I feel kind of stupid asking this question, but upon browsing AD and variations of AC I can not help but wonder if there is also work for AD with higher cardinalities, like $\alpha^\alpha$-games for aribtrary cardinal $\alpha$. For AC we could define variations AC$(a,b)$ with $a,b\in\{\alpha,<\!\alpha,\infty:\alpha\textit{ some cardinal}\}$, where for instance AC$(\omega,\infty)$ is countable choice or AC$(\infty,<\!\omega)$ is choice for finite sets.
Now the axiom of determinateness is orginally defined for $2^\omega$ games (1) but is as axiom of determinacy nowadays more widely spread for $\omega^\omega$ games. I'm wondering if there is work on determinacy for $\alpha^\alpha$ games and arbitrary cardinalities.
From what I understand AC$(\aleph_1,\infty)$ already contradicts AD$(\omega^\omega)$ and AD$(\omega^\omega)$ implies AC$(\omega,\aleph_1)$. I reckon we can not have an axiom of determinacy for all cardinalities at the same time, since AD$(\aleph_1^{\aleph_1})$ assumedly contradicts AD$(\aleph_0^{\aleph_0})$.
Or am I missing something, is AD$(\alpha^\alpha)$ provably contradicting for any $\alpha>\omega$?
(1) Mycielski, Jan; Steinhaus, Hugo, A mathematical axiom contradicting the axiom of choice, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques.. 10, 1-3 (1962). ZBL0106.00804.
See this answer to a question of mine from a while ago. Basically:
and
each provably in ZF. So all that's left, really, is games of length $\alpha$ for $\alpha$ countable, on a countable set $X$.
Well, $X$ might as well now be $\omega$, so really we're talking about countable-length games on the natural numbers. And these rapidly yield strong determinacy principles! For instance, it's a good exercise to show that length-$\omega^2$ determinacy on $\omega$ implies $AD_\mathbb{R}$. Neeman has studied the determinacy of long games, and Martin and Woodin independently showed that ZF + AD + "all sets of reals have scales" implies the following principle:
$$(*)\quad\mbox{For each $\alpha<\omega_1$, every length-$\alpha$ game on $\omega$ is determined.}$$
(See the beginning of this paper of Neeman.) And you may find Neeman's book The Determinacy of Long Games an interesting source in general.
An interesting side question is how strong the Martin-Woodin hypothesis is, in terms of consistency strength. As an upper bound, it is consistent relative to at most the existence of simultaneous limit of strongs and limit of Woodins that determinacy holds and every set of reals is universally Baire and hence is determined and has a scale (this is a result of Larson, Sargsyan, and Wilson); I suspect less is needed, but an obstacle to this is the fact that scales provably fail to exist in $L(\mathbb{R})$ at the $\Pi^2_1$ level.