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Please note that my knowledge of math proofs is little to none. I was wondering what resources, free or paid, would allow me to understand the math proofs specifically related to Game Theory?

I understand that I would need to build my knowledge set step by step and not be able to take huge leaps.

Update: I should elaborate that it is Game Theory with respect to economics (I think that includes Nash Equilibrium?)

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Do you mean combinatorial game theory (Nim, etc.) or game theory with applications to economics and political science (Prisoner's dilemma and Chicken)? –  Joseph Malkevitch Nov 22 '11 at 13:38
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Please see this question: math.stackexchange.com/questions/76096/game-theory-self-study/… Should this be marked as a duplicate? –  Michael Boratko Nov 22 '11 at 14:18
    
Please see Updated post –  user701510 Nov 22 '11 at 22:03
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up vote 3 down vote accepted

The Nash Equilibrium is, at its essence, an application of a fixed point theorem. If you are interested in building the mathematical background to understand Nash Equilibria, you must simply learn the math required to understand fixed point theorems. There is no predefined path for learning about fixed point theorems. One usually learns them in a topology course, but they are neither the focus nor the goal of the course, so you will need to form your own path out of the requirements.

If you have a strong idea about proofs, and a high mathematical maturity, then you could probably work through the standard text in Topology by Munkres right off the bat. If you haven't worked with proofs or elementary set theory in a while, you should get reacquainted with proofs, and I can't imagine a better way to do both than work through Halmos's Naive Set Theory.

In general, check out the topology books listed as an answer to this question. Note that one is free; I would recommend starting with that before purchasing Munkres. If you don't like it, at least it will give you an idea of what subjects you need to know in order to approach Munkres.

After you have understood the ideas of Topology you can move on to a book on game theory which covers the Nash Equilibrium proof. Here are three standards:

Playing for Real: A Text on Game Theory by Ken Binmore
This is a text for advanced undergraduates. Chapter 8 covers what you want to know, however it is limited to the two-dimensional case (which is much simpler to prove, however the ideas are the same in higher dimensions). If this is the book you wanted to go with, I would suggest just buying it and foregoing all the Topology stuff. By omitting the higher-dimensional cases, it is able to stay remarkably self contained.

A Course in Game Theory by Osborne and Rubinstein
This is a graduate level text, and is probably more along the lines of your interest. The only deficiency that I can see is that it does not include a proof of the Kakutani fixed-point theorem! It simply mentions it and uses the consequences to conclude the existence of Nash Equilibria. You still get a good idea of why it happens (the exercises force you to play with the restrictions, and this gives you insight into why it is true - you definitely need the Topology books first for this one though), but if you want a proof then you will not find it here. This book is a standard though, and inexpensive, so it is worth getting so that you can read it along with...

Fixed Point Theorems with Applications to Economics and Game Theory by Kim C. Border
This is really just a collection of notes, but a very good one. It is a standard reference for this subject, and does include a proof of the Kakutani fixed-point theorem. This is not really a good book for self study, however; it is intended as a reference. I would suggest getting this book as a supplement to Osborne and Rubinstein.

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would a math beginner understand the fixed point theorem book you mentioned, or is there something that will help me "bridge" my understanding? –  user701510 Nov 23 '11 at 1:22
    
@user701510 The book I mentioned is definitely not for a beginner. The problem is, with a traditional approach, Nash Equilibrium is not a beginner topic. You would not usually cover it in an undergraduate curriculum at all, so if you are a beginner you are going to have to do a significant amount of work. The path of least resistance, perhaps, would be some sort of elementary proofs class, real analysis, topology and then the Nash Equilibrium, but those cannot be all you study because they depend on context from other classes. –  Michael Boratko Nov 23 '11 at 1:34
    
@user701510 (cont.) For instance, although I suppose it is possible to jump into real analysis without calculus, you would not understand the motivation and as such would not be able to "find your place". How do you know you want to study game theory? I would suggest going through the free book I mentioned above to see if it is really something that interests you, or if the idea of it was more alluring. Often, actually working with a result or topic is much different than a high-level understanding. –  Michael Boratko Nov 23 '11 at 1:36
    
To be honest, I have taken a few calculus classes a few years back, but my understanding of it is hazy since I didn't put my heart into it and didn't see the practical applicability of some of the topics covered in calculus and I haven't taken any classes that forced me to deal with higher level abstract math proofs. I've gone over the basic concepts of game theory in my intermediate Microeconomics classes but have never taken game theory as a separate class. –  user701510 Nov 23 '11 at 5:13
    
[cont] Understanding precise mathematical relationships would help me in my master's degree studies should I choose to get one. I also enjoy learning for its own sake. –  user701510 Nov 23 '11 at 5:13
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