Fundamental mathematical results of game theory

Though strange it seems to be, there are comparatively few main results in mathematical game theory. Among those are Nash theorem (existence of equilibrium in mixed strategies in arbitrary finite game), existence of sequential equilibrium for finite dynamic games, Scarf theorem for cooperative games, and the optimal auction of Roger Myerson. Little bit aside lies the Arrow-Debreu theorem on existence of the Walrasian competitive equilibrium. Almost all these results (apart from Myerson's one) are based on the Kakutani's fixpoint theorem.
In the talk, I first prove the Nash theorem, by reducing it to the Kakutani's fixpoint theorem, the latter one — to the Brower's fixpoint theorem, and the very latter — to the Sperner's Lemma. (As a byproduct, we'll get the equivalence of Brower's theorem and the nontriviality of the $n$-th homotopic group of $n$-sphere.) In the second part of the talk, I will characterize perspectives of Game Theory, and Mathematical economics as a whole.