Exponentially counterintuitive

How can we get better at communicating the idea of accelerating growth?

Suppose a single drop of water falls inside Wembley Stadium. Now imagine that the drop doubles in size every minute from this point onwards. How long before the stadium – all 4 million cubic metres of it – is full to the brim of water?

The answer: 46 minutes¹.

Even if we’re familiar with the concept of exponential growth, it can still stretch our intuition. Something is very small, then quite small, then still pretty small, then SUDDENLY IT’S MASSIVE.

During the COVID pandemic, exponential growth – and its quirks – became a crucial part of public and policy debates. Infectious disease epidemics commonly start with compounding growth: one person might spread an infection to two others, and these two each spread to two others, and so on. From Ebola to COVID, this is why we often see early growth follow such an acceleration.

One of my most widely shared tweets of the entire COVID pandemic – in the early stages of the Alpha wave – made the point that deaths from a pathogen that is 50%  more transmissible (i.e. Alpha) would quickly outstrip deaths from a pathogen that is 50% more fatal². This was because the transmission rate scales case numbers exponentially, but severity scales only case numbers by a fixed amount. Even years later, I still occasionally see adapted visualisations of this tweet pop up on social media.

We see similar compounding growth in other areas of life. If bank savings earn interest, we then earn interest on this interest, and interest on the interest on the interest. Albert Einstein reportedly once called compound interest the ‘Eighth Wonder of the World’ (but probably didn’t). ‘He who understands it, earns it. He who doesn’t, pays it,’ as Einstein (or at least, the person misquoting him) put it. Similarly, COVID policy responses and public debates could often be divided into those who appreciated the implications of exponential growth – and those who didn’t.

In an infectious disease epidemic, failure to account for exponential growth in the early phases can lead people to dramatically underestimate the resources required to control infection. Later on, it can also lead to erroneous underestimation of the impact of control measures. And it’s not just epidemiologists that must think about exponential growth. Whether it’s a start-up analysing user growth and retention, a central bank estimating the risk of financial contagion, or a cybersecurity agency looking at the spread of malware, exponentials crop up across industries.

In a situation where we face exponentially growing problems, we therefore need to think about how we can improve wider understanding of exponential processes.

Learning to be non-linear

Why is exponential growth so counterintuitive? One reason is likely to be familiarity: much of our day-to-day life involves linear growth. If you run a bath for twice as long, you’ll use twice as much water. Run it for three times as long, and you’ll use three times as much. Similarly, how much you pay for petrol scales with how much you put in the tank. None of these tasks involves the shock of an exponentially increasing outcome.

But the linear outlook isn’t necessarily innate to humans. In Alex Bellos’ fascinating book Alex’s Adventures in Numberland, he discusses a series of studies looking at how different groups of people view the relationship between numbers. Whereas most Western adults will position the numbers 1 to 10 on a traditional number line, others view numbers differently. For example, some indigenous Amazonian villagers view numbers in relative terms: the distance from 1 to 2 is the same as the distance from 4 to 8. Similarly, researchers have found that American kindergarteners also place numbers on a line according to their relative values. But by second grade, this viewpoint has made way for a more traditional linear scale.

Communicating exponential growth in a linear culture

From infections and investments to marketing and malware, understanding exponential growth is important to many of the problems people care about. So, here are a few suggestions for how I think researchers can get better at communicating the implications of exponential growth:

• Encourage people to try it out. It’s difficult to communicate exponential growth effectively just by telling people it’s a thing. One of the best ways to get familiarity with a mathematical concept is to put it into practice. For example, if you don’t believe me about Wembley stadium filling with water in 46 minutes, put the numbers into a calculator and see for yourself. Or do a Richard Herring and promise your partner twice as many Ferrero Rocher every year, to get a feel for just how quickly things can get out of hand.

• Put exponential growth on a more natural scale. If disease cases rise from 500 to 1000 in a week, 8000 cases may still seem far away on a traditional linear scale. So rather than saying exponential growth will make it 8 times larger in 3 weeks, we could instead frame the problem in terms of how many times it doubles. We’ve just had a doubling in a week, and it will only require three more doublings – which will take three weeks, if things continue – to get us to 8000. This has the benefit of translating the metric of cases, which scales exponentially over time, into the metric of ‘number of doublings’, which scales linearly with time (i.e. each week means another doubling).

• Rethink visualisations. There can also be advantages to changing the scales of our visualisations. For example, if we plot data on a logarithmic scale (i.e. with the vertical intervals spaced ten-fold apart: 10, 100, 1000 etc.), we can look at the slope of the line to see the rate of growth. Researchers like Oliver Johnson have therefore shared many COVID plots making use of such scales. The visualisations generated by journalists like John Burn-Murdoch are also a good example of how to combine logarithmic scales with the concept of doublings. This approach is particularly powerful if we already have data on situations that have gone downhill exponentially, and hence illustrate trajectories we really don’t want to be on.

  

  Source: Twitter

• Share rules of the thumb. Exponential growth can to tricky to navigate, so it can be helpful to have simple calculations that give a sense of the situation. For example, the ‘rule of 72’ tells us that if something – whether an infection or investment – is growing at x% per week, it will typically take roughly 72/x weeks to double in size. Another useful rule of thumb is the fact that if you double something 10 times, it will end up about a 1000 times larger.

• Encourage people to make predictions. Aversion to imagining the impact of exponential growth can also lead to aversion to remembering this lack of imagination. Hence there can be value to writing down what we think will happen – even if just for our private reflection – then comparing it to the subsequent reality. I’ve sometimes wondered how much better COVID discourse would have been if policymakers and media commentators had routinely done this. It is much easier to understand our inability to appreciate exponential growth if we have documented the predictions in real-time.

• Explain the theory. Why does a 50% more transmissible pathogen lead to more short-term deaths than a 50% more severe one? Working through the logic of counterintuitive features of exponential growth can lead to memorable ‘aha’ moments. What are the conceptual tools we need to make sense of a situation? And what is the most straightforward way we can get these tools into people’s hands? Take that viral post of mine in January 2021, and all the subsequent visualisations it inspired. I hope it went at least some way to helping people appreciate the implications of exponential growth during the Alpha wave. Even if it was perhaps just a drop in the ocean.

1. This assumes a drop of water is 0.05ml. Given the stadium has a volume 4 million cubic metres (i.e. 4 billion litres = 4 x 10⁹), then 0.00005 x 2⁴⁶ = 4 x 10⁹

2. I focused on timescale of weeks in that thread because that was a crucial period for the UK given the vaccine rollout. But theoretical tradeoff could have been different in longer term if no were vaccine available, which I discussed more in this follow up thread. Unfortunately, it would turn out that Alpha was both more transmissible and more severe. But it was still the increased transmissibility that caused more problems.