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.)