I've been wondering about the decision theory of having a "risk profile", e.g. "200 microcovids per week", and at least under some simplistic but not unreasonable assumptions I don't think it makes sense.
Let's say we just have one activity A which yields some fixed positive utility U but has a certain probability p of giving you covid which has a large cost C. Then the expected utility of A is simply E = U - pC. If E > 0 then you should just do A as much as possible, otherwise not at all. There's no cap on total covid risk. (OTOH the "risk profile" method would recommend doing A about 200e-6/p times.)
One counterargument is that the positive utilities don't add up linearly. That is, just because going to a restaurant once is nice, doesn't mean you'd enjoy constantly being in one. But I think this is a red herring -- you could instead imagine a variety of different activities (movies, dating, eating out, parties, etc.) that do add up more or less linearly. You could also just be the type of person who gets the same enjoyment from eating out every day. Either way, the issue of "where does microcovid budget come from?" remains.
I think the crux of the matter is that (consequentialist) decision theory is Markovian. When you make a decision, you only care about the state you're in right now, not in how you got there. So whether or not you did some risky activity yesterday, unless it actually gave you covid, there's no effect on your current state, and so therefore it shouldn't affect your current and future decisions about whether to engage in something risky.
To be clear, my goal isn't to abandon risk budgets -- they seem very sensible to me, and I don't have a good replacement. But, I'd like to know if there is some better model which captures the intuition around risk budgets (or an error in the above reasoning).