I think this question has great potential, but as it is currently phrased it is very confusing (although I admit it is rigorously defined). IIUC, as per @JoelDavidHamkins's comment, this isn't at all about games but about accessibility in a certain graph. Basically you have a group of finite operations which you “complete” into something that might not be a group but is still, at least, an oriented graph (with edges being “accessible in $\leq\omega$ moves”) and you are asking whether any out-neighbor of the origin is also an in-neighbor, or something of the sort. [contd.]
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