Tuesday, November 20, 2012

The Power of Exponential Doubling

By Tom Kando

Here is a  parable: An ancient Eastern Emperor, because he was bored,  ordered his court scholars to invent an exciting  game. One of them came up with the game of  checkers. The emperor  was very pleased. He told the inventor that he would reward him with anything the man wanted. The inventor replied  that his desires were modest. He merely asked  to receive some grains of rice, namely one grain placed on the checker board’s first square, 2 on the second square, 4 on the third, and so on,  doubling the number until the last square of the checker board  was reached.

The king laughed at this humble and silly request, and he sent out some  men to collect the amount of  rice necessary to satisfy the request. Of course, the emperor discovered that  all the rice  in all of the empire’s granaries  was insufficient to meet the inventor’s wish.

The inventor of the game of checkers  was able to trick the emperor because the emperor did not understand the  power  of exponential growth  (whether it is a good idea to trick an emperor, which can cost you your head, is besides the point).

Growth is defined as exponential when the increase of a quantity is proportional to the size of that quantity. Any quantity that grows by a constant percentage grows exponentially. If a population or an economy grows by 3% per year, then each year that 3% represents  a larger absolute value, because each year it gets added to the baseline upon which the next  3% is computed.

In the checkers story, the exponential growth curve is exceptionally steep. The initial quantity doubles each time. Doubling means an  increase of  100% per unit of time. Of course, how rapid the growth is depends on what your time unit is. Something can double in a week, or it can double in a  million years. For example: You borrow   $1,000 from a loan shark, and he charges you 100%  interest every month. A year later, you owe him over a million.

Getting back to the checkers story: Do you have any idea of  how many grains of rice  the emperor would have had to scrounge up to satisfy his inventor’s request?
          Well, if you start with one grain  on the checkerboard’s first square and you double this for each consecutive square, then this is the progression:

square #1 has one  grain.
Square #2 has two  grains.
Square #3 has four, which is two  squared.
Square #4 has eight, which is two  to the third  power.
And so forth.

So square #64 of the checkerboard  has 2  to the 63rd power: This is 9,223,372,036,888,000,000, which is over 9 quintillion grains  of rice - far more than the entire planet could ever produce.

And here is another interesting way to look at this:

Let’s  say that  the emperor orders  his imperial farmers to produce all the rice needed  to satisfy the inventor’s request. Let’s say that the imperial rice fields have 10,000 rice plants, and that each plant  produces 100,000 grains  of rice every year.

Thus,  after one year, the imperial rice fields will have produced 1 billion grains of rice.  At that rate, it will take one thousand years to produce a trillion grains, a million years to grow  a quadrillion grains,  and a billion years for a quintillion. But since the inventor asked  for more than    9 quintillion grains  of rice, it  will take over nine billion years to grow that amount, which is the age of the Universe, all the way back to the Big Bang.

So you see, a silly little checkerboard with 64 squares can yield unfathomably large numbers.


The true importance of this lies in its implications for real life. For example, Thomas Malthus warned us against the potential danger of unchecked population growth. According to the 18th century scholar, unchecked population growth is exponential, (1-2-4-8) whereas the increase in the food supply is only arithmetical (1-2-3-4). There exists therefore the danger of eventual famine and other Malthusian "population checks" such as war and disease - IF the population were to continue to grow exponentially.

Now don’t misunderstand me: I do not subscribe to neo-Malthusian doomsday prophecies. Obviously, the population has NOT continued to grow exponentially. Neo-Malthusians like Paul Ehrlich have been severely (and rightly) criticized (For example by David Deutsch, in The Beginning of Infinity).

It is the same with other exponential curves. Say, you have an increase in the number of poor people, or in unemployment, or inflation, or in global temperature, or in your body weight, or anything else. The increase may be exponential for a while. This may not require alarm, but it does suggest vigilance. Because whatever the growth coefficient may be, in due time it would lead to disaster - a 100% rate of poverty or unemployment, the type of hyper-inflation experienced in Germany in the 1020s, when a loaf of bread cost a trillion Marks, etc.

The point is that exponential growth can be so deceptive, because even if the growth coefficient is as small as 1% or 2%, it eventually leads to astronomical quantities (as the checkerboard story shows). In everyday life, we don’t usually worry about a 1% or 2% increase in something. But sometimes we should.

To be sure, growth rarely remains exponential ad absurdum. I merely point out that it BETTER slow down before it’s too late (which it usually does).