Apple and Microsoft War Now Dictated by Mathematics

| Analysis

The fact that Microsoft's Internet Explorer is steadily declining in market share, monotonically, suggests that there are mathematical market forces at work that cannot be overcome by any action Microsoft takes. The same mathematics applies to Apple's OS market share, and so Microsoft's battle against Firefox is just a warmup.

The most recent data from Net Applications shows that Microsoft's Internet Explorer has dropped to less than 70 percent on the Internet. The decline is a steady, monotonic decline, which begs the question of what market and mathematical forces can lead to that kind of decline.

A monotonic function is one that steadily decreases or increases without reversing. What that seems to suggest is that the market share of IE is outside the control of Microsoft, no matter what improvements they make. Conversely, the monotonic increase in Firefox and Safari suggest the same thing -- market forces are at work.

One way market forces can affect the share of a product is word of mouth and consumer communication on the Internet. It's an effect similar to to how crime increases faster than population growth in a closely bound region. A comparison is worthwhile.

The chart below shows how in a closed population of two people, A can interact with B or B with A. Using the notation of [number of people: number of interactions], the first group is [2:2]. For three people, the result is [3:6]. For four people, the result is [4:12]. Even though the population has only doubled from the first group of two to the final group of four, the number of possible interactions has gone up by a factor of 6. Assuming the probability of any given illegal interaction is a constant, and other factors being equal, the mathematical progression says that the probability of a one-on-one crime goes up faster than the rate of population growth.


Note that this simple analysis doesn't even take into account multiple interactions, such as (A+B) interacting with C. That increases the number of possible interactions even more. In fact, it's a factorial function.

The same thing happens when word of mouth gets around on a particular product, whether it's between IT managers or consumers. As the adoption rate grows, more people talk to more people about their experience and assessment. Instead of a crime happening, as in the example above, a certain percentage of people, call it "p1," will be influenced by their peers during the interaction and make the switch.

My mathematical intuition tells me that this is what is happening with Internet Explorer. It's not happening as fast with on the operating system side because IT managers are fairly fixated on Microsoft and its suite of solutions, but they, and consumers, have an additional degree of freedom to select their browser of choice -- even if they're stuck on Windows.

The fact that there is a monotonic decrease in both Windows and IE market share suggests that there is nothing Microsoft can do, including advertising, to reverse the trend. They can slow it, by reducing the constant p1 via marketing, but it can't be reversed. A similar effect happened to the Netscape Corporation as Microsoft monotonically stole their market share in the late 1990s.

When companies arrive in such a situation, there are two common reactions. One is denial. The other is a radical change in the status quo.

I remember a classic case of denial. When I was at the Oak Ridge National Laboratory in the 1990s, I was a member of an advisory committee for IT managers. We were tracking the purchase of desktop databases by laboratory staff members. The sales data, over a period of about 10 years was roughly as follows:


Our committee recommendation was that Access was coming on strong and would supplant Foxbase. The Laboratory should plan accordingly. Management reaction, upon seeing that chart was basically: "Foxbase has the bigger installed base and would be the standard forever. The data is incomplete."

It was a denial of the mathematics -- in this case, three consecutive Gaussian curves. [Later, MS Access did, in fact, supplant Foxbase well before Foxbase was discontinued.]

The other thing a company could do, and Apple has done this, is to make a radical change. Just as Apple canned OpenDoc, the Newton and Classic, Microsoft could radically rebrand and rethink the browser. That's unlikely, considering their commitment to business -- they've lost a degree of freedom along the way.

When a company can't change and the market forces are driven by the statistics of large populations, there's nothing to be done. Worse, when the statistics contain an implicit factorial function, a market share upheaval happens faster than anticipated. Then the denial morphs into a panic reaction which does more harm than good.

Now is probably the right time for Microsoft to think its way out if this IE market share problem before the same effect happens on the OS side. Apple's market share is also monotonically increasing due to the same market effects: word of mouth, on the Internet or in person.

While the decline in Windows market share is happening at a much smaller rate, similar fundamental forces are at work there as in the browser market. It's just that the mathematical constant, p1, is smaller. Solving the IE vs. Firefox problem will be a warm up for what comes next, an exponentially rising Apple market share that gets out of hand and takes Microsoft by surprise.



Yeah, but there’s one problem Apple would have to overcome for the Macintosh to truly supplant Windows (I’m not speaking browsers here, but the whole OS). Apple simply cannot provide enough hardware to cover everybody. They _will_ have to change how they do things (license) if they truly want to expand this far…. which, for some reason, I doubt Apple truly wishes to do in either case.


Lee Dronick

“They _will_ have to change how they do things (license) if they truly want to expand this far?. which, for some reason, I doubt Apple truly wishes to do in either case.”

I am thinking the same thing. Apple needs to be big enough so that people write Mac software, Mac accessible websites, and so that there are Mac compatible peripheral.


I’d love to hear more of an explanation from JonGI, and Sir Harry Flashman.

Providing enough hardware should be a matter of increasing quantity manufactured - not trivial, but not impossible, either, given a gradual increase.

As for licensing formats, MS charges for CALs - Client Access Licenses, on the server side, while Apple does not;  that should make OS X Server much less expensive for most businesses, and Windows machines can run as clients off of Mac OS X Server. Thus, seems to me, that Apple would have the advantage.  What am I missing?




@JonGI: “Apple simply cannot provide enough hardware to cover everybody.”

Wrong, Apple can easily satisfy additional demand by lining up more contract manufacturers. If there is a great shift in demand from Dells to Macs, Apple will just hire that contract manufacturing capacity that used to build Dulls.

@ Sir Harry Trollman

Are you really that naive? The World Wide Web is platform agnostic (and incidentally the first web pages were developed on NEXT, the precursor to OS X)-with the exception of IE, all of the major browsers are based on the exact same engines and have identical functionality. What works in Firefox or Safari in Windows works exactly the same in OS X. Maybe this should be rephrased as ‘websites that are less Internet Explorer-centric’. IE compatibility as a concern (it breaks CSS like it’s going out of style, no pun intended) is and has long been the bane of every web designer’s existence. And coming from this vantage point, I can tell you that the tide is definitely turning.

Apple computers were the first to implement USB, and have long been the only choice that more often than not don’t even require special drivers. Very, very few peripherals are ‘Windows only’ (even those made by Microsoft work great on the Mac).

Ugh. I won’t even get into the software FUD.

Anyway, interesting article, Mr. Martellaro. Growth of this kind exponential, and once it reaches a tipping point, lookout!

Lee Dronick

“Are you really that naive?”


” ?websites that are less Internet Explorer-centric?. IE compatibility as a concern (it breaks CSS like it?s going out of style, no pun intended) is and has long been the bane of every web designer?s existence. And coming from this vantage point, I can tell you that the tide is definitely turning.”

There is your clue kid.


Is this the new format?  All posts centered?  My God, it’s so hard to read!  Who’s brilliant idea was this?


“@JonGI: ?Apple simply cannot provide enough hardware to cover everybody.?

Wrong, Apple can easily satisfy additional demand by lining up more contract manufacturers. If there is a great shift in demand from Dells to Macs, Apple will just hire that contract manufacturing capacity that used to build Dulls.”

_Everybody_? And you think Apple is _willing_ to serve _everybody_? Apple isn’t even willing to make a tablet or netbook…. They have only a few, specific models—they could never satisfy everybody. One company does not a platform make. Sorry, unless Apple licenses, they won’t go beyond a few percentage points—but just thinking this is, IMO, a mistake. For Jobs and Apple, it’s not about market share. It’s about raw numbers, sell the best, charge the most, and make the most. It’s simple, and it works because of the first factor—sell the best. Market share may be nice, but it’s a side show for apple. Everybody thinks “market share” because of Microsoft, but it’s not about market share, and I hope it never is, because then Apple will cease to be Apple.


Jon T

This Tipping Point has been alluded to for some time now. The mathematics behind it is interesting but the reality is we all experience the word of mouth effect everyday now, especially with iPhone bringing more users over to Macs.

As for meeting demand with production issues.. why not maintain decent margins instead?

It’s a non-issue anyway, China and Taiwan are soon going to have SO much excess manufacturing capacity available that Apple would be able to satisfy demand and get some incredibly good deals too…

Jon T

“Apple isn?t even willing to make a tablet or netbook?”

Errr… Apple already makes the defining tablet, it also doubles as a telephone.

Maybe they will announce a slightly larger one in due course too.


There is nothing in this to suggest that the process magically ends with Apple. Apple will enjoy its heyday, but it will also have a successor.


Microsoft’s failing is its attempt to enter every possible market with every possible product. Apple is not obligated—or stupid enough—to manufacture any particular product, no matter how badly you want it. Just as Apple knows what features not to include, they also know which products not to make.

No matter how badly I want an Apple can opener, Apple won’t make it. However, I wouldn’t be surprised at all to see a Microsoft can opener.

Just because Microsoft has a failing, it doesn’t mean that Apple has to have it too.


“Errr? Apple already makes the defining tablet, it also doubles as a telephone.

Maybe they will announce a slightly larger one in due course too.”

Aha. So you do admit that the iPhone isn’t really the end-all for tablets. That is my point. Apple isn’t trying to be all for everybody.

Macintosh computers are built solely by Apple. The iPhone also, is built solely by Apple. These are products, not platforms, although they do function as platforms. But the truth is, they are products—and the best products you can find…. But as such, they can never replace Microsoft products, because Microsoft merely produces the software that runs on hardware. Others make the hardware. This is both the blessing and curse of Microsoft-based products. This allows a small firm to make a niche product that only a few people will want (a large tablet, for instance), yet still be profitable. It also allows for greater saturation of the market. Of course, it also creates compatibility problems for both the hardware and software manufacturers, but it is this flexibility and customizability where Apple products cannot go—and Apple will _not_ go. That is the way it is. Why try to pretend that Apple can and will replace Microsoft by continuing to produce the entire widget? It ain’t going to happen. Market share beyond the low double-digits won’t happen so long as Apple is happy making the entire widget. Now, if Apple were to decide to license the software, and not do hardware any more (like that’s ever going to happen), you might see the MacOS take over from Microsoft, but I don’t see it happening. It’s far more likely that we will see Linux slipping into Microsoft’s spot. And considering that you get far fewer headaches with Linux than with Microsoft products, and you aren’t paying the Microsoft tax, this is far more likely a scenario than the MacOS taking over the world.

And I won’t complain. I think that Apple being a minority player that is still strong is just the perfect position for Apple, and I don’t get why people think that Apple needs to “win” in the same way that Microsoft “won.” Also, by staying the minority platform, the issues that John discusses in his article above will not, I believe, ever come into play for Apple. IMO, very, very slow market-share growth is far better in the long run—more financially healthy, and better for the customer—than sky-rocketing sales growth. But that’s just my opinion, and what do I know….



The function is n^2 - n, which is quadratic, not factorial. While you can represent it with factorials, the cancellation makes the function quadratic.


OK, JonGI, I think I get what you are saying.  MS does an OS only, there’s lots of hardware manufacturers, they provide lots of variety in products.  Therefore,  MS will continue to command most market share.

Have I got it right?

Seems to me, then, that a lot will depend on i) how things develop, and ii) on how you define market share. 

Given, as you state, this is MS’s strength, as well as their curse, I note that the complexities they have to deal with are sometimes crippling.  To the degree that customers have more and more difficulties caused by this complexity - think spaghetti code, access points for viruses, and other malware, difficulties due to mass demand for backward compatibility, etc.
So a lot depends on how these issues get better or worse for MS, and its customers.

Second, I don’t know how much of the overall market for computers is made up of tablets and other ‘fringe’ products, that Apple - and we agree here, is not likely to manufacture.  If these products make up only, say 10% of the overall market, it seems to me that there’s plenty of room for Apple to increase to, say 50% market share.  I’m not saying this will happen - just that it could, and still maintain ‘making the whole widget’ (and I agree with you about that, too - that Apple will not license the OS.)

Your thoughts?



@LLG Yes, this is essentially what I was saying.

Now, as to market share. As I said, market share isn’t a bit of a concern to me. In fact, I don’t really long for huge market share—and 50% is way beyond what Apple needs. What Apple needs, IMO, is market growth, not market share. Growth, in fact, demands a small market share. Look at what happened to Microsoft. With market saturation, there’s no where to go, but to “create” new market opportunities, etc. It isn’t an enviable position. This is why I said that sloooow growth is what Apple is most likely looking for. I doubt it will happen, if John’s article above is correct, however. grin



Although my head starts to spin, confronted with mathematics of a more advanced kind, I think I can follow the reasoning somewhat. The comments are also interesting and I have often thought that at some point of Appleʼs success, they will need to consider a change in tactics. Of course they can hire manufacturers, as many as they can pay for, but that is not the problem as I see it.

The problem is to develop a variety of devices and enough models of those. So far, they have wisely cut down on the variation and concentrated on a few, essential models of each device. Thus, we have a limited range of desktops, laptops and phones and iPods. No matter how many factories they could get to produce their hardware, they would not hire others to develop them. And I do not think that Jobs or others at Apple seriosly see it as an ideal situation that one brand should not only dominate the OS- and softwaremarket, but also the hardwaremarket.

I tend to believe that Jobs also want to set things straight. That he still feels that what happened way back when, when Microsoft managed to achieve a global domination, was wrong. But I do not think that he is interested in replacing Microsoft with Apple. Thus, those who say that market share is not the main concern of Apple is probably right.

But regardless of what anyone at Apple thinks, I believe that what is important, is that Microsoft is toppled down from their monopolic pedestal. When that happens, it will create leeways for others to get in. As it has been, all gates have been closed for that to happen, except for two, tiny cracks kept open by Apple and the Linux-field.

If Microsoft will shrink further, more opportunities will be opened for others, but not necessarily to create new OSes as we know them. Microsoft, too, cannot be totally counted out. It seems like their new chief software big bang - Ozzie - is introducing a different attitude up at Redmond and that may result in something positive not only for MS, but for all of us. That is, if their tendency from the past, to dominate whatever the cost, is being changed. And with Ozzie at the helm, that might happen, I think.

The game might change in many interesting ways. But whatever happens, the most important thing would be the end of the present situation. One company with a global monopoly. I cannot come to think of any other similar situation. National monopolies, yes, but global?


Thanks, JonGI, for your response.

I agree that market growth is great for Apple, and, of course,that’s exactly what has happened with the iPod and iPhone.  No reason why other products from Apple can’t do the same thing. 

That said, this shareholder also likes market share growth, for Macs, as well as iPhone, etc. And, I see no reason that Apple can’t go up to 50% or more, given the issues with Windows, and the cost advantage running Macs is said to have.  See

In this economy, anything to help lessen costs, especially for small and medium sized business (enterprise has a different set of issues).

So, you can advocate for market growth, and I’ll applaud you - but I also want market share growth.  IMO, it’s good for Apple, and the general economy.



“No matter how badly I want an Apple can opener, Apple won?t make it. However, I wouldn?t be surprised at all to see a Microsoft can opener.”

Microsoft won’t make a can opener unless Apple or Google makes one, first.

Ian Ollmann

The math here is bunk.  Math isn’t really what you want. Try chemistry, statistical mechanics more specifically. For example, the notion that interactions increase exponentially or geometrically is broken. I live in a city of a million people. I don’t have that many interactions. There is only so much time in the day. There is little point in me trying to maintain daily contact with more than a dozen people. I wouldn’t get anything done! So there is a rate limiting phenomenon that takes over long before the “math” explodes here. The reaction is viscosity limited.

Likewise Chemistry tells us that ternary or greater reactions—where more than two reagents come together to react simultaneously—are very rare, and practically never happen, except in cases where two reagents preassociate (e.g. enzyme + substrate 1) through an earlier process. e.g. Among bigamists, how many married both wives at the same time?

The right math to describe this stuff is likely that for thermodynamic equilibria.  We can fit recent market dynamics and figure out various reaction rates for Windows -> MacOS X conversion, and reverse reaction rates.  We can likely extrapolate to figure out the Gibbs free energy of the market at equalibrium—that is an empirical measure of how much better the market perceives one OS is over another. 

It could be sobering.

Ian Ollmann

In any case, there is plenty that Microsoft could do to reverse the trends. They could charge less for the OS.  They could change perceived value of Windows, either by advertising well (e.g. Mojave project), releasing a warmed over version that is not called Vista, or undercut the competition, for example by discontinuing Office for MacOS X. They could make a netbook version of Windows 7 and sell it for cheap.

The risk here for Microsoft is that their strategy of incompatibility will move from asset to liability at some point in the near future, should established MS non-standards move from majority to minority. At that point, I expect the collapse to come quickly, as people stop bending over backwards to work around the Microsoft-only problems.

A quick collapse is also expected from the behavior of chemical equilibria. It doesn’t take much to turn a 70-30 ratio into a 30-70 ratio the other way. Getting the last 10% is really the hard part. Luckily for Apple, that is more than enough volume to keep the company going and nicely profitable.

Log in to comment (TMO, Twitter or Facebook) or Register for a TMO account