How I wish we measured percentage change

JDN 2457415

For today’s post I’m taking a break from issues of global policy to discuss a bit of a mathematical pet peeve. It is an opinion I share with many economists—for instance Miles Kimball has a very nice post about it, complete with some clever analogies to music.

I hate when we talk about percentages in asymmetric terms.

What do I mean by this? Well, here are a few examples.

If my stock portfolio loses 10% one year and then gains 11% the following year, have I gained or lost money? I’ve lost money. Only a little bit—I’m down 0.1%—but still, a loss.

In 2003, Venezuela suffered a depression of -26.7% growth one year, and then an economic boom of 36.1% growth the following year. What was their new GDP, relative to what it was before the depression? Very slightly less than before. (99.8% of its pre-recession value, to be precise.) You would think that falling 27% and rising 36% would leave you about 9% ahead; in fact it leaves you behind.

Would you rather live in a country with 11% inflation and have constant nominal pay, or live in a country with no inflation and take a 10% pay cut? You should prefer the inflation; in that case your real income only falls by 9.9%, instead of 10%.

We often say that the real interest rate is simply the nominal interest rate minus the rate of inflation, but that’s actually only an approximation. If you have 7% inflation and a nominal interest rate of 11%, your real interest rate is not actually 4%; it is 3.74%. If you have 2% inflation and a nominal interest rate of 0%, your real interest rate is not actually -2%; it is -1.96%.

This is what I mean by asymmetric:

Rising 10% and falling 10% do not cancel each other out. To cancel out a fall of 10%, you must actually rise 11.1%.

Gaining 20% and losing 20% do not cancel each other out. To cancel out a loss of 20%, you need a gain of 25%.

Is it starting to bother you yet? It sure bothers me.

Worst of all is the fact that the way we usually measure percentages, losses are bounded at 100% while gains are unbounded. To cancel a loss of 100%, you’d need a gain of infinity.

There are two basic ways of solving this problem: The simple way, and the good way.

The simple way is to just start measuring percentages symmetrically, by including both the starting and ending values in the calculation and averaging them.
That is, instead of using this formula:

% change = 100% * (new – old)/(old)

You use this one:

% change = 100% * (new – old)/((new + old)/2)

In this new system, percentage changes are symmetric.

Suppose a country’s GDP rises from $5 trillion to $6 trillion.

In the old system we’d say it has risen 20%:

100% * ($6 T – $5 T)/($5 T) = 20%

In the symmetric system, we’d say it has risen 18.2%:

100% * ($6 T – $5 T)/($5.5 T) = 18.2%

Suppose it falls back to $5 trillion the next year.

In the old system we’d say it has only fallen 16.7%:

100% * ($5 T – $6 T)/($6 T) = -16.7%

But in the symmetric system, we’d say it has fallen 18.2%.

100% * ($5 T – $6 T)/($5.5 T) = -18.2%

In the old system, the gain of 20% was somehow canceled by a loss of 16.7%. In the symmetric system, the gain of 18.2% was canceled by a loss of 18.2%, just as you’d expect.

This also removes the problem of losses being bounded but gains being unbounded. Now both losses and gains are bounded, at the rather surprising value of 200%.

Formally, that’s because of these limits:
lim_{x rightarrow infty} {(x-1) over {(x+1)/2}} = 2

lim_{x rightarrow infty} {(0-x) over {(x+0)/2}} = -2

It might be easier to intuit these limits with an example. Suppose something explodes from a value of 1 to a value of 10,000,000. In the old system, this means it rose 1,000,000,000%. In the symmetric system, it rose 199.9999%. Like the speed of light, you can approach 200%, but never quite get there.

100% * (10^7 – 1)/(5*10^6 + 0.5) = 199.9999%

Gaining 200% in the symmetric system is gaining an infinite amount. That’s… weird, to say the least. Also, losing everything is now losing… 200%?

This is simple to explain and compute, but it’s ultimately not the best way.

The best way is to use logarithms.

As you may vaguely recall from math classes past, logarithms are the inverse of exponents.

Since 2^4 = 16, log_2 (16) = 4.

The natural logarithm ln() is the most fundamental for deep mathematical reasons I don’t have room to explain right now. It uses the base e, a transcendental number that starts 2.718281828459045…

To the uninitiated, this probably seems like an odd choice—no rational number has a natural logarithm that is itself a rational number (well, other than 1, since ln(1) = 0).

But perhaps it will seem a bit more comfortable once I show you that natural logarithms are remarkably close to percentages, particularly for the small changes in which percentages make sense.

We define something called log points such that the change in log points is 100 times the natural logarithm of the ratio of the two:

log points = 100 * ln(new / old)

This is symmetric because of the following property of logarithms:

ln(a/b) = – ln(b/a)

Let’s return to the country that saw its GDP rise from $5 trillion to $6 trillion.

The logarithmic change is 18.2 log points:

100 * ln($6 T / $5 T) = 100 * ln(1.2) = 18.2

If it falls back to $5 T, the change is -18.2 log points:

100 * ln($5 T / $6 T) = 100 * ln(0.833) = -18.2

Notice how in the symmetric percentage system, it rose and fell 18.2%; and in the logarithmic system, it rose and fell 18.2 log points. They are almost interchangeable, for small percentages.

In this graph, the old value is assumed to be 1. The horizontal axis is the new value, and the vertical axis is the percentage change we would report by each method.

percentage_change_small

The green line is the usual way we measure percentages.

The red curve is the symmetric percentage method.

The blue curve is the logarithmic method.

For percentages within +/- 10%, all three methods are about the same. Then both new methods give about the same answer all the way up to changes of +/- 40%. Since most real changes in economics are within that range, the symmetric method and the logarithmic method are basically interchangeable.

However, for very large changes, even these two methods diverge, and in my opinion the logarithm is to be preferred.

percentage_change_large

The symmetric percentage never gets above 200% or below -200%, while the logarithm is unbounded in both directions.

If you lose everything, the old system would say you have lost 100%. The symmetric system would say you have lost 200%. The logarithmic system would say you have lost infinity log points. If infinity seems a bit too extreme, think of it this way: You have in fact lost everything. No finite proportional gain can ever bring it back. A loss that requires a gain of infinity percent seems like it should be called a loss of infinity percent, doesn’t it? Under the logarithmic system it is.

If you gain an infinite amount, the old system would say you have gained infinity percent. The logarithmic system would also say that you have gained infinity log points. But the symmetric percentage system would say that you have gained 200%. 200%? Counter-intuitive, to say the least.

Log points also have another very nice property that neither the usual system nor the symmetric percentage system have: You can add them.

If you gain 25 log points, lose 15 log points, then gain 10 log points, you have gained 20 log points.

25 – 15 + 10 = 20

Just as you’d expect!

But if you gain 25%, then lose 15%, and then gain 10%, you have gained… 16.9%.

(1 + 0.25)*(1 – 0.15)*(1 + 0.10) = 1.169

If you gain 25% symmetric, lose 15% symmetric, then gain 10% symmetric, that calculation is really a pain. To find the value y that is p symmetric percentage points from the starting value x, you end up needing to solve this equation:

p = 100 * (y – x)/((x+y)/2)

This can be done; it comes out like this:

y = (200 + p)/(200 – p) * x

(This also gives a bit of insight into why it is that the bounds are +/- 200%.)

So by chaining those, we can in fact find out what happens after gaining 25%, losing 15%, then gaining 10% in the symmetric system:

(200 + 25)/(200 – 25)*(200 – 15)/(200 + 15)*(200 + 10)/(200 – 10) = 1.223

Then we can put that back into the symmetric system:

100% * (1.223 – 1)/((1+1.223)/2) = 20.1%

So after all that work, we find out that you have gained 20.1% symmetric. We could almost just add them—because they are so similar to log points—but we can’t quite.

Log points actually turn out to be really convenient, once you get the hang of them. The problem is that there’s a conceptual leap for most people to grasp what a logarithm is in the first place.

In particular, the hardest part to grasp is probably that a doubling is not 100 log points.

It is in fact 69 log points, because ln(2) = 0.69.

(Doubling in the symmetric percentage system is gaining 67%—much closer to the log points than to the usual percentage system.)

Calculation of the new value is a bit more difficult than in the usual system, but not as difficult as in the symmetric percentage system.

If you have a change of p log points from a starting point of x, the ending point y is:

y = e^{p/100} * x

The fact that you can add log points ultimately comes from the way exponents add:

e^{p1/100} * e^{p2/100} = e^{(p1+p2)/100}

Suppose US GDP grew 2% in 2007, then 0% in 2008, then fell 8% in 2009 and rose 4% in 2010 (this is approximately true). Where was it in 2010 relative to 2006? Who knows, right? It turns out to be a net loss of 2.4%; so if it was $15 T before it’s now $14.63 T. If you had just added, you’d think it was only down 2%; you’d have underestimated the loss by $70 billion.

But if it had grown 2 log points, then 0 log points, then fell 8 log points, then rose 4 log points, the answer is easy: It’s down 2 log points. If it was $15 T before, it’s now $14.70 T. Adding gives the correct answer this time.

Thus, instead of saying that the stock market fell 4.3%, we should say it fell 4.4 log points. Instead of saying that GDP is up 1.9%, we should say it is up 1.8 log points. For small changes it won’t even matter; if inflation is 1.4%, it is in fact also 1.4 log points. Log points are a bit harder to conceptualize; but they are symmetric and additive, which other methods are not.

Is this a matter of life and death on a global scale? No.

But I can’t write about those every day, now can I?

The power of exponential growth

JDN 2457390

There’s a famous riddle: If the water in a lakebed doubles in volume every day, and the lakebed started filling on January 1, and is half full on June 17, when will it be full?

The answer is of course June 18—if it doubles every day, it will go from half full to full in a single day.

But most people assume that half the work takes about half the time, so they usually give answers in December. Others try to correct, but don’t go far enough, and say something like October.

Human brains are programmed to understand linear processes. We expect things to come in direct proportion: If you work twice as hard, you expect to get twice as much done. If you study twice as long, you expect to learn twice as much. If you pay twice as much, you expect to get twice as much stuff.

We tend to apply this same intuition to situations where it does not belong, processes that are not actually linear but exponential. As a result, when we extrapolate the slow growth early in the process, we wildly underestimate the total growth in the long run.

For example, suppose we have two countries. Arcadia has a GDP of $100 billion per year, and they grow at 4% per year. Berkland has a GDP of $200 billion, and they grow at 2% per year. Assuming that they maintain these growth rates, how long will it take for Arcadia’s GDP to exceed Berkland’s?

If we do this intuitively, we might sort of guess that at 4% you’d add 100% in 25 years, and at 2% you’d add 100% in 50 years; so it should be something like 75 years, because then Arcadia will have added $300 million while Berkland added $200 million. You might even just fudge the numbers in your head and say “about a century”.

In fact, it is only 35 years. You could solve this exactly by setting (100)(1.04^x) = (200)(1.02^x); but I have an intuitive method that I think may help you to estimate exponential processes in the future.

Divide the percentage into 69. (For some numbers it’s easier to use 70 or 72; remember, these are just to be approximate. The exact figure is 100*ln(2) = 69.3147… and then it wouldn’t be the percentage p but 100*ln(1+p/100); try plotting those and you’ll see why using p works.) This is the time it will take to double.

So at 4%, Arcadia will double in about 17.5 years, quadrupling in 35 years. At 2%, Berkland will double in about 35 years. Thus, in 35 years, Arcadia will quadruple and Berkland will double, so their GDPs will be equal.

Economics is full of exponential processes: Compound interest is exponential, and over moderately long periods GDP and population both tend to grow exponentially. (In fact they grow logistically, which is similar to exponential until it gets very large and begins to slow down. If you smooth out our recessions, you can get a sense that since the 1940s, US GDP growth has slowed down from about 4% per year to about 2% per year.) It is therefore quite important to understand how exponential growth works.

Let’s try another one. If one account has $1 million, growing at 5% per year, and another has $1,000, growing at 10% per year, how long will it take for the second account to have more money in it?

69/5 is about 14, so the first account doubles in 14 years. 69/10 is about 7, so the second account doubles in 7 years. A factor of 1000 is about 10 doublings (2^10 = 1024), so the second account needs to have doubled 10 times more than the first account. Since it doubles twice as often, this means that it must have doubled 20 times while the other doubled 10 times. Therefore, it will take about 140 years.

In fact, it takes 141—so our quick approximation is actually remarkably good.

This example is instructive in another way; 141 years is a pretty long time, isn’t it? You can’t just assume that exponential growth is “as fast as you want it to be”. Once people realize that exponential growth is very fast, they often overcorrect, assuming that exponential growth automatically means growth that is absurdly—or arbitrarily—fast. (XKCD made a similar point in this comic.)

I think the worst examples of this mistake are among Singularitarians. They—correctly—note that computing power has become exponentially greater and cheaper over time, doubling about every 18 months, which has been dubbed Moore’s Law. They assume that this will continue into the indefinite future (this is already problematic; the growth rate seems to be already slowing down). And therefore they conclude there will be a sudden moment, a technological singularity, at which computers will suddenly outstrip humans in every way and bring about a new world order of artificial intelligence basically overnight. They call it a “hard takeoff”; here’s a direct quote:

But many thinkers in this field including Nick Bostrom and Eliezer Yudkowsky worry that AI won’t work like this at all. Instead there could be a “hard takeoff”, a huge subjective discontinuity in the function mapping AI research progress to intelligence as measured in ability-to-get-things-done. If on January 1 you have a toy AI as smart as a cow, one which can identify certain objects in pictures and navigate a complex environment, and on February 1 it’s proved the Riemann hypothesis and started building a ring around the sun, that was a hard takeoff.

Wait… what? For someone like me who understands exponential growth, the last part is a baffling non sequitur. If computers start half as smart as us and double every 18 months, in 18 months, they will be as smart as us. In 36 months, they will be twice as smart as us. Twice as smart as us literally means that two people working together perfectly can match them—certainly a few dozen working realistically can. We’re not in danger of total AI domination from that. With millions of people working against the AI, we should be able to keep up with it for at least another 30 years. So are you assuming that this trend is continuing or not? (Oh, and by the way, we’ve had AIs that can identify objects and navigate complex environments for a couple years now, and so far, no ringworld around the Sun.)

That same essay make a biological argument, which misunderstands human evolution in a way that is surprisingly subtle yet ultimately fundamental:

If you were to come up with a sort of objective zoological IQ based on amount of evolutionary work required to reach a certain level, complexity of brain structures, etc, you might put nematodes at 1, cows at 90, chimps at 99, homo erectus at 99.9, and modern humans at 100. The difference between 99.9 and 100 is the difference between “frequently eaten by lions” and “has to pass anti-poaching laws to prevent all lions from being wiped out”.

No, actually, what makes humans what we are is not that we are 1% smarter than chimpanzees.

First of all, we’re actually more like 200% smarter than chimpanzees, measured by encephalization quotient; they clock in at 2.49 while we hit 7.44. If you simply measure by raw volume, they have about 400 mL to our 1300 mL, so again roughly 3 times as big. But that’s relatively unimportant; with Moore’s Law, tripling only takes about 2.5 years.

But even having triple the brain power is not what makes humans different. It was a necessary condition, but not a sufficient one. Indeed, it was so insufficient that for about 200,000 years we had brains just as powerful as we do now and yet we did basically nothing in technological or economic terms—total, complete stagnation on a global scale. This is a conservative estimate of when we had brains of the same size and structure as we do today.

What makes humans what we are? Cooperation. We are what we are because we are together.
The capacity of human intelligence today is not 1300 mL of brain. It’s more like 1.3 gigaliters of brain, where a gigaliter, a billion liters, is about the volume of the Empire State Building. We have the intellectual capacity we do not because we are individually geniuses, but because we have built institutions of research and education that combine, synthesize, and share the knowledge of billions of people who came before us. Isaac Newton didn’t understand the world as well as the average third-grader in the 21st century does today. Does the third-grader have more brain? Of course not. But they absolutely do have more knowledge.

(I recently finished my first playthrough of Legacy of the Void, in which a central point concerns whether the Protoss should detach themselves from the Khala, a psychic union which combines all their knowledge and experience into one. I won’t spoil the ending, but let me say this: I can understand their hesitation, for it is basically our equivalent of the Khala—first literacy, and now the Internet—that has made us what we are. It would no doubt be the Khala that made them what they are as well.)

Is AI still dangerous? Absolutely. There are all sorts of damaging effects AI could have, culturally, economically, militarily—and some of them are already beginning to happen. I even agree with the basic conclusion of that essay that OpenAI is a bad idea because the cost of making AI available to people who will abuse it or create one that is dangerous is higher than the benefit of making AI available to everyone. But exponential growth not only isn’t the same thing as instantaneous takeoff, it isn’t even compatible with it.

The next time you encounter an example of exponential growth, try this. Don’t just fudge it in your head, don’t overcorrect and assume everything will be fast—just divide the percentage into 69 to see how long it will take to double.