Outside of Austrian and Marxist circles, there is a well known phenomenon that dead people, occasionally, were wrong about things. So it was with Harold Hotelling in his 1929 paper about what is now called Hotelling competition, but that has not kept people from repeating his mistake on reddit, in TED ed videos, and in wikipedia articles.

What's all this about? Well, largely, it's about trying to answer the question: "why do you occasionally see similar businesses located next to each other?". Let me explain Hotelling's original answer to this question. Usually, the example people give to illustrate Hotelling competition is of ice cream vendors on a beach. Imagine there are 2 ice cream vendors on a 1 mile long beach, that there are people evenly distributed along the beach, and that people don't like walking for their ice cream. Where on the beach should each vendor setup shop? Your options are:

1. |------XY------| (both in the middle)

2. |X------------Y| (both at opposite ends)

3. |---X------Y---| (at the 25th and 75th percentiles)

Hotelling correctly points out that Case 3 is socially optimal. In this world, the distance people need to walk for their ice cream is minimized -- nobody ever needs to walk more than a quarter of a mile. A monopolist with 2 ice cream stands would pick this configuration, since said monopolist could charge the highest possible price here (consider that high prices might deter far away customers from bothering to walk over for ice cream). Two non-monopolist firms located here would then get all customers to their extremes and split the middle of the market.

Hotelling goes on to argue that what the vendors actually will end up picking is Case 1 -- both locating in the same spot in the middle. The logic here is basically that even if your 2 vendors start at the collusive optimum (Case 3), each ice cream vendor will have an incentive to move a little closer to the center since doing so lets them capture a larger market share. If vendor X moves a little closer to the center, the number of customers to his left and within his exclusive chunk of the market increases, boosting his market share and thus profits. Vendor Y of course will, for symmetric reasons, have the same incentive and so she also will move to the center. Hence you end up in Case 1 and have the "principle of minimum differentiation".

There's a bit of a problem here, however. We've basically been taking price as constant in the above reasoning. But what if we let vendors X and Y pick their prices? (Note: a more mathematical treatment is available here for those interested.)

First, in Case 1, what prices do vendors X and Y charge? The answer has to be the same price, since if one was a little cheaper then they would capture the whole market. The answer also has to be "the lowest possible price", since the incentive to undercut your competing ice cream vendor (Bertrand competition style) will lead to a downward price spiral that bottoms out at the point where you are both earning 0 profits.

Are there deviations from this situation that allow one of the vendors to earn positive profits? Yes! Starting from Case 1, vendor Y can choose to move a little further to the right and then jack up prices. She will cede market share to vendor X (nobody between them will walk to her), but some of the people to her far right will prefer to buy her more expensive ice cream than schlep all the way to vendor X. In fact, it turns out, the equilibrium you end up in is actually Case 2 -- both ice cream vendors locate on opposite sides of the beach, charge jacked up prices for ice cream, and earn profits off of the people that are willing to put up with their high ice cream prices on the grounds that walking all the way to the other vendor isn't worth any potential savings. So, in this case, you have what is sometimes called the "principle of maximum differentiation".

Hotelling's original analysis went wrong because he had firms choosing their location just based on competition over customers, without paying much attention to price competition. This is all well and good when prices really are fixed (e.g., 2 politicians trying to win an election should both pivot to the center, assuming all voters are required to vote) or when actually there are no transportation costs (i.e., when there is no product differentiation and the situation reduces to Bertrand competition). But the more general result is that firms want to engage in product differentiation (i.e., locate at different points on the beach) because it softens price competition and lets them extract some positive profits using their market power over people with a taste for their differentiated product's characteristics (i.e., over people sitting closer to them on the beach). Indeed, this is a large part of why you see firms engaged in so much work to build up unique products, brands, etc. -- it's all a form of product differentiation that can help them build up some market power.

As a side note, why is it that similar firms often locate in the same place? Well, consider that a key assumption in the Hotelling model is that people are distributed evenly along the beach. If it turns out most people are in one spot on the beach, both ice cream vendors should go near there. Similarly, in real life, you often see several coffee shops located in the same spot when that spot happens to be in a very high traffic location. Sure, it affords them less location based product differentiation, but there is little value in having the brand identity "further away from the subway stop and the office building cluster".

Final caveat: These are just illustrative models. While problems with Hotelling's paper have been long documented across a wide array of papers (for a single example, see this old Econometrica paper), you can rig up a stylized model to deliver Case 1 or something between cases 1 and 2 as the correct solution by playing with all sorts of assumptions. My main point here is that the pat story given in TED video misses entirely that there are incentives for product differentiation and pitches an equilibrium that requires very special assumptions indeed.