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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?

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