Regarding the ‘first mistake’, you correctly show that survival of a species for a billion years requires reaching a low per-period level of risk (averaging roughly 1 in a billion per year). I don’t disagree with that and I doubt Bostrom would either. No complex species has yet survived so long, but that is partly because there have been less than 1 billion years since complex life began. But there are species (or at least families) that have survived almost the whole time, such as the Nautilus (which has survived 500 million years). So risk levels comparable to 1 in a billion per year do occur. For Bostrom’s modelling of the EV of risk reduction to work, he just needs there to be at least a small chance (say 1 in 1 million) that risk declines to such a level or beyond. That sounds eminently plausible to me, and my best guess of this probability would be much higher.
You say that: “there is a clear sense in which the drop in existential risk that Bostrom envisions is not small, but instead very large”. But note that this is not the drop in existential risk that need be caused by the intervention Bostrom is evaluating. He relies on there being at least a slender possibility that risk levels fall to something like those of the safest species on Earth, but the intervention doesn’t need to bring that about.
So on this ‘first mistake’, I agree that it is often also useful to think of things in per-period risk, and that this could provide a sanity check. But in this case, I think Bostrom’s estimate passes that sanity check, so don’t think he has made any kind of mistake here.
Regarding the ‘first mistake’, you correctly show that survival of a species for a billion years requires reaching a low per-period level of risk (averaging roughly 1 in a billion per year). I don’t disagree with that and I doubt Bostrom would either. No complex species has yet survived so long, but that is partly because there have been less than 1 billion years since complex life began. But there are species (or at least families) that have survived almost the whole time, such as the Nautilus (which has survived 500 million years). So risk levels comparable to 1 in a billion per year do occur. For Bostrom’s modelling of the EV of risk reduction to work, he just needs there to be at least a small chance (say 1 in 1 million) that risk declines to such a level or beyond. That sounds eminently plausible to me, and my best guess of this probability would be much higher.
You say that: “there is a clear sense in which the drop in existential risk that Bostrom envisions is not small, but instead very large”. But note that this is not the drop in existential risk that need be caused by the intervention Bostrom is evaluating. He relies on there being at least a slender possibility that risk levels fall to something like those of the safest species on Earth, but the intervention doesn’t need to bring that about.
So on this ‘first mistake’, I agree that it is often also useful to think of things in per-period risk, and that this could provide a sanity check. But in this case, I think Bostrom’s estimate passes that sanity check, so don’t think he has made any kind of mistake here.