The Importance of Thinking Exponentially

(Note to all the math pedants out there: I know that there is a difference mathematically between an exponential curve and a power law curve, but throughout this article I'm going to be using "exponential" in the more colloquial sense -- to describe any rapidly increasing non-linear curve, whether exponential, power law, or something else).

I came across an interesting journal article recently, although it was published back in 2020. Titled "Diverging Sensemaking Frames During the Initial Phases of the COVID-19 Outbreak in Denmark," it examines the differences between Denmark politicians and scientists in their approach to dealing with the pandemic using the lens of Dave Snowden's Cynefin framework.

Before I dig into how all this relates to the title of my own article, let me first give a quick overview of the Cynefin framework for those who might not have heard of it. Google "Cynefin" and you will find countless blog posts and YouTube videos explaining it. Most will have a diagram that looks something like this:

Cynefin is intended as an aid to decision making and problem solving. Different kinds of problems demand different approaches to solving them. Obvious (sometimes categorized as simple) problems are the kind often dealt with by tech support people: well-understood problems that can be solved just by asking a few questions or following a decision tree. Complicated problems are ones that can be solved by experts through the process of gathering and analyzing data. Complex problems often have fuzzy boundaries or many interacting parts, and require experimentation to understand. And chaotic problems are catastrophes where the most important thing is to do something quickly and worry about the consequences later -- any action is better than none at all.

So if you misunderstand the type of problem you are dealing with, you could be banging your head against the wall for a long time as you try to solve it. In the case described in the Danish research paper, this manifested itself as a clash between government officials and health experts as to how quick and severe the actions to mitigate the spread of the virus needed to be. Politicians were acting from within the Chaotic domain, wanting to take quick, dramatic action, while health experts were in the Complex domain, urging them to wait until more evidence was available. This made me wonder whether a better understanding of exponential growth might be a useful indicator to help quickly classify problems.

We humans are notoriously bad at comprehending magnitudes outside those we encounter in our everyday lives. We all have a very good understanding of the difference between a hundred dollars and a thousand. But when it comes to a million or billion or trillion, all the numbers start to blur together.

The same with time: Although it's hard to believe, the Apollo 11 Moon landing is as far back from now as World War I was to the people watching the Moon landing at that time. Cleopatra is 600 years closer to us than she was to the people who built the Great Pyramid. Our movies show a T. Rex attacking a Brontosaurus or fighting with a Stegosaurus, but that was impossible because T. Rex lived 90 million years after the others died off. In our minds, anything after a few generations just gets amalgamated into "the past."

And the same with distance: We like to think it wouldn't be that hard to send a spaceship to the nearest star, but the Voyager spacecraft were launched in the 1970s and are just now reaching the edge of our own solar system. But thanks to evolution, we are only good at easily visualizing these measurements when they are numbers we can count, sizes about the same size as us, times about the same as our lifespan, and distances we could reasonably travel in a day. (YouTube has some good videos to help you conceptualize size (here), time (here), and distance (here and here).

For as bad as we are at conceptualizing scales and magnitudes, we are even worse at recognizing how fast those numbers can grow. The same evolutionary forces that conditioned us to understand only a certain range of numbers also conditioned us to think linearly about how they accumulate. If you think of some of the big problems today such as global warming, COVID-19, even planning for retirement, they often stem from our inability to understand exponential growth.

There is a double whammy when we delay acting while in an exponential situation. Not only do things get rapidly worse the longer we wait, but the options we have also decrease. Suppose I'm cooking in the kitchen and the grease in my pan catches fire. At this point, I could try to smother it or maybe grab the fire extinguisher or even call the fire department. But if I wait until the fire jumps to the kitchen cabinets, my smothering option rapidly declines. And if I wait until the whole kitchen is ablaze, my only option now is to call the fire department.

So think about the three examples I gave above. Nobel Prize winner Svante Arrhenius was the first to recognize the effects of carbon dioxide on atmospheric warming back in 1896. Carl Sagan warned the U.S. Congress about it in 1985. If our brains were conditioned to think exponentially about the problem, we might have had 50-100 years to actively work on it in ways that were much less disruptive to the world economy than solutions that are now being proposed.

A 25-year-old saving $300/month at 8% interest can have over a million dollars by age 65. But waiting just five years to start saving requires them to save $450/month. And if they wait until age 40, it's going to require over $1,000/month.

In the early stages of a pandemic, contact tracing can be a very effective means of stamping it out if done quickly and thoroughly, and coupled with isolation of those infected. But once the virus spreads to hundreds of thousands or millions of people, contact tracing is like a fire extinguisher against a house fire.

But the big question is: How do we know that we really are actually in the early stages of a pandemic, or for that matter any other exponentially growing situation? How do we know if we're in the Cynefin chaotic domain or in the complex domain?

Well, we don't. That's why things are by definition complex or chaotic. But we do have two questions we can ask ourselves that can help us make a decision as to how to proceed:

1. Do we know anything about this situation that might lead us to suspect that it can grow exponentially?
2. What are the potential consequences if we guess wrong?

Imagine we are at the earliest stages of a situation represented by this graph:

To simplify, I've shown three possible situations:

1. The problem grows exponentially, in which case it is important to act as quickly as possible for all the reasons outlined previously.
2. The problem grows linearly, in which case we have some time to think about it, do some testing, and choose the most effective response scaled to what we know the situation will be like at some point in the future.
3. The problem appears but is static (like a chronic ache), in which case we can prioritize it against all of our other problems as to when or even if we choose to deal with it.

Of course, in the early stages of the problem it's really difficult to see where the trend line is headed. This is why the two questions I mentioned are so crucial. Figuring out whether you are in a potentially exponential situation can be aided by a healthy dose of Bayesian thinking. I won't go into Bayesian thinking here, but you can google it or read this to get a better understanding.

So question #1: Is what we're dealing with exponential? If you are contemplating saving for retirement, you know it is exponential because compound interest is built right into the system. If you are dealing with a new virus, your prior experience should tell you that it could "grow virally" (duh!). Global warming is more complicated, but we have a lot of evidence that increased temperature could create cascading effects (decreased reflectivity due to melting ice, increasing decomposition of matter that was previously frozen tundra, etc.). So we have good prior evidence that global temperature could grow exponentially.

The second question asks what the cost is for being wrong. Almost always, the cost of assuming something is exponential when it is not is less than assuming something is not exponential when it really is. So in the case of the Danish research paper I mentioned at the start of this article, the best course of action is to assume this problem can grow exponentially and therefore act as if we are in the chaotic quadrant, taking action as quickly as possible. However, the way we move from Chaotic to Complex is to evaluate the actions we take, learn from their success or failure (revising our priors using Bayesian thinking) and move forward with that new information. So for example in the case of COVID-19, treatments such as ivermectin or hydroxychloroquine need not be dismissed early on in the Chaotic confusion when no one knows for sure what works and what doesn't. But once the evidence shows that they don't work, they need to be quickly abandoned so that resources can be allocated to treatments that are more likely to be effective.

To sum up, one way that the Danish government could have determined whether to operate in a chaotic or complex domain would have been to look at the likelihood of this problem being an exponential one. Given that anything dealing with viruses is most likely exponential in nature, the assumption should be that we are in the Chaotic domain and actions should be taken quickly. Having said that, it is equally important that the efficacy of the actions taken be assessed as quickly as possible so that ineffective actions can be eliminated while effective actions are continued and expanded.