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Since my University library lists Conway's "Winning ways for your mathematical plays in the section "recreational mathematics" alongside books on origami and puzzles, I wondered to what extent game theory is abranch of "serious" mathematics. I'm aware of the fact that there is a lot of theoretical background for game theory (surreal Numbers, nimbers etc.), but on the other hand, game theory seems to be a bit seperated from the rest of mathematics, in the sense that I know neither any applications of game theory (or nimbers or Numbers) to other areas of mathematics or applications of mighty theorems from, say, number theory or topology, to combinatorial games.

I would be glad if you could give me examples which prove my perception wrong.

(Note: I'm talking about combinatorial game theory, e. g. chess and morris, not about economic game theory, e. g. Prisoner's dilemma.)

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There's some group theory in there, evidently. – Alexander Gruber Jun 9 '13 at 16:47
I would say that Kakutani's Theorem is a good example of an application of topology to game theory. – Dan Rust Jun 9 '13 at 17:00
@Daniel: this is classical game theory, not combinatorial game theory. The two subjects are very different. – Qiaochu Yuan Jun 9 '13 at 20:12

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Aviezri S. Fraenkel begins his paper ‘Combinatorial Game Theory Foundations Applied to Digraph Kernels’, The Electronic Journal of Combinatorics, Volume $4$, Issue $2$ ($1997$) as follows:

Modern combinatorial game theory has largely been a parasite: it drew tools and results from fields such as logic, computational complexity, graph and matroid theory, combinatorics, algebra and number theory to generate results for itself. More recently, it has also begun to contribute back to some of its benefactors, such as to surreal numbers, a subject created by John Conway [Con1976], and to linear error-correcting codes (which is linear algebra) [CoS1986], [Con1990], [BrP1993], [Fra1996].

As the title indicates, the paper gives an application to graph theory.

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