QUESTION

Let M be an inner model (of height Ord) containing all the reals. For each $X \in M$, define $S_X = \{x \in X^\omega : x_I \in M \land x_{II} \not \in M\}$. ($x_I$ is the set of plays in $x$ by player I: i.e. the even indices of $A_I$. Similarly for $x_{II}$.)

I am wondering about the consistency strength of the following statement: for each $X \in M$ and $A \subset X^\omega \cap M$, $G_X(A \cup S_X)$ is determined.

(We can add a unary predicate to $ZFC$ and ask about the existence of such an $M$; but I am also curious about whether $M = L(\mathbb{R})$ is possible.)

MOTIVATION

For the statement of AD, games are restricted to be played on natural numbers, terminating after $\omega$ plays. In that case payoffs correspond to subsets of $\omega^\omega$.

There are various ways of generalizing this; for example, given $X$ and $A \subset X^\omega$, one can define the game $G_X(A)$ where each player takes turns choosing from $X$, and where the payoff is given by $A$.

With that definition of "game," it is not possible that every game in M is determined in M. Indeed, suppose $AD^M$ holds: I define a nondetermined game played on $\omega_1$ (following Kanamori).

On the first move, player I picks some $\alpha < \omega_1$. On the subsequent moves, player II picks elements of 0,1. Player II wins iff his/her sequence of bits encodes $\alpha$.

Letting A be the payoff corresponding to the above, then $G_{\omega_1}(A)$ cannot be determined in M. But it is certainly determined in V!

Note that if $S_X$ is nonempty, then for any $A \subset X^\omega \cap M$, player II has an easy winning strategy in $V$ for $G_X(A)$, which is neutralized by adding $S_X$. (This made my previous question If M is an inner model containing all the reals, might every game in M be determined in V? not quite what I was after. Although it is still interesting)

SUBQUESTIONS

Note that the hypothesis implies $AD^M$ since every real is in $M$.

(a) Is this hypothesis outright inconsistent?

(b) Is this hypothesis equivalent to $AD^M$, or to, say, $AD^M_{\mathbb{R}}$?

(c) Does this hypothesis depend on the precise generalization of games I use? (Another generalization would be to allow ordinal-length games, for example.)