Friday, March 07, 2008

Extremes Meets

In economic theory, there's a problem called "Hotelling's law". Hotelling is a famous mathematician's name.

What is the "Hotelling's law"? Here's one example:

Suppose that there are two competing shops located along the length of a street running north and south. Each shop owner wants to locate his shop such that he maximises his own market share by drawing the largest number of customers. (In this example, the shop itself is the 'product' considered.)

Customers are spread equally along the street. Suppose, finally, that each customer will always choose the nearest shop.

For a single shop, the optimal location is precisely halfway along the length of the street. The shop owner is indifferent about the location of the shop since it will draw all customers to it, by default. From the point of view of a social welfare function that tries to minimize the sum of square distances that people need to walk, the optimal point is halfway along the length of the street.

Hotelling's law predicts that a street with two shops will also find both shops right next to each other at the same halfway point. Each shop will serve half the market; one will draw customers from the north, the other all customers from the south.

A careful reader should notice that the point where the two shops get together is the Nash equilibrium. This is not just only the problem of shop location. We can see similar problems in our life. What is it, for example? That's your question.

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