Any color you want so long as it is black

Author: Carlos Gomez Grajales

Last winter I bought my first tablet. I thought it would be easy to find my perfect travel companion but the decision was indeed harder that I thought. The reason? Well, almost every device looked exactly the same. And I'm not cranky; some other folks have also noticed that variety is not a very real option for consumers. My real options were the screen size (tablets usually have a screen size between 8" and 10". I know, a 5 centimeter difference is all we got), and the software, which can be either iOS or Android, but that was it. Processing power, storage capacity and everything else looked exactly the same. I even needed to read some specialized websites to discover what was really different! And guess what; those differences are in some really tiny details that are not always relevant to consumers.

In the end I bought one (I’d tell you which one but that would be a commercial and none of the manufacturers reached my price), but I was left with a weird feeling. The options were not as diverse as I thought they would be, at first I felt like all these tablet companies were selling the same product. And it gets worse because, as you may have noticed by now, that type of behavior is not exclusive to tablet producers. MP3 players are quite similar, so are movie theaters. I'm not a car expert but I can easily see that some companies launch the same vehicles with different names. Why is this happening? Are our entrepreneurs losing the ability to create? Are new products gone for good? Well, as usual, statisticians have the answer.

Harold Hotelling was an American statistician, born in 1895 and commonly known for his many contributions to statistical science. Among his many additions to statistics we can mention the Canonical Correlation Analysis and Hotelling's T-square distribution, including its use for hypothesis testing. Hotelling's T-square provides a multivariate approach to the t-test, allowing multiple comparisons of means between two populations. Yet despite all his contributions to our science, Hotelling's main interest was not in statistics but in economics, where he devised analytical models to explain the reasoning behind some economic rarities, such as the fact that every SUV looks exactly the same.

In 1929, Hotelling published an article in Economic Journal that explored this subject. In these pages, the statistician devised an economic puzzle somewhat similar to this: There are two "places of business" located along a line which may be Main Street in a town or a transcontinental railroad. Where should each retail stand be located in order to get the most business? For the sake of this explanation, let's think that the two places of business are two hot dog carts located in a street. To simplify the idea, let's assume that both hot dog sellers have the exact same price, ingredients and taste, so the only difference between them being the location of the stands, so that people who want a hot dog buy from the closest stand. So, where would you place your stand?

Most people would think that the best possible solution for both cart owners would be one that provides each business man an equal share of the market by locating the stands as illustrated in the next figure:

Images by rollingrck and Javiertz/Wikimedia.

By locating the carts exactly at ¼ and ¾ of the street length, both businesses have the same share of costumers in the street. Remember that a costumer will buy on each stand depending solely on the distance between him and the carts, so any customer in the red area will buy from the red stand and any customer in the green area will buy green hot dogs (presumably eco-dogs).

You may think that this configuration would be the optimal, but Hotelling will insist that you were wrong. What would happen if the red hot dog cart decided to step a little bit to the left? In that case, his share of the market would increase. Yes, those customers in the extreme right would need to walk more but who cares, it's not like they are switching to green just for that. The next picture shows how by moving a little to the left, the red hot dog cart can steal green's customers:

Images by rollingrck and Javiertz/Wikimedia.

So red wins! No, claimed Hotelling, since the green cart can easily use the same technique and regain his customers back by moving his cart a little bit to the right. So, is there an optimal point where both vendors would be satisfied and decide that they would do not better by moving they carts? Hotelling answer was definitive but weird: the optimal solution is for both vendors to stand side by side in the middle of the street, as presented in this picture:

Images by rollingrck and Javiertz/Wikimedia.

Why? Simply because with this configuration the cart owners have no further incentive to keep moving. What would be later defined as Hotelling's law, establishes that in many markets it is rational for producers to make their products as similar as possible. At this point you could have already noticed something really, really sad. This solution is in no way better for the consumers. The customers who want a hot dog now have to walk even more in order to get one. If you are in either extreme of the street you will save your gym fee since the distance to the carts has dramatically increased, but that is how the free economy works. That is why when I buy tablets nowadays I don't get too many options (black or white are not real alternatives). "Buyers are confronted everywhere with an excessive sameness," Hotelling himself wrote. This insightful observation was made by a statistician about 80 years ago.

Hotelling's article provides an explanation to those twilight zone events we experience in life, such as that coffee shop located exactly in front of other one, but it gets worse than that. These principles apply not only to every single hamburger you've ever eaten; Hotelling also noticed that this love for the middle also applies to politics. Let's assume the street does not represent actual distances but ideological ones. So instead of left or right of the street we have liberal or conservative thoughts, which are in fact also called left and right (which also proves that hot dogs are the best possible analogy for political parties). How can a candidate/politician/party move to the center? Well, by avoiding extremist ideas, or unappealing legislation, even by denying his relationships with not so “centered” companions. Hotelling himself described it clearly in his article: politicians "reply ambiguously to questions, refuse to take a definite stand in any controversy for fear of losing votes.” Sound familiar?

Hotelling's law can explain why candidates avoid controversial matters before elections. By doing something extremely liberal, you move to towards the left, losing “customers” in the conservative section. If you are a renowned liberal you already have the left side in your pocket, so there's no need in convincing those voters. It is always more tempting, more productive to flirt with the “middle”. Hotelling explained a reason why we aren't getting an immigration reform in the US prior to the elections, and he did it well before Barack Obama was born. Although Hotelling's law has limits, since not every political party and every company are exactly the same, it positively explains common phenomenons usually found in real life.

Hotelling is just another example of a statistician who contributed to develop knowledge in many other fields, which is the one of the things I love the most of statistics: you can use it everywhere to learn something about everything. I certainly hope statisticians can still contribute significant results in many more fields. That way maybe someday they will be able to tell us how on earth Nyan Cat managed to get over 80 million views on youtube.