Yes, essentially preventing extinction “pays off” more in the low risk situation because the effects ripple on for longer.
Mathematically, if the value of one century is v, the “standard” chance of extinction is r, and the rate of extinction just for this century is d, then the expected value of the remaining world will be
v(1−d)+v(1−d)(1−r)+v(1−d)(1−r)2+...
= v(1−d)/r (using geometric sums).
In the world where background risk is 20%, but we reduce this century risk from 20% to 10%, the total value goes from 4*v to 4.5*v.
In the world where background risk is 2%, but we reduce this century risk from 20% to 10%, the total value goes from 49*v to 49.5*v.
In both cases, our intervention has added 0.5v to the total value.
Yes, essentially preventing extinction “pays off” more in the low risk situation because the effects ripple on for longer.
Mathematically, if the value of one century is v, the “standard” chance of extinction is r, and the rate of extinction just for this century is d, then the expected value of the remaining world will be
v(1−d)+v(1−d)(1−r)+v(1−d)(1−r)2+...
= v(1−d)/r (using geometric sums).
In the world where background risk is 20%, but we reduce this century risk from 20% to 10%, the total value goes from 4*v to 4.5*v.
In the world where background risk is 2%, but we reduce this century risk from 20% to 10%, the total value goes from 49*v to 49.5*v.
In both cases, our intervention has added 0.5v to the total value.