I think there's an ambiguity here:

• if X*_i presses "sim" and X*_i receives .2 utility, then "sim" is the dominant choice and there's nothing more to say.

• if X*_i presses "sim" and X gets .2 utility, then each X' (X or X*_i) faces a variant of the Absent Minded Driver.

The second interpretation seems though to be contraddicted by OP words:

For every agent it is true that she does not gain anything from the utility of another agent despite the fact she and the other agents are identical!