# Strong determinacy principles for “small” sets

In the course of a (fun but silly) project I'm working on, I've run into the following class of determinacy notions:

Suppose I have a "reasonably definable" class $\mathcal{C}$ of games (in the usual sense - perfect information, two player, length $\omega$, on some appropriately definable set) - e.g. all games, or "Banach-Mazur" games (players alternate finite sequences, and I wins if the real so built is in a predetermined payoff set), or so on. Then say that a transitive set $S$ is thoroughly $\mathcal{C}$-determined if every game in $\mathcal{C}$ which exists in $HOD(S\cup\{S\})$ is "determined at $S$": there is a strategy $\Sigma\in S$ such that, for every strategy $\Pi\in S$ for the opposing player, $V$ thinks $\Sigma$ beats $\Pi$. Note that the strategies are taken from $S$, but the plays are evaluated in the actual universe $V$.

Basically, the point is that not only does $S$ think every game is determined, but every game which can be "reasonably" defined in terms of $S$ is also determined inside $S$, even if $S$ can't really see the game directly.

Now for the class of all games, this isn't a very interesting notion: we trivially have that any thoroughly determined set contains every HOD real. As a computability theorist, that's about where my interest wanes (at least right now . . .).

For more limited classes of games, though, things are more interesting. I've been playing around with classes of games for which thoroughly determined sets of small height (e.g. vastly below $\omega_1^L$, regardless of what large cardinal properties $V$ satisfies) exist, and I've managed to show a couple things. My question, now that I'm writing this all up, is whether this notion has already been studied, and if so, what's known about it.

(I'm particularly worried that what I've proved are trivial consequences of known facts, and I'd like to check that before going through the labor of writing a full paper.)