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I have recently become interested in game theory by way of John Conway's on Numbers and Games. Having virtually no prior knowledge of game theory, what is the best place to start?

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Community Wiki is the best place to start. To clarify, I mean that you should tick the "community wiki" box before posting a question like this. However, you're in luck. You can edit your question and tick it now. –  Harry Gindi Dec 8 '09 at 22:32
    
I'm confused, why should this be a community wiki question? –  Noah Snyder Dec 8 '09 at 23:37
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"What is the best place to start?" It's a community wiki question because it's a question with no definite answer. It's polling the community about something. I was under the impression that that kind of question fell under "Should be community wiki." –  Harry Gindi Dec 8 '09 at 23:47
    
Just for reference, I'm going by the discussion at tea.mathoverflow.net/discussion/6/… –  Harry Gindi Dec 8 '09 at 23:58
    
Thank you everyone. It sounds like Winning Ways and rereading portions of ONAG as I go is the place to start, and I will move on from there. –  user2421 Dec 9 '09 at 6:56

5 Answers 5

up vote 7 down vote accepted

Winning Ways for your Mathematical Plays (in four volumes) has an enormous amount of stuff about combinatorial games. But most of it you probably won't be interested in for a while. There are a few quickly diverging directions one could study in combinatorial games. Here are some that come to mind immediately, and a possible list of topics to study in each:

1) Impartial games. Read a bit of On Numbers and Games so that you know how to read the notation and understand game equivalence and addition. Then learn the winning strategy for Nim, read the relevant bits of Chapter 3 and all of Chapter 4 of Winning Ways. After that, if you like the infinite theory, Lenstra has a paper called "On the algebraic closure of two" which is really nice. If you like the finite theory, learn about nim multiplication from ONAG, and then read Conway and Sloane's paper "Lexicographic codes: error-correcting codes from game theory." I think this part of the theory is the most interesting.

2) (Surreal) numbers. Again, learn how to read the notation and about game equivalence and addition. (You will need this for everything.) Then read the first part of ONAG. Then, perhaps learn about real-closed fields in general; you can make most of real analysis work over the Field of surreal numbers. (A Field is something like a field, but it has a proper class of objects instead of a set.)

3) Weird games, for example from Hackenbush and Domineering. Read Volume 1 of Winning Ways. The stuff on thermography and all-small games is quite interesting.

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Extra sources:

"Lessons In Play" fills in most of the mathematical details for volume one of "Winning Ways";

"Games of No Chance 3" is out and is available online through MSRI publications;

Aaron Siegel's upcoming book on combinatorial games will include all the advances in the theory, he's responsible for many, and will be a good companion to "On Numbers and Games"

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Aaron's book, "Combinatorial game theory" has just appeared. –  Andres Caicedo Nov 11 '13 at 4:08

First, "game theory" (as used in economics) and "combinatorial game theory" are two completely different things.

Second, I think the best place to start learning combinatorial game theory is still Winning Ways for your Mathematical Plays by Berlekamp, Conway, and Guy.

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In addition to Winning Ways for your Mathematical Plays, I also recommend the books Games of No Chance and More Games of No Chance. Many of the articles from Games of No Chance are online.

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If you have played Go, then you may enjoy Mathematical Go: Chilling Gets the Last Point by Berlekamp and Wolfe. Some go players, especially Bill Spight, have developed combinatorial game theory within go further.

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Thanks. I play Go very badly, but it is interesting. I will look into it. –  user2421 Jan 23 '10 at 0:49

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