On a long beach, two ice-cream carts split the sunbathers between them. Where should each one stand? Anyone who has ever seen two petrol stations, two pharmacies or two fast-food outlets facing each other across the same junction has already seen the answer. A 1929 paper by the American economist Harold Hotelling explains why it happens, and why it is not good news for the customers.
Below, sunbathers are spread evenly along the beach and each walks to the nearest cart. Drag either cart. The colours show whose customers are whose.
Start the carts at the quarter points, one quarter of the way from each end. Each gets exactly half the customers, and nobody has to walk far. Now drag one cart a little toward the other. It steals a slice of the rival's customers and keeps all of its own, because the people behind it still have nowhere closer to go.
That is the whole mechanism. Standing next to your rival, on the side facing the larger crowd, always beats standing further away. The rival sees the same logic and steps over in turn. Press "Let them compete" and watch the carts take turns hopping toward each other until they stand almost back to back in the middle of the beach, each with half the customers: the same income as before, for no gain.
With the carts in the middle, every customer on the far ends of the beach has a long walk. Measured as an average, the walk is a quarter of the beach length. At the quarter points it is only an eighth, half as far. Neither cart gains from the clustering: each still has half the market. The customers lose, and the carts cannot get back to the better arrangement on their own, because whichever cart moves out first hands its rival the middle ground.
Hotelling's paper, "Stability in Competition", appeared in the Economic Journal in 1929. His own picture was a main street with two shops and customers spread evenly along it, and he pointed out that competition can make sellers' products too similar. The beach and the ice cream came later, in textbook retellings.
Economists soon noticed that the same pull appears anywhere rivals chase a spread-out audience. Two political parties aiming at the voters in the centre of a left-right scale tend to drift toward each other, a version known as the Hotelling-Downs model. Competing shops, brands and even TV channels can end up with products that look almost interchangeable.
This tidy result needs strict assumptions: fixed prices, evenly spread customers who always walk to the nearest seller, and exactly two sellers. Change one and the picture shifts. In 1979 Claude d'Aspremont, Jean Gabszewicz and Jacques-François Thisse published a paper in Econometrica titled "On Hotelling's 'Stability in Competition'", showing that Hotelling's original argument does not hold once the sellers also compete on price. Add a third cart and there is no stable arrangement at all: if two carts stand together, the newcomer can squeeze in just beside one of them and take its outer customers, and the squeezed cart then hops over the top, round and round. Even so, the carts still tend to jostle near the centre.
So the lesson is a careful one. The middle is a magnet for competitors, but the magnet is not the best place for the people being served. The next time two near-identical shops open side by side, you are looking at Hotelling's beach.