Thanks, Adam. The risks do not even have to be very close to 0. 10 % is enough. For 2 risks r1 and r2, the probability of survival is p_s_i = (1 - r1)*(1 - r2). If the risks change by Delta_r1 and Delta_r2, the probability of survival becomes p_s_f = (1 - r1 - Delta_r1)*(1 - r2 - Delta_r2) = p_s - (1 - r2)*Delta_r1 - (1 - r1)*Delta_r2 + Delta_r1*Delta_r2. So the increase in the probability of survival is Delta_p_s = p_s_f—p_s_i = -(1 - r2)*Delta_r1 - (1 - r1)*Delta_r2 + Delta_r1*Delta_r2. If both risks are at most 10 %, which I would say very much holds in reality, 1 - r1 and 1 - r2 are roughly 1. In addition, for relatively small allocations of money, the last term is negligible. So the increase in the probability of survival is approximately -Delta_r1 - Delta_r2, which is the sum of the decreases in risks 1 and 2.
I like your idea of doing a back-of-the-envelope calculation assuming that there are just two risks. Suppose that risk #1 has survival probability s1 and risk #2 has survival probability s2. Assume s1 < s2. Let’s compare two alternative interventions:
Intervention 1: Increase the survival probability for risk #1 by some small amount x.
Intervention 2: Increase the survival probability for risk #2 by that same amount x.
Now we have:
Intervention 1 will increase the overall survival probability from s1*s2 to (s1+x)*s2. That is an increase of x*s2.
Intervention 2 will increase the overall survival probability from s1*s2 to s1*(s2+x). That is an increase of x*s1.
So the increase in overall survival probability produced by intervention #1 is 100 * (s2/​s1 − 1) percent greater than the increase in overall survival probability produced by intervention #2.
Plugging in some toy numbers:
Suppose that risk #1 has survival probability 90% and risk #2 has survival probability 99%. Then an intervention that increases s1 by a small amount will produce 10% more increase in overall survival probability than an intervention that increases s2 by that same small amount.
If risk #1 had survival probability 80% instead, then intervention 1 would produce approximately 24% more increase in overall survival probability than intervention 2.
So a risk’s base survival probability does make some difference to how much one increases overall survival probability by mitigating that risk. But overall I agree with you that for risks with survival probability greater than 90%, the difference is modest.
Thanks, Adam. The risks do not even have to be very close to 0. 10 % is enough. For 2 risks r1 and r2, the probability of survival is p_s_i = (1 - r1)*(1 - r2). If the risks change by Delta_r1 and Delta_r2, the probability of survival becomes p_s_f = (1 - r1 - Delta_r1)*(1 - r2 - Delta_r2) = p_s - (1 - r2)*Delta_r1 - (1 - r1)*Delta_r2 + Delta_r1*Delta_r2. So the increase in the probability of survival is Delta_p_s = p_s_f—p_s_i = -(1 - r2)*Delta_r1 - (1 - r1)*Delta_r2 + Delta_r1*Delta_r2. If both risks are at most 10 %, which I would say very much holds in reality, 1 - r1 and 1 - r2 are roughly 1. In addition, for relatively small allocations of money, the last term is negligible. So the increase in the probability of survival is approximately -Delta_r1 - Delta_r2, which is the sum of the decreases in risks 1 and 2.
I like your idea of doing a back-of-the-envelope calculation assuming that there are just two risks. Suppose that risk #1 has survival probability s1 and risk #2 has survival probability s2. Assume s1 < s2. Let’s compare two alternative interventions:
Intervention 1: Increase the survival probability for risk #1 by some small amount x.
Intervention 2: Increase the survival probability for risk #2 by that same amount x.
Now we have:
Intervention 1 will increase the overall survival probability from s1*s2 to (s1+x)*s2. That is an increase of x*s2.
Intervention 2 will increase the overall survival probability from s1*s2 to s1*(s2+x). That is an increase of x*s1.
So the increase in overall survival probability produced by intervention #1 is 100 * (s2/​s1 − 1) percent greater than the increase in overall survival probability produced by intervention #2.
Plugging in some toy numbers:
Suppose that risk #1 has survival probability 90% and risk #2 has survival probability 99%. Then an intervention that increases s1 by a small amount will produce 10% more increase in overall survival probability than an intervention that increases s2 by that same small amount.
If risk #1 had survival probability 80% instead, then intervention 1 would produce approximately 24% more increase in overall survival probability than intervention 2.
So a risk’s base survival probability does make some difference to how much one increases overall survival probability by mitigating that risk. But overall I agree with you that for risks with survival probability greater than 90%, the difference is modest.
Thanks for the elaboration, Adam! That makes sense.