This introduces another factor we need to control for. Yes, if you really are better than the alternative CEO you might sell more cigarettes, and yes the board clearly thought you were the best choice for CEO—but what if they’re wrong? We need to adjust by the probability that you are indeed the best choice for CEO, conditional on the board thinking you were.
This seems like a pretty hard probability to estimate. My guess is it is quite low—I would expect many potential applicants, and a relatively poor ability to discriminate between them—but in lieu of actual analysis lets just say 50%.
You seem to shift here between p(Best among applicants) and p(Better than the guy who would have been hired in lieu of you). Guesstimating 50% for the former sounds reasonable-ish to me, but I would guess it’s substantially higher for the latter.
Maybe this comes out in the wash, since the difference between you and your actual replacement is smaller in expectation than the difference between you and the best among all the applicants.
Slight quibble:
You seem to shift here between p(Best among applicants) and p(Better than the guy who would have been hired in lieu of you). Guesstimating 50% for the former sounds reasonable-ish to me, but I would guess it’s substantially higher for the latter.
Maybe this comes out in the wash, since the difference between you and your actual replacement is smaller in expectation than the difference between you and the best among all the applicants.