Exponential growth and panic thresholds
TLDR; A thought experiment originally described in stadium form by Albert Bartlett, physicist at the University of Colorado Boulder, who built it around Folsom Field, the stadium on his own campus. Bartlett used the example as an analogy for population growth, but it also applies to any exponential growth scenario, including AI.
If you have one drop of water in a stadium, and then 2 drops and then 4 drops, when should you panic, and when do people actually panic?
Worth doing the actual arithmetic first, because the folk version of this puzzle undersells it.
A drop is about 0.05 mL. A large stadium bowl is roughly a million cubic meters. That’s about 44 doublings from one drop to full. And the milestones are brutally back-loaded:
- Doubling 27: a 1mm film across the pitch. Grass looks damp.
- Doubling 34: ankle deep. Ten doublings left.
- Doubling 38: over your head on the pitch — and the stadium is 1.6% full.
- Doubling 43: half full. One period left.
So the first two-thirds of the process is invisible, and the moment the water is deep enough to drown you is the moment the stadium is still 98% empty.
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When you should panic
The mathematically correct trigger isn’t a water level at all — it’s the moment you can confirm the doubling and measure the period. After that, the level tells you nothing you didn’t already know, because the deadline is fixed and the only variable is how long your response takes.
If your response takes N doubling-periods to mount, you must start N doublings before the end. That’s the whole calculation. Need six periods to build a drain? Act at doubling 38 — at 1.6% full. Need ten? Act at ankle-deep, when the stadium is one-thousandth full and you will look completely insane.
Note what this means: the required panic point depends on your response latency, not on the severity of the threat. Slow institutions need to panic earlier than fast ones, about the identical problem.
When people actually panic
Somewhere around 25–50% full — one or two periods from the end, which is to say far too late to matter.
A few reasons this happens, and one of them is respectable:
Exponential growth is experienced as linear from the inside. Each period delivers the same relative change — always exactly double. Subjectively, nothing is accelerating. The sensation of “this is picking up speed” never arrives, because in ratio terms it never does.
Wagenaar’s work in the 70s and the COVID-era replications both found people extrapolate exponentials roughly linearly from the visible portion, and badly under-forecast.
We perceive levels, not rates. The level is the salient percept, and it stays negligible for 85% of the process.
Forty minutes of nothing happening is real evidence. This is the respectable one. Most doubling processes stop — bacteria exhaust nutrients, epidemics run out of susceptibles, fads saturate. Logistic curves are indistinguishable from exponentials early on, and the person who panicked at doubling 10 has cried wolf many times before, correctly-ish. Distinguishing “genuinely exponential” from “will bend soon” is legitimately hard while you’re still in the flat part.
What the asymmetry does is make that reasoning insufficient anyway. Being early costs you one wasted mobilization. Being late costs you everything, and “late” arrives with no warning — the system goes from ankle-deep to full in six minutes.
The characteristic signature isn’t gradually rising concern. It’s complacency, complacency, complacency, then total panic — and no intermediate state.
