Recall how games work. Let $X$ be a set (the "game space") and $\alpha$ an ordinal (the "game clock"). Alice and Bob take turns naming elements of $X$. We write them down in order until we have a sequence of length $\alpha$ in $X$, i.e. a point $f \in X^\alpha$. If $S \subseteq X^\alpha$ (the "game"), we say that Alice has won the $S$-game if $f \in S$, and Bob has won if $f \not \in S$. We say that the game $S$ is determined if either Alice or Bob has a winning strategy -- they can guarantee a win no matter what moves the opponent plays.
Recall that the Axiom of Determinacy (AD) says that every game in $S \subseteq \omega^\omega$ is determined. It is inconsistent with ZFC, but consistent with ZF relative to large cardinal hypotheses. More generally, if $X$ is a set and $\alpha$ an ordinal, let $AD(X,\alpha)$ be the assertion that every game $S \subseteq X^\alpha$ is determined. According to Cantor's Attic, if you make either $X$ or $\alpha$ just a little bigger than $\omega$, then $AD(X,\alpha)$ is already inconsistent with ZF.
Question: Are there examples of sets $X$ and ordinals $\alpha$ such that $AD(X,\alpha)$ is inconsistent with ZF, but consistent with some flavor of constructive set theory, where we drop the law of excluded middle?
(Conversely: are the refutations of $AD(X,\alpha)$ constructive?)
One might speculate that if $AD(X,\alpha)$ is consistent in some constructive set theory, then it might have high consistency strength, just as $AD(\omega,\omega)$ has high consistency strength in ZF.
