COVID: Stop Arguing About the Physics; The Maths Has It

Acceptable Risk

Well, the “it’s ok” brigade put me in a lecturing mood. So here it comes: You might think some measures you can take to “protect yourself” from COVID have a very small effect on your chances of getting it or even of spreading it. You are definitely right, as your chances of getting it right now are quite small as it is (depending a little where you are and when exactly we are talking about of course), and you can’t spread it unless you got it. But that is missing the point, because it’s not about “protecting yourself” only, and it’s most definitely not about “protecting yourself right now”, and it’s also not just about “protecting others just in case you have it and you don’t even know”. If it was, I might not even adjust my own behaviour much to be honest: Right now, today, the risk dial for almost everyone of us personally might read “Acceptable”. I get that. But that does not mean you shouldn’t aim at tiny reductions in what is an already tiny risk for you today. It isn’t just about “you today”, it’s about us all in the future. The maths doesn’t lie, and it’s really simple, so bear with me:

Compounding small (changes of) risks

If everyone reduces their chances of getting COVID tomorrow, say, from 0.01% to 0.008% (that’s a one in 50,000 swing in probability, a reduction to only 4/5ths of the tiny risk that it was to begin with), then this tiny swing (if everyone did it) will more than halve the R factor for all. Why? Because the time from when you catch the virus to the time when you are infections is several days, say n days for argument’s sake, then your contribution to the R factor decreases by a factor of (4/5)n . If, say, this period between catching the virus and being able to pass it on were four days for you, then your “personal” contribution to the R factor of the society you live in reduces by more than half, just by making a tiny change to an already acceptable risk. If almost everyone did that, this tiny reduction in everyone’s “personal” risk is reasonably likely to make the difference between the magical reproduction factor R being larger than one or smaller than one, between an outright epidemic and practically snuffing out the virus. But there’s more: It reduces everyone’s chances to get it in two months by a factor of — well, if you followed me this far, you can work it out — (4/5)61 which is somewhat close to 1/1,000,000. Everyone’s chances to be infected in 2 months would drop by literally 99.99988%. Yes, this truly is the difference between an epidemic and snuffing out the virus! I don’t know if Einstein really did say that nothing in the universe is as powerful as the power of compounding, but this is the power of compounding.

The signalling value of masks

And while we are at it, here’s a different point altogether: any discussion which focuses on the “physical” effectiveness of masks misses an important point. Even if everyone wore a useless knitted mask, these would still act as a reminder to everyone, all day long, to do things they should (wash their hands, keep a distance), and avoid things they shouldn’t do (shake hands, form “random” crowds at bus-stops or in front of the ATM, whatever…), and it reminds everyone to adjust their behaviour to their constantly changing risk levels (random example: save your hand sanitiser when on a hike in the country; You are going to need it on the crowded bus home). So even knitted masks would still have a public health effect! If we all just wore a covid reminder sticker on our foreheads, that too would have a beneficial effect.

And because of the extreme power of compounding, even just this “signalling” value of masks will have a real public health effect before too long. People today were joking on Facebook about Malta’s health minister wearing a mask in front of his video conferencing laptop. I hope to have demonstrated that this is in fact beneficial to us all in a very real, if indirect way. We should all wear the effing mask like it’s a flag, because it actually is. It’s right there in your face! People won’t miss it. Stop worrying about the physics and just care about the maths. It may not have made a blind bit of difference to the health minister’s risks that he wore his mask there and then, and it will have made close to zero difference to the (maybe zero) people near him at that point. It will have made zilch of a physical difference to his viewers on the other end of the line as well, just like it may only make a tiny difference to you today if you wear one today. But it will also have been a visual confirmation to all about what is being talked about and what is being taken seriously. It sends a non-verbal message that is still being received sub-consciously even while he was anyway talking about COVID (I presume). This behaviour, this signalling effect, will have made a massive difference to everyone two months down the line but only if most of us keep doing it. Not because the mask will have actually protected many of us from an infection we would otherwise have actually got. For each one of us individually, that ‘actual’, physical prevention event is unlikely to occur. But because everyone’s mask reminds everyone to be mindful, every time they see one, that COVID is still with us, it changes their other behaviours in small ways, maybe even only for a minute or two. That’s what the health minister’s mask will have done to his viewers on the other end of a virtual meeting. Pointing out that his wearing it was physically pointless is literally missing a point here.

Public health measures and game theory

The compounding effect of these very small behavioural changes also applies to all kinds of other risk-reduction measures set by your regional chieftains. You just have to understand that these measures aren’t there to protect you, today. You have to stop thinking about the physics and start thinking of the maths. And the question isn’t “should we really put the brakes on the economy to reduce tomorrow’s numbers by maybe 100?”. Your risk today and tomorrow is already low and they know it too. Yet they are doing exactly what a rational and benevolent dictator(1) would do in the game theory image of a COVID-behaviour game.
The behavioural choices we have in the face of the COVID PHEIC form a multi-player prisoner’s dilemma (and thus create a free-rider problem)(2). Such a multi-player prisoners’ dilemma (“MPD”) exists when (i) cooperation is always costly to the individual, but yet (ii) cooperation of all is still net beneficial to the collective. Because the selfish choice in an MPD is never to cooperate under any circumstances, the problem of how to get people to cooperate is fundamentally about how to get people to give up their personal liberties(!), and no-one should apologise for that. There is a part of the political spectrum who thinks that almost any sacrifice of (even personal) liberty is too much. Strangely, often the very same people have no problem at all with governments asking for personal sacrifice in other domains where it’s for the common good, such as public service or military service. But the game theory is the same and makes no distinction between a COVID-liberties-jockey and a draft dodger: they are both free-riders in an MPD game. Make no mistake, these public health measures are akin to a mass mobilisation effort, and they face the same problems any mass mobilisation effort faces. The right and “nice” thing to do is to comply, yet selfishness dictates non-compliance (in game theorists parlance, defection is a “dominant strategy”).

But is it just about compliance of each and every person? Is that all you — individually — should do? No, no, and no! You now see that these health measures aren’t there to protect you today, they are bout bringing the R factor down. So the question you must ask is “what else can Little Me do to bring the R factor down?”. Well, it turns out there is an even easier and more effective behaviour than compliance itself, and that is to “punish”(3) non-compliance of others around you. There are multiple legal and even polite ways of doing that (as well as sneaky and hilarious ones too), so I’m not going to suggest specifics and leave it to your creativity. Societies find multiple ways of making known draft dodgers pay, for example, and that is exactly what evolutionary game theory would predict to happen. Over time, societies find ways to avoid the “tragedy of the commons”, but sometimes only after staring into the abyss. But what game theory tells us about MPDs is clear: The “winning” societies will be the ones who demand and enforce compliance. You aren’t always going to get it voluntarily, yet you need it. The maths don’t lie(4).

(1) the game theory terminology here is telling: a rational and benevolent dictator is present when there is an individual agent or body who has the ability to set and enforce behaviour which would make the participants of a coordination game achieve the collectively optimal outcome. The fact that it is usually called “dictator” already implies that in the absence of one it is the individuals’ “liberties” which will result in a collectively sub-optimal outcome. Indeed, the only mathematically ‘interesing’ coordination games are those which have coordination ‘problems’, meaning the preferences of the individuals detract from achieving the collectively optimal state of cooperation.

(2) Strictly speaking, this isn’t always true. At some level of infections per capita it becomes optimal even from a selfish perspective to restrict one’s liberties and stop ignoring COVID. The point of this article, however, is that long before we even reach that point, when you shouldn’t even be very worried about your personal risks (yet), social distancing, mask-wearing, and the planting of ever-present behavioural clues for everyone are collectively beneficial even if they are individually costly.

(3) again, a telling choice of language often used in this context in game theory: “punishing” in this context means any act which lowers their “score”, i.e. the “enjoyment” of the liberties they might take. Inflicting the risk of financial penalties, reputational penalties, or penalties in terms of loss of other liberties are viable strategies for anyone: dictators, but also less powerful legislators and even other “common” players can implement some or all of them to lower other players’ incentive to take liberties/non-cooperate.

(4) As a technical side-note, the draft dodger’s game isn’t always an MPD. Under some conditions, it can turn into what’s called a “stag hunt”, namely if from a purely selfish perspective I’m better off joining the military campaign anyway if also the army is large enough (i.e. if there aren’t too many draft dodgers already). In that case the selfish incentive is to do whatever everyone else does, no matter what that is: If everyone joins the army, then so will I. If no-one does, then neither will I. But the parallels we draw between COVID-ignorers and draft dodgers remain valid even when the draft-dodger game turns into a stag hunt: punishment of defectors is still necessary to reduce the risk of the outcome where everyone is dodging the draft (and is therefore better off continuing to do just that).

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