# Fine-grained complexity for game-type problems

My specific question is the following. Consider the following problem that I call Strange-TQBF:

there is a Boolean function $$f(x_1, \ldots, x_n)$$ and two players Alice and Bob.

They take turns selecting variables and assigning values to them. After $$n$$ steps Alice wins if the value of $$f$$ (according to the assignment) is equal to $$1$$ and loses otherwise.

It is possible to determine the winner in this game in time $$3^n \cdot \text{poly}(n)$$ by using dynamic programming. The idea is to solve this problem for all subgames.

Questions: Is there an algorithm that solves the problem in time $$3^{n- \varepsilon} \cdot \text{poly}(n)$$?

Is it possible to prove that there is no algorithm that solves this problem in time $$2^{n+\varepsilon}>0$$ under some hypothesis like SETH?

Also, a general question:

Are there any results in fine-grained complexity for games? Can we prove (under some assumptions) for some game that the best possible way to solve the game is just to consider all sub-games?