How would a 2D model of Hotelling's law outweigh the benefits of a 1D model?

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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?

Giskard · Accepted Answer

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.

How would a 2D model of Hotelling's law outweigh the benefits of a 1D model?

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