, from what i've found out, the simplest way to examine the probabilities of board spaces requires at least 40 different variables in the Matrix. In many of the papers I've read about this application to Monopoly, the writers have used computer programming and coding to find the results they did. So I was wondering, would there be any way to make this exploration simpler, such that an average (I'm no prodigy) Maths HL student can do it. I'm quite interested in this whole topic, but I could change it if it is necessary. 

 

I'm a little bit stressed because I'm leaving on holiday in three days, and I would have liked to sort out my topic and all my materials by then so I could write it up on the plane and any spare time I have, as I may not have wifi during my trip. I still have a bit over three weeks of holidays left, but I really need to sort it out soon. Any suggestions?

 

Thanks

[ctrls]

I haven't studied Markov chains at all, so I don't really know what kind of calculations and working you'll have to do. That said, it should be possible to change around the topic a bit to be feasible, since the main issue seems to be having too many variables. In particular, two things come to mind:

 

Firstly, you could alter the game to make it simpler and reduce the number of variables. You don't really loose much for example without having all 40 states, rather this could be reduced to something a lot smaller - you could later consider how larger boards would change your results. Admittedly it may seem a bit artificial with only 5 or so states for example, but at the very least it may be a good starting to point to get a feel for what's going on.

 

Secondly, there are resources that can help with the raw computation. I don't know what exactly you'd need to calculate, but there are online calculators which can compute things like determinants, inverses, eigenvalues, solve systems of equations, etc. Even with the above simplifications you'll probably still have to work with matrices that are fairly large, if you were to do them by hand. A quick google search for example gave me this, which can do up to 32x32 matrices.

Edited July 22, 2015 by ctrls

[Armin]

Hey Rahul,

I have recently started to think about my IA and I had the same idea as yours too with the exception that I know computer programming and so I can write an algorithm to simulate games and compare that to the theoretical results I would have calculated with the Markov chains. Have you been able to calculate it? If so could you give me any pointers/tips?

Thanks

[kw0573]

May this be of some inspiration to you. Note that since the math principles are same for 4 squares or 40 squares or 40000 squares, it may be better to work with Markhov chains of a smaller size.