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Let $M$ be a polynomial-time deterministic machine that can ask questions to some oracle $A$. Initially $A$ is empty but this is can be changed after a game that will be described below. Let $x$ be some string.

Consider the following Alice and Bob game. Initially, Alice and Bob have $m_A$ and $m_B$ dollars respectively. Alice wants $M^A(x)=1$ and Bob wants $M^A(x)=0$.

At every step of the game a player can add one string to $A$; this costs one dollar. Also a player can miss his or her step.

The play ends if both players spends all money or if some player missed step when he or she in a losing position (that defines by the current value of $M^A(x)$).

Question: is the problem of defining the winner of this game for given $M, x, m_A, m_B$ is an

EXPSPACE-complete task?

Note that $M$ can ask (for belonging to $A$) only strings of polynomial length so there is no sense for Alice or Bob to add more longer strings to $A$. Hence, this problem is in EXPSPACE.

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I don't have an exact characterization but it's unlikely this problem is EXPSPACE-complete. Suppose $M^{\Sigma^*}(x)$ accepts and let $S$ be the polynomial-size set of strings queries by this machine. If I understand the game right, Alice can win by playing every string in $S$. The only way to prevent this is if $m_A$ is polynomially-bounded but that would put the game somewhere inside the exponential-time hierarchy (or lower).

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