Why two ice-cream stands end up side by side in the middle of the beach, and why a standoff in which neither dares to move is called a Nash equilibrium.
The video builds the intuition; sources and limits are set out in the article. English- and Japanese-captioned versions of the vertical film are still on the to-do list.
Why do rival shops so often open next to each other? Hotelling (1929) answered with competition along a line.
Take a 100-metre beach with one visitor every metre. Two ice-cream stands, A and B, sell the same flavour at the same price, and visitors follow one rule: buy from whoever is nearer.
- With the stands at 25 m and 75 m, each gets half, and no one walks more than 25 m.
- If you were A, you would step toward the centre: you lose no one on your left and win a few more in the middle. B thinks the same.
- Step by step, both end up side by side in the middle. Sales are still split in half, but visitors at the ends now walk twice as far.
- Now whoever steps back alone loses customers, so neither moves. Such a standoff is a Nash equilibrium: stable, yet not necessarily best for everyone.
Mind the assumptions: two sellers, fixed prices, and visitors who buy however far they must walk. If the stands can also compete on price, or customers stop buying when the walk is long, the result can reverse and sellers deliberately keep apart (d’Aspremont, Gabszewicz and Thisse 1979). Clustering in real cities has other causes as well.
Japanese version
The same film with Japanese captions.
Sources: Hotelling (1929), Stability in Competition; d’Aspremont, Gabszewicz and Thisse (1979).