As far as I know, there are researches about "combinatorial game theory" and "game theory in optimization" which are generally "unrelated" branches.
Is there any connection between these two branches?
Are there common tools that can be applied to both branches?
Up to now, I haven't seen combinatorial tools in games in optimization or something similar that connects these two types of game?
As far as I know, there is absolutely zero connection between combinatorial game theory (CGT) and what I will call "economic game theory". This is quite unusual in mathematics. Normally, even seemingly unrelated fields in math like calculus and combinatorics will have commonalities and applications to each other, but this is just not the case with the two game theories.
Both fields are about games, but the games are very different in nature. Games in CGT are perfect information, zero-sum (purely competitive), and players take turns. Games in economic game theory are imperfect information, non-zero sum (so there is a mix between cooperation and competition), and players usually decide their moves simultaneously.
CGT is just such a strange field that it rarely finds common ground with anything else. CGT normally involves concepts that only make sense in that field; concepts like temperature theory and atomic weights were invented for CGT, and unlike anything else. When conventional math does show up in CGT, it is usually set theory and algebra, especially group theory and Galois theory. On the other hand, economic game theory is focused on more applied problems, employing methods from convex optimization, differential equations and analysis (especially fixed point theorems).