Mathematical Proof of the Need for Anti-Monopoly Laws

Someone on Slashdot recently asked the question of why AMD could get away with suing Intel over their agreement with Skype that limited non-Intel CPUs to 5-way conference calls. This person's issue was that people make exclusive agreements all the time. My original thought was simply: "It's because Intel controls the market". I quickly realized, however, that this answer wasn't enough. Everybody knows that Intel controls the market. The real question was: "Why do we treat market leaders differently?". I think that many people have 'a gut feeling' that banning preferential agreements by dominant market players is a good thing, but it's very difficult for most people to explain precisely why. It turns out that the answer to the question lies in the mathematics of market control.

Preferential and exclusionary agreements are relatively common. Cereal companies vie for preferential placement in supermarket aisles; Coke and Pepsi are in a dogfight for exclusive deals in schools; Credit card companies offer huge incentives for exclusive rights to events like The Olympics. Everywhere you look, you can find examples of companies fighting for some sort of preferential treatment which allows them to get the attention of the consumer.

For the most part, these deals appear to be harmless. Especially where someone is attempting to break into a market, they simply allow a small player to get the attention of the consumer in some specific (but usually small) part of the market.

The rules change slightly when you've got a near-monopoly. This is part of what tripped up Microsoft in their anti-trust trials.

The heart of the problem is that it's far easier to convince someone to exclude "the competition" from the market when the competition has a disproportionately small portion of the market. This is because discriminating against the market leader will have a larger effect on your company than discriminating against a small player.

As a thought experiment, and for the ease of math, let's say that the general market is 90% Intel, 10% AMD, and Skype's customer base reflects that -- in other words, Skype's customer base is 90 parts Intel users and 10 parts AMD. If Intel has to pay $10million to compensate Skype for market opportunities lost by excluding AMD using customers then AMD would have to pay $90million to get Skype to go the other way and exclude Intel users.  Add to that the fact that Intel has 9 times as much income from their larger market share and you have a 81-1 difference in impact on their profit margins (9 times the income and 1/9 the cost to do the same thing). Or - - to put it another way, between gross profits and market share, Intel could afford to buy off 81 market 'slots[1]' for every one that AMD could afford to.

It's basically a market-ratio squared relationship. If the market ratio is increased to from 90%/10% TO 95%/5% (19-1 ratio) the advantage increases from 81-1 (9²) to 361-1 (19²). As you can see, this can easily spiral into a near-absolute market ownership, denying customers any real choice in the market no matter how good the competition is. (MS/Linux, anybody?)

Although the problem gets a bit more difficult as you introduce multiple market players, I think that it's clear that you can easily get into this death-spiral as soon as one of the market players exceeds 50% of the market.

It's actually a worse than ratio squared relationship because we haven't taken into account things like the probability that, if Intel has a 81-1 ratio of market-exclusionary agreements, they can now exact a higher profit margin without significantly affecting customers' willingness (or, ability) to buy AMD. That, however is harder to quantitize, so I'll only mention it as an aggravating factor, rather than including it in my math.

About the only real way to avoid this problem is to create artificial rules designed to stop such market-warping agreements when the market gets too lopsided, to prevent market choice from getting totally destroyed. I would also suggest that this simple bit of mathematics indicates that the 'dominant market position' required to block such agreements should kick in (at the very latest) by the time a market player reaches 50%, and arguably much earlier.

Table of relative advantage in preferential agreements vs market share:
(the actual formula is: marketshare=sqrt(advantage)/(1+sqrt(advantage))  )

Advantage  Market share
1-1 	   50.00%
2-1 	   58.58%
3-1 	   63.40%
4-1 	   66.67%
5-1 	   69.10%
10-1       75.97%
20-1       81.73%
50-1       87.61%
100-1      90.91%
1000-1     96.93%
10000-1    99.01%

Comments? Most notably, do I lose you? If I do lose you, at what point do you get lost? Suggestions are gratefully accepted.