# 54

A friend is hosting a party tonight! It'll cost me 200 microCOVIDs, which, as a healthy thirty-something, I very cautiously estimate to cost about 2 micromorts,[1] which is roughly equivalent to 1 hour out of my remaining life expectancy. But I'm super excited for this party; I'd happily burn an hour of my life driving there and back. Should I go?

Unless something subtle is going on... obviously yes, right?

"What about your visit to that coffee shop earlier today? Don't you think that's decision-relevant?"

The coffee shop? I'm confused. Why would that be decision-relevant?

"Well, you accumulated about 150 microCOVIDs there."

Sure, but that's not really relevant to the decision theory of whether to attend the party, is it? The party is equally enjoyable either way, and the costs of multiple COVID exposures add pretty much linearly by the axiom of independence, so previously-incurred risks are irrelevant to my future decisions. (If the coffee shop had been a week ago, sure, I'd be inflicting some of those microCOVIDs on my fellow partygoers, which, sure, could be decision-relevant, I haven't done the math; but it seems very unlikely to me that I'll become a full-fledged germ factory between this morning and this evening, so I think that consideration is insignificant in this case.)

"But you try to maintain a 200-microCOVID-per-week risk budget, don't you?"

Sure, but... hmm.

"So there's something wrong with your assertion that previously-incurred risks are irrelevant to future decisions."

...or something is wrong with the idea of risk budgeting.

"...hmm."

...so, which is it?

## Zero externalities

Maybe the answer will be clearer if we simplify the problem: let's get rid of externalities. Suppose I am the world's most boring superhero, Captain Can't-Transmit-COVID. I can still catch it just like anybody else, but the cost is borne by me alone, not my friends or housemates.

Now is my coffee-shop visit decision-relevant for whether I should attend the party?

I can't think of a single reason it would be.[2] This strongly suggests that "risk budgeting" has something to do with the risk of transmitting COVID to other people.

## Externalities to an enthusiastic optimizer

Let's add an externality back in: suppose I can only transmit COVID to my partner. Perhaps risk budgets will fall naturally out of the arrangements we make.

Me: So. My best estimate for partner-to-vaccinated-partner transmission, given how often we see each other, is 30% over two weeks, so for every 10 microCOVIDs I accrue, 3 spill over onto you.

My partner, the enthusiastic optimizer: Indeed. Using the 100-to-1-to-0.5 exchange rate between microCOVIDs, micromorts, and life-hours, this means that you cost me about one hour of my life for every 600 microCOVIDs you accumulate. However, a good Coasean would point out that this externality is the result of a joint decision -- your decision to take risks, plus my decision to date you during your infectious period -- so I'm willing to split that cost 50/50. Post-dinner cleanup usually takes half an hour; how about, for every 600 microCOVIDs you accrue 2-12 days before a date, you handle cleanup on a night when it would normally fall on me?

Me: And vice-versa? Every 600 microCOVIDs that you accumulate, you--

Them: Of course.

Me: Sounds fair!

Them: Oh, and-- each of us has other connections, housemates and such, who might not want to interact with us when we're high-risk. The math isn't clear to me off the bat, but we might ballpark that as doubling the cost of each of our microCOVIDs. Change from 600 microCOVIDs per dishes to 300?

Me: Sure!

Hmm. No risk budgets in sight.

## Externalities requiring costly discussions

Suppose I can also transmit to my housemate?

Me: So. My best estimate for housemate-to-vaccinated-housemate transmission is 30% over two weeks, so for every 10 microCOVIDs I accrue, 3 spill over onto you.

My housemate, the sick and tired of everything about COVID: ...okay.

Me: Alex and I worked out this system where, for every 300 microCOVIDs each of us accumulates in the infectiousness window before a date, we owe the other a get-out-of-doing-the-dishes token, to compensate them for the risk we're imposing--

Them:

Me: --and your facial expression right now confirms my expectation that you would hate this. Can you elaborate on why?

Them: Just... constantly thinking about COVID, weighing every social interaction, opening microcovid.org every time I go outside, keeping track of how many microCOVIDs I'm getting from you so that I can report that number to the two other people I regularly interact with... it would ruin my days. I just really, really want to not think about it anymore.

Me: Okay, sure, I can sympathize with that. I... I'm afraid that the "don't think about it at all" strategy has really high costs for me, since I'll need to make pessimistic assumptions about your COVID risk in order to uphold my agreements with Alex and other folks...

Them: I know, I know. Ugh. How about... I'll just... I won't keep precise counts of my microCOVIDs, but I'll keep enough of an eye on things to know the rough order of magnitude of my risk, and... I don't want to have to keep you constantly updated, but how about, you can safely use 200 microCOVIDs as your pessimistic person-risk for me, and if I need to do anything that puts me over that threshold, I'll let you know.

Me: Let's see, if I assume you always have 200, then I'll have 60 from you on each my dates with Alex, so this arrangement will cost me an average of 6 minutes of dishes-doing per date... yeah, all right, I think I can make this work.

That's starting to sound like a budget!

So, at least in this case, it seems like risk-budgeting is a technique that lets you reduce the amount of time you spend computing/communicating risks, at the cost of sometimes making suboptimal decisions (e.g. skipping an awesome party) because you're thinking in terms of "whether I'll exceed my budget" instead of the underlying costs/benefits.

### Further consequences

So far, we've only explained why people who hate doing risk-discussions keep budgets; but I know several non-risk-discussion-hating people who keep budgets. Why would they do that? Hmm...

My housemate, later: Hey, I was thinking about what you said, about your arrangement with Alex and "every 300 microCOVIDs" -- it sounds like you're keeping the option open to sometimes take on pretty large amounts of risk.

Me: Yeah, probably not often, but it's a possibility.

Them: And... you taking 300 microCOVIDs would give me about 100, a significant chunk of my budget, possibly forcing me to have risk-management conversations I find painful with the other people I've told they can safely bound my risk at 200.

Me: Hmm. Yeah, that sounds accurate. So, if I ever plan to go above... 200, say, which is about 60 for you... then perhaps I should let you know -- which imposes a cost on you, for the emotional toll of both that conversation and the downstream conversations you might have to have with those other people.

Them: Yeah.

Me: So, figuring that those conversations will be about... two hours of unpleasantness, all told?... a cost which I'm willing to split with you 50/50... if I ever go above 200, then I need to save you one hour of time. Payable by doing chores?

Them: Yeah, that sounds good.

So now I have something resembling a budget, even though I, personally, have no particular distaste for risk-discussions.

## Conclusions

• For people whom risk-discussions cause significant pain, risk budgets serve as a tool for reducing the amount you have to talk with people about COVID: set a limit on how much risk you will ever take on, and let other people assume you're at that level. When you go over that limit, you tell your contacts about it, and endure the painful discussions.

• For people who both associate closely with people whom risk-discussions cause significant pain, something that resembles a risk budget falls straightforwardly out of cold hard utility-maximization: if you go above a certain threshold, then you have to have notify a close-contact risk-discussion-hater; which causes them pain; which you probably either feel bad about or compensate them for; which reduces your utility.

• For people who aren't close contacts with risk-discussion-haters, I don't think risk budgets make sense. Instead, you simply track your microCOVIDs and share your risk info with your contacts, so everybody can do normal cost/benefit analyses like they're hopefully used to, without "budgets" appearing anywhere in the calculus.[3] You probably do not have enough close contacts for the communication overhead to be significant.

1. 100-to-1 is a guess at what my audience, i.e. you, thinks the microCOVID-to-micromort exchange rate is, judging by things like microcovid.org's recommended budget of 200/week. As best I can figure, though, for a healthy 30-year-old the actual exchange rate is about an order of magnitude higher, even including non-death outcomes like long COVID. I'm writing this footnote to assuage my guilt over contributing to a false appearance of consensus around "100 to 1" when I don't endorse it. ↩︎

2. Edit: okay, I thought of a reason: maybe I don't trust myself to make smart cost-benefit assessments on a case-by-case basis, and I think my brain will routinely exaggerate the benefits of risky activities, and my psychology is such that drawing a bright line and saying "this is my budget, I can't go over this" will at least bound how much risk my lying brain can trick me into. But I think that few of the people I see maintaining "risk budgets" would give this as the reason. ↩︎

3. Well, maybe your friends will have "under-300-microCOVID-people-only" parties, and those will add discontinuities to your utility function that look kind of like budgets, but I think the similarity is only superficial. ↩︎

# 54

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[-]gjm3y90

I'd make an analogy with ordinary budgets.

One sort of budget is where you have a rigidly fixed amount of money to spend -- your income is X per month, you need to spend something like X per month in order to have a decent life, you don't have a lot of savings, so you will be spending about X per month and you just have to figure out what on. But even if you are comfortably out of paycheck-to-paycheck mode, you may still sometimes think in terms of budgets.

Suppose I'm buying a computer, a car, a camera, a bicycle, a painting to hang on the wall. There's a big difference between how much I could spend without breaking the bank and how much I'm actually willing to spend. In principle the right way to make my choice is to figure out what utility I'll get from each possibility, figure out what utility I'll get from having any given amount more savings, and choose whatever maximizes the total. In practice that's rather difficult. A common solution is to first pick some amount of money that seems reasonable: "I'll allow myself to spend up to \$600 on this", "I expect to end up paying about \$3000", etc. And then to go shopping and be guided by that budget. You might end up substantially over or under that initial budget -- maybe it turns out you underestimated what you have to spend to get a good enough car, maybe you hadn't known that there's an unknown artist who paints things exactly to your taste and sells them cheap, maybe you see a particular thing that you fall in love with. But the budget is still a useful tool for focusing your search.

What justifies all this is that there's some sort of tradeoff between cost and value-to-you, and you can make a rough estimate of where on that curve without doing expected-utility calculations; e.g., to whatever extent your utility function is like others', you'll probably want to spend about the same amount on this thing as other people of roughly similar wealth and interests spend.

I think risk budgeting (and, in particular, COVID-19 risk budgeting) is just like this. It's not that you have a rigidly fixed amount of risk you can take. (I mean, in some sense you surely do, because if you do things with a 10% chance of death every week then you won't last long, just as even a multimillionaire with a large salary can't literally spend unlimited amounts on everything, but you're mostly operating with levels of risk that aren't like that, just as the multimillionaire is mostly not going to run out of money even if they buy a fancier car.) But risk has a utility cost, just as stuff has a financial cost, and rather than trying to do expected utility calculations every time you do anything it may be easier to set a rough amount of risk you're willing to incur in a given context.

Just as with financial budgeting, sometimes you'll see something particularly attractive and say "screw the budget". There's a party you think you'll really enjoy (translation: you expect the utility/risk tradeoff for this to be much more favourable than usual) so your usual budgeting heuristic needs to be adjusted. You should still look at the risks, but this is a one-off so you're willing to take the trouble to think explicitly about the expected utilities and decide: yes, it's fine.

That's exactly like the situation where you're shopping, you have a rough budget in mind, and you see something way over budget you really like, so you think "OK, is this really worth it for me?" and start doing calculations like "if I buy this and make up for it by not buying X, Y, and Z, is that a good tradeoff?" or "do I value this as much as X, Y, and Z which I spend similar amounts on?" or "if I buy this and don't do anything to make up for it, it means I have to wait a week longer before I retire; is it worth a week of work?".

(As with all budgeting, you should really be deciding policies rather than initial purchase choices; think not "how much do I want to go to this particular party?" but "if I decide to go to this particular party and am consistent, at what rate will I be going to such parties, and how much overall rate of risk does that expose me to, and how do I feel about that?".)

This analogy with ordinary budgets feels solid enough to me that I think it's sufficient to explain why risk budgeting (in general, or for COVID-19 in particular) is a thing; the specific phenomenon mentioned here -- the fact that your own risks may spread to other people who don't want to have to think about COVID-19 risks all the time -- is one more reason, but I don't think it's the main one.

Thanks for the thoughtful counterargument!

Things I think we agree on:

• you should really be deciding policies rather than initial purchase choices

Yes, absolutely, strong agreement.

• "Deciding how to accumulate COVID risk" closely resembles "deciding how to spend a small fraction of your money," but not "deciding how to spend a large fraction of your money": when money is tight, the territory contains a threshold that's costly to go over, so your decision-making process should also contain a threshold that shouldn't be gone over, i.e. a budget; but there is no such nonlinearity when accumulating (normal amounts of) COVID risk, or when spending a small fraction of your money.

• In principle the right way to make my choice [of what to buy] is to figure out what utility I'll get from each possibility, figure out what utility I'll get from having any given amount more savings, and choose whatever maximizes the total... A common solution is to first pick some amount of money that seems reasonable... And then to go shopping and be guided by that budget.

Actually, I'm not sure I disagree with any of your explicit claims. The only claim I think we might disagree on is something like "budgeting is a good strategy even when costs/benefits add pretty much linearly," as in the 'spend a small fraction of your money' or 'accumulate COVID risk' scenarios: I perceive you as agreeing with that statement, whereas I disagree with it (because it encourages you to think in terms of "whether I'll exceed my budget" instead of the ground truth).

If you do endorse that bolded statement, I'm curious why. I read your comment as explaining why people do budget in low-stakes scenarios, but not why that's optimal. (My best guess at your answer, reading between the lines, is "because it saves a lot of error-prone calculation," which, hmm, doesn't speak to me personally, but people differ, and maybe I overestimate my own ability to do good cost/benefit calculations.)

(Don't get me wrong, I do sometimes do something that looks like budgeting, as you describe, when I'm spending small amounts of money; but I view it as a bad habit that I want to break myself of -- with a proper eye towards TDT, though, of course.)

[-]gjm3y30

(I don't know why I wrote "initial purchase choices" when I meant "individual purchase choices", but obviously it was comprehensible anyway.)

As for whether budgeting is ever a good idea when the amounts are small enough for utility to be close to linear -- I think it does two useful things: it saves cognitive effort, and it may help you resist spending more than, on careful and sober reflection, you would want to. How often those are worth the utility-loss from using a cheap approximation will vary.

More discussion here.

Fantastic. Thanks so much for that link -- I found that whole thread very enlightening.

(If the coffee shop had been a week ago, sure, I'd be inflicting some of those microCOVIDs on my fellow partygoers, which, sure, could be decision-relevant, I haven't done the math; but it seems very unlikely to me that I'll become a full-fledged germ factory between this morning and this evening, so I think that consideration is insignificant in this case.)

Between morning and evening seems negligible indeed, but it's worth noting that the Delta variant has a much shorter incubation period. There are cases where it takes only about 24h for people to infect others.

Yes, agreed! An earlier draft had the exposure happening "yesterday" instead of "this morning," but, yeah, I wanted to make it clearer-cut in the face of the reports I've heard that Delta has very short incubation periods some nonzero fraction of the time.

It seems to me that if I make some reasonable-ish assumptions, then 2 micromorts is equivalent to needing to drive for an hour at a random time in my life. I expect the value of my time to change over my life, but I'm not sure in which direction. So equating 2 micromorts with driving for an hour tonight is probably not a great estimate.

How do you deal with this? Have you thought about it and concluded that the value of your time today is a good estimate of the average value over your life? Or are you assuming that the value of your time won't change by more than, say, a factor of 2 over your life?

That's a great point! My rough model is that I'll probably live 60 more years, and the last ~20 years will be ~50% degraded, so by 60 remaining life-years are only 50 QALYs. But... as you point out, on the other hand, my time might be worth more in 10 years, because I'll have more metis, or something. Hmm.

(Another factor: if your model is that awesome life-extension tech / friendly AI will come before the end of your natural life, then dying young is a tragedy, since it means you'll miss the Rapture; in which case, 1 micromort should perhaps be feared many times more than this simple model suggests. I... haven't figured out how to feel about this small-probability-of-astronomical-payoff sort of argument.)

I agree with your points about risk budgets not being logically necessary because risks are generally linear and independent. But I think about risk budgets as a supplement to the normal decision-making process, because people aren't good at making one-off decisions in the abstract.

If the party costs 1 hour of expected life, I'll go. If it's 2 hours, would I still go? 3 hours? It's hard to really wrestle with that decision and figure out where the tradeoffs aren't worth it. "1 hour" is already a more concrete framing than "2 micromorts", but you're still trading off a concrete outcome (going to a party) with something pretty abstract (1 hour expected value of life), so it's hard to make logically consistent decisions that don't vary with your mood, etc.

If, instead, you set an overall weekly risk budget, you can then trade off between specific concrete choices. "If I go to this party, I can't go to the grocery store. Which would I rather do?"

So it helps to make yourself more consistent and make your decisions more concrete. Help you "set policies" (and be consistent with them) vs. "making initial decisions" to gjm's point in the other comment.

Would be interested in your thoughts!

• Hmm! I think the main crux of our disagreement is over "how abstract is '1 hour of life expectancy'?": you view it as pretty abstract, and I view it as pretty concrete.

The reason I view it as concrete is: I equate "1 hour of life expectancy" to "1 hour spent driving," since I mildly dislike driving. That makes it pretty concrete for me. So, if there's a party that I'm pretty excited about, how far would I be willing to drive in order to attend? 45 minutes each way, maybe? So "a party I'm pretty excited about" is worth about 3 micromorts to me.

Does this... sound sane?

• I'm in a house that budgets pretty aggressively, so, in practice, I budget, and maybe I'm wrong about how this would go; but, if I ditched budgeting entirely, and I was consistently bad at assessing tradeoffs, I would expect that I could look back after two weeks and say, "Whoa, I've taken on 50 life-hours of risk over the last two weeks, but I don't think I've gotten 50 hours of life-satisfaction-doubling joy or utility out of seeing people. Evidently, I have a strong bias towards taking more risk than I should. I'd better retrospect on what I've been taking risk doing, and figure out what activities I'm overvaluing."

Or maybe I'm overestimating my own rationality!

I've also seen a couple of variations on risk budgets in group houses, along the lines of: the house has a total risk budget, and then distributes that budget among its members (and maybe gives them some way to trade). In the case where the house has at least one risk-discussion-hater in it, this might make sense; but if everybody is an enthusiastic cost/benefit analyzer, I strongly suspect that it's optimal to ditch the budget, figure out how many housemates will get sick if a single person gets sick (e.g. if housemate-to-housemate transmission is 30%, then in a 3-person household, one sick person will get an average of about 0.60 housemates sick), and use that to institute a Pigouvian tax on microCOVIDs, exactly as in the "enthusiastic optimizer" example.

[-]gjm3y30

Pedantic note: if housemate-to-housemate transmission is 30%, then there's a 30% chance that you infect each of your two housemates, which indeed gives 0.6 sick housemates -- but in the case where you infected exactly one of them (0.42 of the time) there's then a 30% chance that they infect the other one after all, giving an extra 0.126 sick housemates for a total of 0.726.

(Well, maybe. Perhaps that 30% figure is partly because some people are just harder to infect, so that conditional on your having infected A but not B, B is then less likely to get it from A.)

Pedantry appreciated; you are quite right!

I was planning to say this too.

IIRC, covid risk budgets arose in a group housing context - the idea was that it was an equitable way to balance the risk that your activities were presenting to your housemates, and to prevent the more risk-loving housemates from unfairly exposing risk-averse housemates to undue dangers.

If you're not concerned about the risk to people close to you, or if they're defectors who are doing whatever they want anyway, then a covid risk budget makes less sense, and OP's cost-benefit analysis makes more sense.  Of course, as pointed out upthread, if you don't trust yourself to make reasonable long-term decisions in the moment, then committing to a budget is a pretty good way of lowering overall risk.