Thanks for mentioning this. I know this wasn't put very nicely.
Imagine you were a very selfish person X only caring about yourself. If I make a really good copy of X which is then placed 100 meters next to X, then this copy X* only cares about the spatiotemporal dots of what we define X. Both agents, X and X, are identical if we formalize their algorithms incorporating indexical information. If we don't do that then a disparity remains, namely that X is different to X* in that, intrinsically, X only cares about the set of spatiotemporal dots constituting X. The same goes for X* accordingly. But this semantical issue doesn't seem to be relevant for the decision problem itself. The kind of similarity that is of interest here seems to be the one that determines similiar behavior in such games. (Probably you could set up games where the non-indexical formalization of the agents X and X* are relevantly different, I merely claim that this game is not one of them)