# Honors project idea for linear algebra

At my institution there is an Honors program which encourages (or requires) students to petition classes for Honors. Basically, what this amounts to is the student has to write a 10 page (not set in stone, this is just ballpark) paper on some topic loosely associated to the course. I'm teaching linear algebra this semester and I have an Honors student making a request to do such a project. Now, my default response is to ask them to do some project on tensors and multilinear algebra as I know a little about all that. However, this student also has interest in Game Theory and was interested in a project about Markov chains etc. The problem for me, I don't have a good sense of what would make a nice project in the direction of Game Theory. So finally, the question:

Question: what would a good book, pdf, video series etc. be to provide a framework for an expository piece on Game Theory, or related topics, for someone who is currently learning linear algebra ?

I'm interested in book suggestions, but, also templates for study if you have some passion for this topic. At this point, the student has already studied Gaussian elimination, matrix algebra and after next week, linear transformations from $\mathbb{R}^n$ to $\mathbb{R}^m$. The next third of the course is basic vector space theory from subspaces to a bit quotient spaces. Then the final third of the semester we study eigenvectors, orthogonality, spectral theorem and a somewhat non-rigorous treatment of the real Jordan form.

• Evolutionary games might be an interesting and manageable topic for 10-page paper, see here and here. An excellent introductory book on GT is "A primer in Game Theory" by R. Gibbons – Rusan Kax Jan 31 '15 at 19:10
• @RusanKax You might want to consider making your comment a answer if you want to elaborate just a tad more. – Chris C Jan 31 '15 at 20:28
• As he done not have much experience with Linear Algebra, may I suggest that they first consider investigating game theory networks through Netlogo. Netlogo is a platform designed specifically for this sort of thing, and supports investigating both linear and non-linear game theory networks (among other things). I understand that this may take it away from your specialisation, but I found it much easier to understand Game Theory and Markov chains using Netlogo, and then as my research continued, the Linear Algebra perspective became understandable. – Richard Jan 31 '15 at 23:05
• As a project suggestion, it is fascinating to study network evolution of agents playing games such as Prisoners Dilemma using competing strategies. There are many simple games that can be modelled, and rules for how agents breed/die or adjust their strategies. There is also lots of literature on Hawk/Dove games and more complex mixes of strategies. I'm not sure to what extent these can be modelled with linear systems, as I believe the standard approach is the more flexible Agent Based Modelling. – Richard Jan 31 '15 at 23:12
• @Richard thanks for your comments and answer. Certainly they may be of use to my student. – James S. Cook Feb 4 '15 at 6:26

It would make a nice project to use payoff matrices to describe the Prisoner's Dilemma, and to use that to explore the Nash equilibrium. The Kahn Academy's 10-minute video is a place to start:

• thanks for pointing me towards this. I do think this is close to the right idea for a project. However, I'm not convinced it's linear algebra just yet... maybe it is not reasonable to demand both game theory and linear algebraic content? Or maybe I should just focus the project on Markov chains? I mean, these Pay-off matrices and the calculations, it seems the matrices are just an organizational device. Do I misread it, is there a nice matrix algebraic structure here I miss? – James S. Cook Feb 4 '15 at 6:33
• You are correct in that at the start, matrices are just organizational devices. But there is more L.A. content if one goes deeper. E.g., Li, Natara. "Some Matrix Techniques in Game Theory" PDF download – Joseph O'Rourke Feb 4 '15 at 12:39

I have looked at a few basic courses which use video lecture series, and you might be able to pick something useful out of them. I don't know what level you are looking at, so they may be too basic for your requirements.

Stanford offers a video lecture series on Game Theory (https://class.coursera.org/gametheory-004/lecture you will need to register for access) though as an intro course I don't think it covered Markov chains. They also have an advanced computational course which also uses video lectures which is not offered on Coursera and which was advertised to people who had done the first course. I didn't do that second course so I'm unsure of the syllabus.

University of Tokyo also offers Game Theory on Coursera, but I have not looked at it.

Other video lecture series that I have found useful is Stanford's Social Network Analysis, also through Coursera. I can't remember if that made links to Game Theory, or whether I got those linked from independent study after the course. Either way, SNA introduces you to NetLogo which makes it easy to investigate Game Theory concepts in networks and Markov processes.

Looking through networked game theory simulations on NetLogo has also been extremely useful. For some investigations, finding the endpoint of the network's evolution is isomorphic to finding the dominant eigenvector, though the software just as easily models non-linear game theory processes.

Model Thinking, also through Coursera, gives very brief overviews of many topics, including Markov processes and Game theory, though I can't remember if they link the two concepts.

All of the above are video lecture series which require minimal prerequisites. There are other courses which I have not looked at which may be helpful to you, and there may be something on the EdX platform, but most of the courses that I have done have been on Coursera, so that limits the range of my recommendations.

• Perhaps I'll have to steer him to Markov chains, for lack of a new idea, maybe that is the right idea... – James S. Cook Feb 4 '15 at 6:36
• @JamesS.Cook in case you are interested, I've just watched a videos on using Markov chains for sampling in a Monte Carlo algorithm, and the Metropolis algorithm is analysed using Linear Algebra, proving correctness by using the eigenvector to calculate the final state, and secondary eigenvalues to calculate the rate of convergence. It is the first week of the Statistical Mechanics course on Coursera. Not Game Theory, but I found it fascinating. – Richard Feb 9 '15 at 22:57

James,

If you have the book, you could have your student read in David Poole's Linear Algebra, chapter 2.4, there is a section on Finite Linear Games as related to systems of equations. There is also a section on Markov Chains.

I could send you a pdf of that (those?) section(s) if you are interested.

Zach

• I'll wait to see what my student decides, but, I might take you up on it, I'm not sure if I have Poole sitting around here or not... thanks for the suggestion. – James S. Cook Feb 9 '15 at 21:32