This seems analogous to the other sort of budget.
Suppose I decide that I'm allowed to spend £500/year on books and no more. This isn't because once I've spent that much I get no further value from them, and it isn't because if I spend more then I will literally run out of money and starve. It's because there's some degree of diminishing returns from buying more books, and there are other things I want to be able to spend money on, and putting a hard limit on how much I spend is a nice simple system that approximates the Right Thing. It doesn't approximate it very closely, but that's OK: the difference between approximating it very closely and approximating it rather crudely is small.
(I don't, in fact, put a ceiling on how much I'm allowed to spend on books.)
So, how does this relate to Covid-19 risk budgeting? I think the key claim I want to make is that I don't believe you when you say "you could instead imagine a variety of different activities [...] that do add up more or less linearly". I mean, of course you could imagine that there is such a variety of different activities, but I don't believe that you can imagine a specific set of such activities that actually do add up nearly linearly. At least, not when things are fairly bad Plague-wise.
The reason is that (so at least it seems to me) social activities (which are the things people bother with microcovid-budgets for) typically bring positive utility in two different ways. First, the fun of doing whatever thing it is: the pleasure of eating well-prepared food, the enjoyment of walking somewhere with nice scenery, the appreciation of the music at a concert, etc. Second, the extra gain of doing it with someone else. And, while some people get more out of being with other people than others do, I'm pretty sure that gain doesn't scale up linearly with the time you spend with those other people. Not even if you're doing a variety of different things with them. And when the Plague is bad, I think that (at least for the sort of person who is likely to be drawing up microcovid-budgets in the first place) there isn't enough of the first kind of utility to justify the risk: that is, the expectation in the OP here is consistently negative. But there is, at least sometimes, enough of the second, because for most people going completely without human contact is terrible. But that one has diminishing returns.
Those diminishing returns surely don't look at all the way they would need to for a risk budget to be exactly The Right Thing. (That would mean something like marginal utility = constant when t<T, and zero when t>T.) But they might have the property I claimed financial budgets sometimes have: that the utility difference between having a budget and doing all the utility calculations right is small. Do they?
I think they plausibly do if three things hold. (1) That for the activities being considered, the utility gain that matters is the one that comes from doing them with someone else -- that the utility from the thing-in-itself is much less. (2) That, on the margin, this utility gain is roughly proportional to the risk, at least conditional on the way you will actually behave. (3) That the actual risk budget figure you choose is somewhere near to optimal, given that you're risk-budgeting.
If those three things are true, then 1 and 2 mean that total utility is roughly some function of total risk; if that were exactly true then optimal behaviour would be continuing to socialize until reaching some risk level and then stopping, which is exactly what risk-budgeting does; if it's roughly true then I think that means that that behaviour (with the correct threshold) isn't too far from optimal; and then 3 means that you're near to the correct threshold.
I think 1 is plausible. (Because for many people the gain from having some social activity is very large, and microcovid-budgeters are generally being cautious enough that they have rather little social activity.)
I think 2 is plausible. (Because if you're microcovid-budgeting you will be picking your activities and how you do them so as to maximize social-utility-per-unit-risk, which means that most of the things you do will have about the same social-utility-per-unit-risk. Also, for obvious reasons there is a correlation between the two -- e.g., the more people you're with, the higher the risk but also the more social interaction you get.)
I think 3 is plausible, but this is the place where I think things are most likely to go wrong. (Not least because I don't have any argument for its plausibility that I really trust. But I think I think that while people are terrible at judging risk, it's mostly the actual risk estimation that's bad rather than the ability to make reasonable judgements of what utility gains are worth what risks.)
On further reflection I agree that diminishing return are pretty important. One consequence of them is that there is effectively a cap P on the total positive utility in a given time T. That turns into a risk cap of P/C per time period T.
I think gjm made some good points.
But I also want to note that having a budget is most important for coordination. I think this is what microcovid was originally designed for – you have a bunch of roommates, or people in a quaranbubble, and you want to agree on how you interact with the world. Giving everyone a budget is easier than a more complicated set of rules.
If you're living on your own or with one person (i.e. close friend or romantic partner) who's easy to stay in sync with, then it's less important, unless you find it helpful for your own thinking.
A risk budget makes much more sense, once we consider it an exposure budget and consider logical decision theory. You and a community of identically thinking friends are deciding how much exposure between each other to tolerate. To the extent that your community is very large, homogeneous and hardly ever get exposed to outsiders, you have a threshold between exponential growth and exponential decay. Now if hypothetically some people got more utility from exposure, and you could perfectly coordinate, then those who gain more utility from interactions would interact more (assuming fungible utility.)
Mod note: I frontpaged this (despite a policy not usually frontpaging covid content) because I think "how to think about microcovids" is actually fairly confusing and it could use some more dedicated discussion, and because I think that translates a bit into general thinking outside the domain of covid.
I think a large factor for people making decisions around covid risk is not just the risk they are posing to themselves, but also the risk they are imposing on others. Insofar as "risk I impose on others" enters my utility function, this is going to change a lot of your conclusions pretty quickly. The reason being that "risk imposed on others" is growing super-linearly in most activities.
E.g. If I go to a restaurant and then meet a friend, I've incurred much more risk to the friend than if I didn't go to the restaurant. If I then meet a third friend separately, the risk to that friend is increased by each of the prior interactions, and so forth. Each additional activity poses additional risk to all the people involved in later activities. This is classic exponential growth type stuff, but all we need is super-linear growth in risk.
Once you have something (bad) growing super-linearly like that, it should be pretty straightforward to see that even if the net utility from each of two different actions is positive, the utility from doing both actions may be less than the utility of doing just one. Insofar as the thing that is growing super-linearly is about 'micro-covids', it makes sense that some things are better and worse on that scale (eating inside a restaurant vs going on a walk with a friend), and so accounting for that differential cost makes sense. And now we're firmly in 'budget' territory -- different costs for different activities all of which i like, but with some kind of max on how much I can reasonably spend.
You can sum expected utility of doing your your activity A K times like so:
Calculate expected utility of doing activity A K times with K chosen in such way such that p is approximately 0.5
Then your calculation becomes: E = KU-C/2. Is this E still positive?
Another objection is that you assume that C is fixed, but it is actually a function of utilities of all the activities Ai with positive utility which you will be unable to do if infected.
My main argument in favour of risk profiles is to think in terms of the frequency in which events go wrong. It is true that me not getting Covid yesterday should not impact my decisions today, however making choices that yield a probability of a bad event of p over the next month means I'll have that bad event happen once every 1/p months, on average. Due to loss aversion I might want to cap that frequency, even at the cost of reduced expected utility, hence I'll give myself a risk profile. This goes into the psychological effects of loss, which tend to overweight positive outcomes. Any thoughts?
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).