Ok, so say you have a fixed budget. Then you want to maximise mean(total effect), which is equal to mean(budget/cost * unit effect)
… I agree.
Also, infinite expected values come from having some chance of doing the thing an infinite number of times, where the problem is clearly the assumption that the effect is equal to budget/cost * unit effect when this is actually true only in the limit of small numbers of additional interventions.
Also, Lorenzo’s proposal is ok when cost and effect are independent, while the error he identifies is still an error in this case.
The below is a reply to a previous version of the above comment.
I do not think we want to maximise mean(“effect”—“cost”).
Note “effect” and “cost” have different units, so they cannot be combined in that way. “Effect” refers to the outcome, whereas “cost” corresponds to the amount of resources we have to spend.
One might want to include “-cost” due to the desire of accounting for the counterfactual, but this is supposed to be included in “effect” = “factual effect”—“counterfactual effect”.
We want to maximise mean(“effect”) for “cost” ⇐ “maximum cost” (see this comment).
Ok, so say you have a fixed budget. Then you want to maximise mean(total effect), which is equal to mean(budget/cost * unit effect)
… I agree.
Also, infinite expected values come from having some chance of doing the thing an infinite number of times, where the problem is clearly the assumption that the effect is equal to budget/cost * unit effect when this is actually true only in the limit of small numbers of additional interventions.
Also, Lorenzo’s proposal is ok when cost and effect are independent, while the error he identifies is still an error in this case.
The below is a reply to a previous version of the above comment.
I do not think we want to maximise mean(“effect”—“cost”).
Note “effect” and “cost” have different units, so they cannot be combined in that way. “Effect” refers to the outcome, whereas “cost” corresponds to the amount of resources we have to spend.
One might want to include “-cost” due to the desire of accounting for the counterfactual, but this is supposed to be included in “effect” = “factual effect”—“counterfactual effect”.
We want to maximise mean(“effect”) for “cost” ⇐ “maximum cost” (see this comment).
Yeah, I was mentally substituting “effect” for “good” and “cost” for “bad”