This week in Occupy Math, we proudly announce a book published by Dr. Andrew McEachern, GAME THEORY: A Classical Introduction, Mathematical Games, and the Tournament. Game theory is a formal structure for studying and resolving conflict and encouraging cooperation that rephrases cooperation and conflict as a game. Andrew developed and taught a course in game theory taken by advanced students from many programs while he was at Queens University. This book is a text based on the course he taught and it is part of an effort to bring textbook prices under control. The book introduces the classical analysis used in game theory — his exposition of The Lady or the Tiger is wonderful — but Andrew also introduces material outside of the standard game theory fare. These include the math behind the fraction teaching game that Dr. McEachern and Occupy Math are developing and techniques for designing fair, balanced tournaments for anything from Prisoner’s Dilemma to Basketball. The book is a text for a course for non-majors that nevertheless has a solid mathematical foundation. We now ask Dr. McEachern a few questions about his book.
Dr. McEachern, why did you write this book?
There are already excellent game theory textbooks out there, the ones written by Herbert Gintis come to mind, but they tend to focus on one or two areas specifically. When I was teaching the course I took inspiration from several sources like classical game theory, combinatorial game theory, mathematical biology, combinatorics, graph theory, and I’m sure there are other things I could add to the list if I thought longer. So I had course notes from all over the place and I wanted to put it all together in one place so that my students and I had a single place to go to learn about the variety of really interesting topics that are out there and under the game theory umbrella.
Can you give us an example of an interesting topic from the book?
One of my favorite topics in the book is the sections on the vaccination game, which I have never seen in a game theory textbook to date. The basic problem is, given a group of people, how many are likely to vaccinate under certain conditions? It’s a bit of a long story, mathematically speaking, but the short answer is that it mostly depends on how expensive people perceive getting the vaccine to be. I find this interesting since it has actual applications in medicine. If public health officials can control the perceived cost of a vaccine, then it can also have an impact on how many people will get that vaccine.
You wrote this book based on material you used to teach a class a couple times. What was your student’s reaction to the material?
It was varied, but mostly positive. Some students preferred the classical game theory and all of the mathematics that went with it, others preferred the combinatorial game theory, and others still preferred the section on tournaments. One of the things I was trying to do was to present a broad overview of game theory and all that goes with it, rather than drilling down into one specific area that might turn off 80% of the class.
Your book explains how to construct balanced tournaments, but you also used tournaments in class. Can you tell us about that?
I ran three tournaments in my class. The first was a novel tournament that involved the Iterated Prisoner’s Dilemma and elements of a Risk-style board. That tournament ran for about a month and the students who participated really enjoyed it and were also very mad at me for the number of hours they used up trying to figure out how to win. The second tournament used the Enhanced Ultimatum Game, detailed in the book, and the students enjoyed that one as well, although it took a great deal less time. The last tournament I ran was a Goofspiel tournament, which is a fascinating card game. This last tournament actually had an interesting ending. One of the teams competing actually bribed a majority of teams they faced with cookies to throw the tournament, allowing them to win. Some people might have been angry about that, but I actually encourage lateral thinking in general, and I never specified that bribery as a strategy was forbidden. In general most of the people who participated had a lot of fun, and I would certainly run those tournaments again, or something similar, the next time I teach the course.
Did using tournaments as part of the class increase your student’s engagement?
I wish I had a way to measure it, but if I could I would guess that my student engagement went up by an order of magnitude when the tournaments were added to the class. I had several students say afterwards that they understood a game or topic much better by having played it in the tournament.
You told me the book was incomplete, like a door into a new place that’s still growing. Can you give us a sense of what discoveries are still out there?
Game theory is such a diverse field that draws from so many other areas that having a complete book would be impossible. This book is an attempt to just show a little bit of all the areas I’ve seen that were interesting to me. As for what discoveries are still out there, I am going to try to keep this short. The last time I taught the course, which was the same time I was finishing this book, I opened the first class with a discussion about self-driving cars and an old thought experiment called the Trolley Problem. The problem boils down to making a decision that will cause either one or more deaths. It was initially used in ethics discussions, but this problem is no longer just in the purview of philosophers.
With self-driving cars right around the corner, there are going to be situations in which a programmer is going to have to give a car instructions about how to deal with a situation that has unavoidable injury or possibly death involved. If you were driving and suddenly a group of people stepped in front of you, would you just keep going or swerve into a wall to save their lives? When the car is doing the driving, the person that programs it has to make that decision when he writes the control program. Keep in mind that the car has much better reflexes than even an Olympic athlete so the total risk to everyone is lower. It is not easy to answer the life-and-death question, but I think game theory has some tools to at least help us approach the problem and to understand what decision we have made when we write the code driving our cars.
There has been a lot of work recently trying to combine psychology and game theory approaches to answer questions about how people will behave under certain situations. The question of altruism existing in what economists tell us should be a selfish world is one that has recently been attempted. Since psychology and neuroscience are opening doors that were once closed, this is an exciting time to bring some mathematical structure to these areas and apply some game theory to see if we can get some meaningful results. This dovetails a little with medicine, and in using game theory to figure out how we can get people to adopt a certain view about things like vaccines, or what to do during a viral outbreak.
Game theory also has a home in the many, many card games in the world, and games in general. Recently a machine learning system was created that can now beat the best Texas Hold’Em players in the world; Chess and Go have been similarly conquered. Using game theory techniques combined with machine learning has opened a vast area of research, of which we are still just scratching the surface. This might be my favorite area to explore and see what is happening out there.
Who is this book for?
This book is not intended for a rigorous mathematics student who wants proof in every statement. It is intended for those students with a decent mathematics background, basic calculus and statistics, that want to see a branch of mathematics in a very different context than what they are used to. There are a lot of examples that work mostly on intuition and logic, and the book is about thinking things through rather than sheets and sheets of calculations.
Thank you for taking the time to answer Occupy Math’s questions!
This book is the first one in a series published by Morgan and Claypool on Computational Intelligence in Games. Morgan and Claypool sell their books in large blocks (as e-books with unlimited use) to universities and other institutions. This means that if you assign Andrew’s book — and your institution has subscribed to Morgan and Claypool’s synthesis series — then there is unlimited use of the e-book without an additional charge for your students and institutional colleagues. Occupy Math is the series editor, in part to support this strategy for radically reducing textbook prices to students.
If you’re one of those people who don’t like e-books, paper copies of Dr. McEachern’s book are available for $44.95, a good deal less than the usual price of a textbook. In the near future we will announce a book in the series on representation in the prisoner’s dilemma and an inexpensive calculus text produced by Occupy Math and his collaborators. Occupy Math is now officially in the textbook business. If you have a book that might fit in the computational intelligence in games area, please drop Occupy Math a note at: firstname.lastname@example.org. If you think cheaper textbooks might be a good idea (or not?), please comment or tweet! This is a discussion that Occupy Math wants to have.
I hope to see you here again,
University of Guelph,
Department of Mathematics and Statistics