Sigmoids vs Exponentials

In a previous post, I argued that we humans suffer from a destructive oversimplifying habit of linear extrapolation.  This professor argues the same point, but he falls into the next logical trap, thinking that exponential extrapolation solves the problem.

From my earlier post, it should be clear that pure exponential growth never happens in reality because the drivers of growth, including resources and incentives, decline (or fail to keep pace with the growth) as time goes on.  This is true even if those resources are themselves growing but at a sub-exponential rate.

The bacteria example that the professor gives, while showing the stark differences between linear and exponential growth, is glaringly flawed.  Specifically, bacteria cannot propagate without a rich substrate of biological material to draw from.  If you seeded a sterile jar with a population of bacteria, not only would it not grow, it would not survive.  And even if you pumped in nutrients and sunlight in the right quantities to induce population growth, the growth would decelerate long before the jar was full, due to overcrowding.  In fact, the jar would never become totally full.

The point I am making is that the professor’s examples are all constructed or taken out of their larger context to illustrate the mathematics of exponential growth while ignoring the thruth that exponential growth does not exist in real-world systems.  At one point he says, “Now with all that history of growth they expected the growth would just go on forever.  Fortunately, it stopped.  Not because anyone understood the arithmetic, it stopped for other reasons, but let’s ask what if.  Suppose that growth had continued….”  But, there are no systems (other than mathematical ones) with the pure conditions required to sustain exponential growth, so it’s silly to ever suppose that growth continues unabated.

The real question is whether there are systems in the real world that exhibit super-linear growth for a period of time such that will have a negative outcome for us before the growth becomes contained of its own accord.  To this, the professor does make a very salient point around 7 minutes into part 2.  Namely, that in such instances, we can either let the system “choose for us” the details of how growth will slow, or we can take a proactive stance and curb the growth artificially.  A great example of this being done in practice occurred in China when laws and incentives were put in place to limit each family to a single child instead of letting growth be curbed by mass starvation and disease.

Sometimes the difference between between true exponential growth and the pseudo-exponential growth period in the accelerating portion of the sigmoid curve is inconsequential.  But sometimes it’s not.  For instance, look at the divergence between the two in this comparison.

Even more importantly, our expectation of the long-terms consequences of not intervening can meaningfully color how we choose to intervene.  Our choices of how to respond are rarely binary, and different choices come with different costs and expected outcomes.  If you truly believe that you are on an exponential curve, the cost of getting off that curve is never a consideration because you have no choice: you have to get off or doom is inevitable.  You might argue that becoming overly alarmed is better than not being alarmed enough.  But I would argue that seeing the truth and future more clearly is better than either.

Hat tip: Kim Scheinberg