Thought Toys · Strategy & computation · Exhibit 106
Two ice-cream carts on one beach. Every sunbather walks to whichever is nearer, and each cart only ever moves where the customers are. Nobody colludes and nobody is told to imitate anyone. They end up standing side by side anyway.
Drag a cart anywhere along the beach cart sunbather
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The line is a beach and the dots are sunbathers, spread evenly along it. Each amber marker is an ice-cream cart, and the shaded stretch under it is its territory: everyone standing closer to it than to any other cart. Drag a cart and watch the boundary move.
Press let them compete. Each cart takes a turn moving to whichever spot would win it the most customers. That is the only rule. No cart is copying the other and neither one wants to be next to its rival.
They end up next to each other regardless. Standing at the quarter mark, a cart owns everything to its left for free — those customers have nowhere nearer to go. So it can always gain by stepping toward its rival, and so can the rival.
The bottom bars are the bill. At the huddle the average sunbather walks a quarter of the beach; spread evenly they would walk an eighth. Competition doubles the walking, and no cart did anything but pursue its own customers. Both bars average over the sunbathers who actually buy, which only matters once the second slider makes some of them give up.
Now set the carts to three. There is no resting place at all — the one in the middle is always being crushed, always bolts for the outside, and the shuffle never ends. Set it to four and calm returns, as two pairs. And drag the second slider down: once a sunbather will give up rather than walk, the carts move apart, because a cart at the edge can no longer take the customers behind it for granted. The huddle was never competition itself. It was captive demand.
improve/verify/106-hotelling.js, 34 checks) checks the share rule against
customer-by-customer counting, runs the dynamics from 40 random starts, sweeps 861 two-cart
configurations to confirm the middle is the only resting place, and confirms the 2×
by closed form and by independent quadrature. Four negative controls: a sweep of 3276 three-cart
configurations finds no equilibrium (and after 200 rounds somebody can still gain 48
points of share); four carts do have one, as pairs at ¼ and ¾, so three is
the oddity rather than the model; capping how far customers walk drives the carts apart
instead of together; and bunching the crowd off-centre moves the equilibrium to the crowd's
median, not the beach's middle — which is the median-voter result, and rules out
“they meet in the middle because the middle is the middle”.
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