## A new platform classification

When looking at the Race to a Billion data I noticed that some platform adoption ramps look quite different from others. It’s not just a matter of “slope” or rate of growth but a distinctly different shape.

If you look at the graph below you might see the difference yourself.

First, note that the graph is logarithmic. A straight line on a log chart implies exponential growth (the y values are a constant to the power of the x value). However, none of the graphs show straight lines. None of the platforms follow exponential growth[1].

Second, we can eliminate linear or logarithmic functions. They simply do not fit.

That leaves: polynomial or power. It’s easy to test these alternatives for all the functions and it becomes clear that most of the platforms split into one of these two classes.

Here are the classes[2]:

Note that I added the formulas for the selected platforms.

I named these classes after their function types: “Power platforms” or “Second Order platforms[3].

This classification has two benefits:

1. It allows a clearer distinction in scope. Both may exhibit signs of network effects, but Power platforms seem to exhibit “global” or unlimited reach while Second Order platforms may have a strong upper bound on reach.
2. It allows more reliable forecasting. If a platform classifies as Power, the forecast could be created with fewer data points.
3. It allows us to guess at the causes of entry into each class for any potential platform challenger.

To illustrate the second benefit, I took the Android, iOS and Facebook power functions and modeled (i.e. extrapolated) them for a few quarters into the future. The result is shown below:

The forecast shows the following:

• Android will reach 10 billion activations 10 years after its release (mid 2018)
• iOS will reach 1 billion units sold by end of next year and 3 billion units by its 10th anniversary in 2017.
• Facebook will reach 2 billion users by 2015 after the end of its 10th year.
Notes:
1. Except perhaps in the very early periods []
2. Some of the platforms had to be dropped because there was no easy fit. My observation is that some of these platforms went through some crisis or change in strategy during their history which caused the pattern of growth to change []
3. Note that all Polynomial functions found above are second order []
• willo

Fascinating Horace, thank you.

• Dick Applebaum

So, 300 million iOS devices in the 14 months Nov 2013 through Dec 2014.

Seems attainable!

Care to break down predictions by device?

• studuncan

It’s not a prediction, it’s an extrapolation.

• Rubio

I would say 200m iPhones and 100m iPads approx

• Peter

If you believe each platform sometime reaches end-of-life, I think a (generalised) logistic S curve will fit best (http://en.wikipedia.org/wiki/Generalised_logistic_function). With a few data points you can try to estimate the maximum of each platform.

• Glaurung-Quena

“Android will reach 10 billion activations 10 years after its release (mid 2018)”

Considering that that’s significantly more devices than there are people in the world (and one or two billion of those people will be using IOS instead), I suspect that Android’s growth curve is going to flatten out before then.

• Activations include replacements and the potential of more than one device per person.

• Sacto_Joe

Has any definitive survey ever been conducted to verify these counts? Also, intuitively, it would seem that iOS devices are gong to hang around longer. How would that affect the curve? Or has that already been taken into account?

• obarthelemy

Not sure why/how iOS devices should last longer ? The oldest devices between me and brothers/parents is my HTC HD2 (upgraded to some CM), then my Ainol Novo8 (upgraded to 4.1), then my Nook Color (upgraded to 4.3). My sister in law’s iP3 something is a distant 4th (and is in its 3rd incarnation after 2 falls), and their iPad 1 5th.

• Jessica Darko

Actual Androids sold = Activations / X

X is greater than ten, I assert.

This is why it’s simply not accurate to use activations as a proxy for devices sold… especially when you’re comparing it to audited, sworn under penalty of perjury, numbers like Apple devices sold (as reported to the SEC each quarter.)

Until you can nail down what X is in the formula above, it’s absurd to use this figure. I know that android zealots want to insist that X is one and they will cite claims by some person or other to “prove” it, but there’s a huge difference between a claim of a PR person (or known liar) and a number published under penalty of perjury.

• Care to explain why X is greater than 10? I have personally activated three Android devices (two purchased and one replacement) and purchased three iPhones (two purchased and one replacement).

In either case, I am counted 3 times as part of 1 billion “Android activations” or the (soon to be) 1 billion “iPhones sold.”

• Glaurung-Quena

“Activations include replacements and the potential of more than one device per person.”

I disagree. Multiple devices per person in wealthy countries will be offset by the 2-3 billion people in the world who aren’t wealthy enough to afford more than one device per household.

Figure half of the world’s 2 billion children are going to be given no device or only non-smart devices. Another billion people with limited access to electricity will probably stick with non-smart devices due to battery life issues. Allow 1 billion non-Android users, and the pool of people using Android is limited to 4-5 billion people. To get to 10 billion activations, at least 1 in every 2 android devices will have to end up in the scrap heap.

• peter

The active installed base will obviously be smaller than that; those 2007 iPhones and early Androids will have been retired by now. So the curve will continue for quite a bit.

I would suspect that the active installed base would follow the S-curve mentioned below. Perhaps use a guess that high end phones last 36 months and low end phones 24 month (or something). (Now if only I could fit S-curves in Excel charts.)

• You can fit to a (simple) logistic curve by massaging the data into a linear relationship and then do least squares. You’ll have to guess the maximum “carrying capacity” for the installed base though.

• Jessica Darko

Where we do have market data, “Activations” does not reconcile with it, so we know for a fact that the “activations” numbers are grossly inflated.

More importantly, you point out that these numbers far exceed saturation… but the people who use the number want to have it both ways.

When they claim that Android is winning or developers should build for the platform, they claim there are 1 billion people actively using android right now.

When you point out that the rates exceed reason and will soon exceed population, they claim that these numbers include historical past devices that have been retired.

I’m certain that for every android device that has actually been sold, there are many activations counted, probably more than a dozen.

But you will never get a number from google for “devices using the latest version of android that have checked in with our servers in the last 3 months”.

This stat would show the real addressable android market.

And it would probably be 1/10th of the same stat for iOS.

• obarthelemy

Actually, I was kidding about aliens. I’ve got 8 or 9 Android devices but I might be pathological; most people around me do have a phone plus a tablet (or at least a shared one), and quite a few also have an Android desktop.

• sscutchen

Aren’t Power fits simply fits where the upper bounds are not yet evident? Seems like the fit for the early portion of the polynomial curves (before the upper bound began to show iteself) would have been classified as power curves.

• That’s right. Note that the time frame makes a difference. The classification makes sense only within a decade-long time frame. Beyond a decade the assumption is that the rules, or business models for sustaining a platform, will change.

• Another assumption worth pointing out is that platform lifetimes operate on the same absolute scale. Indexing to the same start date eliminates offset differences, but not scale differences.

But each of these platforms may be operating at a different scale according to the dynamics for that product / market.

For example, iPod’s lifetime might be 10 years and Facebook’s 20 years.

• obarthelemy

Better yet: about 60 quarters after launch, Android will reach 20b users, which means not only will we have met extraterrestrial life, but they, too, will love Android !

(they’ll probably be very poor ETs, or may not have fingertips. Nor sense of style)

• Jessica Darko

Or put another way, it’s proof that “activations” vastly overcounts actual sales, at best, and is a made up number most likely.

All actual market data shows that the activations outruns sales by an order of magnitude or more.

• obarthelemy

Source ?

• iRush

Healthy scepticism of dubious source!

• Tatil_S

Healthy? 🙂

• Tatil_S

Droids using Androids. 🙂

• This is interesting, but the built in assumptions make me uncomfortable.

One assumption that strikes me: a single model applies to life of the platform. I think it is very likely that different models would be better fit at different stages in the growth of the platform.

For example, maybe power is a better fit to early stage platforms and poly is better for later stage platforms where constraints start to kick in. This makes some intuitive sense.

If that were true then applying the models globally to to-date platform lifetimes would just tell you which “mode” was dominant. So all of the first group are externally constrained and the second group are not.

This might explain the bad fit in the early stages of iPod.

And BBY / Symbian in the second group might be platforms that failed before they started to feel external constraints.

• obarthelemy

I play a kinda of card game where there are really two level of bets: for low bets, you try and work out how many points you’ll make (plus, signalling). For high bets, it’s better to work backwards from the total number of possible points, how many you won’t be able to grab. Maybe the same hybrid method would work best for sales predictions too.

• stefnagel

Lovely math. I think about it along with the relative annual revenue of iOS units vs. Android activations, including apps, other digital media, other shopping.

• Rubio

That means approx. 300m iOS devices 2017q1. 1000m devices for the whole year! Wow!! Can’t wait to see whether that happens!

• Caleb

The Apple TV adaption rate has the most unique shape, but disappeared after the first graph.

• Scott Sterling

The chart extrapolates that Android will sell 9 billion devices in the next 5 years. For this to occur, a scenario like this is necessary:

Assume the average life of a device is 3 years;

Assume 2 billion people buy an Android device in year 1 (we are already in year 1).
Assume a different 2 billion people buy an Android device in year 2.
Assume another different 2 billion people buy an Android device in year 3.
Then in year 4 the first two billion people replace their Android device.
And in year 5 the second two billion people replace their Android device.

All we require is 6 billion unique Android device buyers.

And that assumes they are all repeat buyers, because if not we probably need 8 billion humans. Or you could say that every single person buys two devices. Then we only need 4 billion humans to make the chart come true.

• Kenton Douglas

Interesting analysis. Why not include browsers as a platform in the ‘Race to a Billion’? For me they form the next baseline of computing in the Internet age (ahead the underlying OS). Google were quoting 750m active Chrome users back in the Spring.

• Walt French

“essentially, all models are wrong, but some are useful.” — George Box

Ironically, I built a similar power model and emailed it to Horace just before reading this post. “Ironically,” because I’m not comfortable with the approach as used here.

First, I’ll note that eventually, we should expect ALL these products to lose their power characterization and become “second order” products (unless they grow more slowly than human population/income). All second order models turn downward and decline at the same rate that they grew. The model might predict a peak usage, but I’m skeptical about the symmetry.

Second, trend models have the unfortunate feature of being right until they’re not, and it’s the times that a product deviates from its trend—when the model starts being wrong—that is most interesting. Why did the product go from a power model having a better fit, to the 2nd order model looking better? All we know is that something has capped, or slowed the growth. A competitive product has come on strong? A global recession has devastated the target market? Photos of lunches have become passé? The model tells us nothing that the eye doesn’t.

A better use for these might be as a “is something going wrong?” standard—if a series starts deviating from the trend, it might signal there are important new forces at work. In US economics, we have a notion of “potential GDP”—how much production we could achieve if we had all our factories at work and qualified workers on the job. It’s often estimated by a trend. But when the economy starts heading away from that line, it does so bigtime, and THAT’s worth watching. (We’re now stuck well below that trend line, running in parallel in a way that we may never catch up to our old “potential”). Again, the trend line tells us nothing about WHY the Great Recession hit, but it gives us a measure of how seriously something new is at play.

* * * * *

“All models are wrong.” Still, without putting on too many new variables (on which ALSO look up a Box quote), they ought to predict well out of sample. Estimate the growth pattern on one set of data, see how well it tracks to some new data that it wasn’t “trained” to fit. The iTunes line shows that there are multiple factors at play: Americans’ love for music; the standards for buying vs what I understand is universal ripping-off in Asia; the number of Americans who can afford a nice device. These all change very slowly and trend lines ought to work. The amount of discretionary income and surveys of well-being; the availability of competitive ecosystems: these can change pretty quickly and the trend is not your friend in understanding the fluctuations over quarters or years.

* * * * *

Net-net, the models might quantify the rate of change nicely, but “how fast facebook reaches a billion” is probably much less important than “are the last 4 quarters of iPhone sales a permanent or temporary aberration in Apple’s ability to satisfy?” Personally, I’m much happier with unquantified, but “structural” models such as the extent of disruption. I would like even better an examination of say, what it’d take for Microsoft to get back in the game, to change the balance in tablets, for example. And the non-structural trends don’t answer any of those extremely important and interesting questions.

PS: yes, I describe myself as a quant. I’ve been building and using models for almost 50 years—most recently, this afternoon—and am not dissing models, including/especially the very minimalist ones here.

• breakfast

My base level takeaway is this…you can use the trends of each of these ‘platforms’ to understand what the likely addressable markets are for these platforms. As these leading platforms are global in nature but have not addressed the entire population, i can surmise that when a platform growth starts to flatten, saturation has been approached. While many models will use a base of population or subscribers, these lines highlight the true addressable market regardless of poorly linked factors such as population or income.

• BoydWaters

Lots of comments here, implying that extrapolation yields nonsense numbers.

But I have recently attempted to count the number of “microcomputers” that I own — using a narrow definition of “microcomputer” from 1978, the year I started playing this game. I can quickly get numbers that would sound like nonsense. Are there 20 computers in my car? 200?

Sounds like a silly game.

But I use this game to think about the size of the addressable market. Of course there will be more active “mobile phones” than humans some day.