Pure Mathematical Applications of Advanced Game Theory? Games appear in pure mathematics, for example, Ehrenfeucht–Fraïssé game (in mathematical logic) and Banach–Mazur game (in topology).
But the Game Theory behind those applications is not so deep, and we don't need to know some fundamental theorems for them. Maybe except Zarmelo Theorem.
Are there applications of advanced (anything behind the basic definitions) game theory ideas in pure mathematics?
Thanks!
 A: You can read about the application of Blackwell's approachability theory, a nontrivial area in game theory, to normal numbers in https://www.jstor.org/stable/30035680?seq=1#page_scan_tab_contents
A: I would nominate the following paper of Victoria Gitman and Joel David Hamkins for this purpose:
https://arxiv.org/abs/1509.01099.

The principle of open determinacy for class games---two-player games of perfect information with plays of length $\omega$, where the moves are chosen from a possibly proper class, such as games on the ordinals---is not provable in Zermelo-Fraenkel set theory ZFC or G\"odel-Bernays set theory GBC, if these theories are consistent, because provably in ZFC there is a definable open proper class game with no definable winning strategy. In fact, the principle of open determinacy and even merely clopen determinacy for class games implies Con(ZFC) and iterated instances Con(Con(ZFC)) and more, because it implies that there is a satisfaction class for first-order truth, and indeed a transfinite tower of truth predicates $\text{Tr}_\alpha$ for iterated truth-about-truth, relative to any class parameter. This is perhaps explained, in light of the Tarskian recursive definition of truth, by the more general fact that the principle of clopen determinacy is exactly equivalent over GBC to the principle of transfinite recursion over well-founded class relations. Meanwhile, the principle of open determinacy for class games is provable in the stronger theory GBC+$\Pi^1_1$-comprehension, a proper fragment of Kelley-Morse set theory KM.

A: The purity of math is in the eye of the beholder, but maybe some of the following examples qualify:
There is a proof of Kolmogorov's strong law of large numbers in the book Probability and Finance by Shafer and Vovk that uses the determinacy of quasi-Borel games. The book generally shows how one can use game theory to prove probabilistic results.
Versions of the minimax theorem can be used to prove results in convex analysis. There is even a journal called Minimax Theory and its Applications. It is probably the most useful mathematical tool that came out of game theory.
Konrad Podczeck and I have a purification theorem for measure-valued maps whose proof is at least heavily based on game theoretic intuitions.
I would say the most useful applications of game theory to other areas of mathematics are based on zero-sum games, which are of least interest from the perspective of game theory as a tool of social sciences. 
A: Well, here is one. (A game-theoretic proof of Erdos-Feller-Kolmogorov-Petrowsky law of the iterated logarithm for fair-coin tossing, 2014).
And this:
Tanaka, Kazuyuki, A game-theoretic proof of analytic Ramsey theorem, Z. Math. Logik Grundlagen Math. 38, No.4, 301-304 (1992). ZBL0798.03050.
And this very nice post by Francois Dorais (who used to be an active contributor here) on Fraisse's theorem.
And this ancient wisdom:
Marshall, A.W.; Olkin, I., Game theoretic proof that Chebyshev inequalities are sharp, Pac. J. Math. 11, 1421-1429 (1961). ZBL0118.13703.
