As an example, let’s say you have three interventions with that distribution, and they turn out to be perfectly distributed, you have total cost=$11,010 and total effect=3 so, as a funder that cares about expected value, $3670 is the value you care about.
That’s true if you spend money that way, but why would you spend money that way? Why would you spend less on the interventions that are more cost-effective? It makes more sense to spend a fixed budget. Given a 1⁄3 chance that the cost per life saved is $10, $1000, or $10,000, and you spend $29.67, then you save 1 life in expectation (= 1⁄3 * (29.67 / 10 + 29.67 / 1000 + 29.67 / 10,000)).
Not sure how useful it is as an intuition pump, but here is an even more extreme/absurd example: if there is a 0.001% chance that the cost is 0 and a 99.999% chance that the cost is $1T, mean(effect/cost) would be ∞
That’s a feature, not a bug. If something has positive value and zero cost, then you should spend zero dollars/resources to invoke the effect infinitely many times and produce infinite value (with probability 0.00001).
That’s true if you spend money that way, but why would you spend money that way? Why would you spend less on the interventions that are more cost-effective? It makes more sense to spend a fixed budget. Given a 1⁄3 chance that the cost per life saved is $10, $1000, or $10,000, and you spend $29.67, then you save 1 life in expectation (= 1⁄3 * (29.67 / 10 + 29.67 / 1000 + 29.67 / 10,000)).
That’s a feature, not a bug. If something has positive value and zero cost, then you should spend zero dollars/resources to invoke the effect infinitely many times and produce infinite value (with probability 0.00001).