Probability, polarization and professions of nihilism

There’s a frustrating tendency among decision makers and commentators these days to adopt a position of hopelessness in the face of difficult problems. The public policy response to COVID-19 in the US is a good example of the phenomenon. A recent specific example, which generated a flurry of discussion on social media, is David Leonhardt’s piece in the New York Times on the 9th of March.

Leonhardt expressed skepticism about the value of masking in light of the infectiousness of the Omicron variants. His argument amounts to the proposition that, since the probability of getting infected isn’t numerically much smaller if you wear a mask than if you don’t, wearing a mask is pointless. Here is what he wrote:

“I’ve come to think of the point this way: Imagine that you carry around a six-sided die that determines whether you contract Covid, and you must roll it every time you enter an indoor space with other people. Without a mask, you will get Covid if you roll a one or a two. With a mask, you will get Covid only if you roll a one.

You can probably see the problem: Either way, you’ll almost certainly get Covid.

This analogy exaggerates your chances of getting infected, but it still highlights the basic reason that masks and distancing have had a limited effect”.

The nihilistic message “either way you’ll almost certainly get COVID” encapsulates the problem. It seems to be rooted in a kind of fever of thinking in absolutes that has invaded public life in the last couple of decades. Telling people that there’s no point in doing something because it’s not completely effective reflects a belief (maybe it’s just an adopted position?) that the world is composed entirely of situations that are either one thing or another: You’re either a Republican or a Democrat; you’re for us or against us; if you don’t agree with me, you believe the opposite of what I believe; if masking isn’t 100% effective, it’s useless.

Putting this sort of message in a public policy debate is corrosive; especially since mask wearing has been heavily politicized. Portraying it as an all-or-nothing tactic only gives encouragement to the anti-social factions who put their peculiar, and carefully circumscribed, definition of personal liberty ahead of the collective good. Beyond this recklessness there’s also the problem that Leonhardt’s questionable logic is cloaked in a pseudo-mathematical argument about probability; I’d guess that his numerical ruminations sound just believable enough to make the argument persuasive to the unwary.

Let’s tease apart Leonhardt’s model. Later on I’ll make a slightly more realistic version of it, but for now I want to focus on the issue as he presented it. Part of the problem with his analysis is that it breaks Einstein’s dictum to make our models as simple as possible, but no simpler. This is also where it connects with my point about the pervasive all-or-nothing mania that seems to have invaded our lives; his division of outcomes into useful and worthless is overly simple; a position he backs up with an overly-simple analysis of some completely hypothetical probabilities.

At this point I have to issue a warning: in what follows there is some maths. If you don’t want to read the mathematical reasoning, you can go straight to the charts (scroll down) which show the outcomes.

Leonhardt’s model deals with what are known as conditional probabilities — the chance that something happens given that some other stuff is true; for example, the probability that you get COVID-19 given that you wear a mask. The standard notation for expressing conditional probabilities uses the vertical bar, |, to represent the word “given”. We can rewrite the statement about COVID and masks as:

Pr(COVID|mask), where Pr() means the probability of whatever is in the parentheses, and “COVID|mask” means catch COVID given you wear a mask.

With a six-sided die Leonhardt’s model amounts to:
Pr(COVID|mask) = 1/6
Pr(COVID|no mask) = 2/6 = 1/3.

On this basis, which he concedes has unrealistically high probabilities, he concludes that it’s pointless to wear a mask because the difference between 1 in 6 (= 0.167) and 2/6 (=0.333) won’t make any practical difference; in his words “either way you’ll almost certainly get COVID”.

It sounds plausible, doesn’t it? Your intuition might well agree with his assertion that the difference doesn’t amount to anything worthwhile. But if we slow down our thinking (as Nobel laureate Daniel Kahneman suggests we should, when dealing with a problem like this) the facts behind the intuition start to tell a different story.

Suppose Leonhardt’s example actually was a good approximation to reality. What would the outcome look like for two groups of people in that world, one group who did wear masks and one who didn’t? Let’s run the Gedankenexperiment and imagine we could do a matched cohort study in that world.

We imagine the availability of two groups of individuals, 100 in each group, who are perfectly matched in every sense so that there are no possible confounding factors in play. Furthermore, these two groups are going to behave identically in our imagined world, inasmuch as they are going to participate in identical numbers of identical social events with exactly the same probability of catching COVID-19, except the difference caused by the choice to wear a mask, or not. And, those differences are going to be determined by Leonhardt’s dice rolling.

First, the cohort who are not wearing masks. Each time one of these people enters a social situation the probability that they catch COVID is 0.333, so the probability that they do not catch COVID is 1–0.333 = 0.667. In the strange world of our Gedankenexperiment it’s 0.667 for the first event, and for the second, the third and so on. After two bouts of social interaction the probability that a non-masker is still uninfected is 0.667 multiplied by 0.667, or 0.667 squared. Generalizing this, we can see that after n social exposure events the chances that a non-masker is still COVID-free is 0.667 to the power n, and the probability that they have contracted COVID after n bouts of interaction is 1–(0.667 to the power n). Even for relatively small values of n, 0.667 to the n is almost zero, so the probability of catching COVID is close to 1: 1 minus something close to zero, is almost 1.

For the maskers the reasoning is exactly the same, except that everywhere we wrote 0.667 in the previous paragraph we plug in the value 0.833 (i.e. 1–0.167). The chart below shows the expected outcome for Leonhardt’s dice-rolling world. The gap between the blue line and the red line is the gain, in still-healthy people, in mask-wearing versus not, as the number of encounters with risky situations (n) increases.

At the right-hand end of the graph, we see the outcome that Leonhardt is focused on — in both cases, eventually nearly everyone ends up infected. When the number of encounters is relatively low, wearing a mask has a benefit across the population. How much benefit? This much:

Of course, the social benefit doesn’t guarantee anything at an individual level, but it reduces the burden of disease at a social level, and it provides a social-level goal to which we can all contribute. At the extreme left-hand end, the two lines meet at the point where nobody interacts with anyone else (0 social encounters, n=0) so there’s no transmission and everyone who is healthy stays healthy.

One of the problems is that messages about averages across a population don’t speak to individuals, especially individuals raised in a society focused on personal liberty, “freedom”, and pushing back against government interference (as they might see it) in daily life; people are inclined to say, “OK, great, but what about me as an individual? — I don’t experience the “average” outcome; no-one does”. How to make public policy work for a reluctant public is a whole other conversation, but since the crux of the problem might lie in individual outcomes, for now let’s take a look at what happens at an individual level in our cohorts of 100. We’ll do this by rolling Leonhardt’s dice for each person in turn. Actually, rather than dice, it’s easier to think about this in terms of a kind of card game, but I will set up the game to preserve the chances captured in Leonhardt’s dice.

First, we give each of our 100 maskers and non-maskers a “mask status card” with their “chance” value (0.167 or 0.333 respectively) written on it. Then, they each visit 20 social situations. At every event there is a second card waiting for each person. The cards are for pairs of people, with one masker and one non-masker named on each card. On the card there is a second number, drawn at random in the range from 0 to 1. Every number in that range is equally likely to be on each card. The player compares the number on the card they are given at the door, with their mask status card (again, 0.167 for maskers, 0.333 for non-maskers). If the number on the card they get at the door is larger than the number on their mask status card they stay healthy; if the number is equal to or smaller than their status card, their number’s up (literally) and they get COVID at that event. We keep track of how many events people go to before they catch COVID and track the cumulative number infected as the number of events increases. The results look like this:

In the model world, among non-maskers, about 40% are infected after just one social event, and nearly 60% are infected after two events. All 100 are infected after 13 events. In contrast, among maskers, 60% don’t get infected until after 6 events, and by the end of the sequence of 20 events there are still 4 who remain healthy (they are the little uptick in the blue line at the right-hand end, because I piled them all onto event 21 — it might have taken even longer for some of that final group to succumb). I’m not sure I’d conclude that this is a pointless difference; and remember this is, in Leonhardt’s own words, an “analogy [that] exaggerates your chances of getting infected”.

Rather than illustrating that wearing a mask is pointless I’d say what the example shows is that if you’re going to make up numbers and add hypothetical dice rolling to your argument to give your pre-chosen point of view a veneer of rigor, but you don’t actually think through your example, you might as well do without the mathematical distraction and cut straight to the naked opinion; which in in this case appears to be I don’t believe there’s any point in wearing a mask.

All of this is bad enough, but the main problem isn’t that Leonhardt used some mathematical thinking to look at the issue. The problem is that his model is too simple to be useful, at least in the way he used it. It makes two huge assumptions that can lead us (and him, apparently) astray. First, there is a crucial assumption that mask-wearers put themselves in just as many risky situations as non-maskers. Secondly, it assumes that the situations which mask-wearers put themselves in are just as risky as ones chosen by non-maskers. In truth, lots of risk-averse people combine all three risk mitigation tactics; (1) they wear masks, (2) cut down on mixing with other people (i.e. go into social situations less often), and (3) avoid high risk situations (for example, by avoiding stores, bars and restaurants that do not enforce mask mandates).

The reason that I’m labouring these points is that in Leonhardt’s description of the problem with masking he said:

“I’ve come to think of the point this way: Imagine that you carry around a six-sided die that determines whether you contract Covid, and you must roll it every time you enter an indoor space with other people.”

But, in the remainder of what he says he ignores the number of visits to indoor spaces and focuses only on what epidemiologists call the instantaneous probability of infection, associated with each single indoor visit (the thing that he captures in his dice rolling). As we have seen, the conclusion that there’s no difference between 0.167 and 0.333 only applies if there are enough visits to risky places to allow the cumulative probability of infection to equalize. It’s not sufficient to consider only the instantaneous probability of infection when we are thinking about the implications of behavior.

I’ve added a little complexity to Leonhardt’s model to adjust it to look at what happens when people who wear masks also reduce their frequency of social interaction and avoid the riskiest situations. To keep things simple, I assume that mask wearers cut the frequency of their social exposure events in half (compared with non mask-wearers). To reduce the chances that mask wearers are exposed to as many high risk situations, we select the “cards” for the events from the range 0.1 to 1.0 (instead of 0 to 1), so there is less chance than in the original model of an event having a card value less than the value on the mask status card. Keeping everything else the same as before and adding in these new effects the results for the new model look like this:

Now the difference between maskers and non-maskers looks a lot more dramatic. As a reminder, what has changed in this version of the game is that (1) maskers only put themselves in half of the potential social situations that non-maskers do, and (2) they avoid high risk situations, reducing their chance of infection. By the time the masking cohort gets to their 10th interaction (at the point when 20 possible social events have happened) only 50% of them are infected. The remaining 50% (still uninfected at the end of our Gedankenexperiment) is piled up on event 21. Since the trend for their infection rate is pretty close to linear, it might well have taken something like the equivalent of 40 non-masking events to get all the maskers infected.

At the time all the non-maskers are infected (i.e. when they have visited 13 social exposure situations) the masking cohort has an infection percentage of roughly 30%. The point, here, is clear. The combination of a few simple behavioral changes, none of which appears to do very much on its own, can result in quite marked differences in outcome. This is why the nihilistic messaging about the pointlessness of these individual steps is so frustrating; they don’t have to be perfect to be useful, especially when used in combination.

As I write this, the USA has all but given up any pretense of trying to cut infection rates through collective behaviour, and mask mandates have been dropped almost everywhere. In a rush to press the population back to a semblance of the old normal the cynics have scored a victory for a nihilistic, impoverished perspective that will likely damage the lives of the poorest and most vulnerable most. In a society which has greater wealth inequality than at almost any point in history, and in which the cost of healthcare can bankrupt people, it’s not just a bureaucratic failure of public policy that people have been persuaded out of taking simple preventative measures to protect themselves, it’s a failure of morality.

If you want the R code to recreate the figures or play around with the models, you can download it here:
https://github.com/nmcr01/Masking

Afterthought: It’s often said that putting maths in anything intended for absorption by policy makers and the public reduces the audience by 50% for each equation used. I’m not convinced that that’s even close to being true; it certainly doesn’t line up with my experience. In the specific case of the messaging about COVID and preventative actions, I think that we missed a trick in not using simple general models to show how different types of action combine. The general model we’ve been exploring in this post is:

Pr(COVID|mask) = 1-(1-p)^n,

where, p, is the instantaneous probability of getting COVID in a single social event (i.e. Leonhardt’s dice rolls) and ^n means “to the power n”, with n being the number of social exposure events experienced. Even if we don’t try to understand the equation in detail, we can still use it to point out that actions that affect the value of p (such as wearing a mask) interact with the number of times we roll the dice, n. This serves to reinforce the idea that we can’t, as Leonhardt tried to, consider one without the other.