The point is that if the universe is entirely deterministic then it is entirely described by its starting conditions: from the point of view of entities within it, it makes no difference if you run the simulation or not, they exist from their own perspective. Running the simulation only matters to you. So in the case where you have recursive simulations of the same universe like in the story, even if you turn of your simulation, implying the universe 'above' turns off theirs, you will continue to exist just fine, as far as you can see. That is the core idea of permutation city: they set up a simulation containing themselves and then don't bother to continue to run it: it doesn't matter to the simulated version of themselves (of course, you can ask the question whether they would need to set up the simulation in the first place. And then things get more crazy).
