I'm not sure if this goes here, or on a math exchange. I could move it if you guys want...

Let's examine Hotelling's law on a 1D plain with two shops.

Both shops would do society a favour if placed on the 1/4 mark. But since both owners want to make money, the two shops will most likely be placed in the middle, right next to each other.

There have also been numerous mathematical formula's to analyse Hotelling's law in 1D with more than two players.

However, would this model work in a specific 2D plane with n players?

And if so, how would the information-benefits of a 2D model outweigh the ones of a 1D model concerning the real word?

Would a 2D model better help shop owners find a more suitable location?


There are several scientific papers analyzing this on topologies other than the unit interval. A 2D example is Spatial competition of firms in a two-dimensional bounded market by Aoyagi and Okabe. Unfortunately there are usually no equilibria in these cases. The intuition behind this is that the consumers arriving from different directions are usually not "balanced", and one could gain by deviating to a direction where a higher number of consumers are coming from.

Also, Hotelling models are not used in practice (to my knowledge) when one is assessing where to open a new shop.

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  • $\begingroup$ Really? I thought it was a consideration when businesses open new stores... business-achievers.com/general/… $\endgroup$ – user24424 Dec 2 '19 at 1:37
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    $\begingroup$ This is a blog post. Writing "Busy intersections are generally good." does not get as many clicks. In some industries the clustering effect is relevant, e.g. you can have a "shopping street" for tourists. $\endgroup$ – Giskard Dec 2 '19 at 6:29
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    $\begingroup$ More of a side note, AFAIK there is not really an equilibrium in Hotelling's model either as was pointed out by d'Aspremont: jstor.org/stable/1911955?seq=1 $\endgroup$ – Maarten Punt Dec 2 '19 at 11:02
  • $\begingroup$ @MaartenPunt I included this in the first draft of my answer but then I decided it would obfuscate my point :) $\endgroup$ – Giskard Dec 2 '19 at 11:39

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