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## October 03, 2006

Feedback Loops And Exponential Growth | Environment Essays |

*[We live in a world of exponential growth, but most of us find it hard to grasp what that really implies. Here's a post on the subject I originally posted more than two years ago. It's worth revisiting.]*

There's an old story you've probably heard. Unless you understand this story, you won't understand many of the problems that face us in the coming century.

In an ancient kingdom, a clever man saves the life of the king's daughter. The grateful king, wishing to reward the man, offers to grant him whatever he asks for, within reason. The man, being clever, says give me one grain of rice for the first square of a chess board, 2 for the second, 4 for the third, etc., doubling the number of grains of rice for each of the 64 squares on the board. The king, who's not so clever, thinks he's gotten off easy and readily agrees.

How much rice has the king just agreed to give the man? If a sack of rice holds 18 million grains, the king has agreed to fork over more than one *trillion* sacks. The king is ruined. End of story.

The king did not understand **exponential growth**, which is growth by a constant *percentage* (100% in our story) at each step. The number of grains starts small and grows slowly at first, but soon you're doubling larger and larger numbers:

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, etc.

20 squares yield about a million grains, 30 squares yield a billion, 40 squares a trillion, 50 a thousand trillion, 60 a million trillion.

What makes the growth so explosive is the fact that the output of each step is fed back in as the input to the next step. This is what's known as a **positive feedback loop**. ("Positive" not in the sense of "good", but rather in the sense of "leading to increase".) Population growth is an example. Each generation's children (the output) become the next generation's parents (the input to the next step). In the absence of outside constraints, positive feedback loops give results like in the story of the rice.

The idea of a positive feedback loop — and the kind of ever-accelerating growth that can result — can be applied in a variety of settings, from population growth, to global warming mechanisms, to the way the situation in Iraq is spinning out of control, with chaos leading to greater resistance leading to more chaos, etc. Positive feedback also explains the exponential growth of technical and scientific knowledge, since each new technical effort builds on what has been discovered or invented in the past.

**Unfortunately, we seem to have a built-in bias, either neurological or learned, that makes us extrapolate linearly into the future. I.e., we tend, instinctively, to estimate future change as involving the same size steps in absolute terms — not percentage terms — as in the past, so exponential growth always seems to take us by surprise. That's why, when the king saw that in the first several steps the change was just a few grains of rice, he unconsciously assumed that at every step the change would be just a few grains of rice. We all make this mistake. And it's a very dangerous mistake to make at this point in human history.**

Here's a riddle.

Suppose someone puts a few bacteria in a petri dish at noon on Monday. Suppose further that the bacteria grow at a rate that causes their population to double every hour. Suppose finally that the growth is such that the petri dish is completely full of bacteria at noon on Wednesday.

Question: When is the petri dish half full?

Click the link below for the answer.

Many people say Tuesday at noon — halfway between noon Monday and noon Wednesday. That's *linear* thinking, and it's incorrect. Since the population doubles each hour, the dish is half full *just one hour before it's full*, i.e., at 11 AM on Wednesday. From noon Monday to 11 AM Wednesday, the bacteria have plenty of room to spare. No worries. Then *wham!*

What's the point? **When growth is exponential, or when powerful feedback loops are present, we can think everything's going along fine until just before we hit the wall. Many of the problems that face us — problems of population growth, resource depletion, environmental degradation, political instability — involve exactly this kind of exponential growth caused by powerful feedback loops. If we don't have an appropriate mental model of what that means, we'll be complacent right up to the moment when we hit the wall — hard — and, like the king, lose everything.**

**It's crucial that we overcome our linear bias. We're living in a world of exponential growth, and our petri dish is filling rapidly.**

Posted by Jonathan at October 3, 2006 10:42 PM