A system with no stable state to be steered into.
The commons had a rhythm to keep, and steering meant holding a population on it. This one has nothing to keep, and that is the point of it: can a controlled minority stabilise a system with no rest point at all?
Vendors pick a spot on a beach. Beachgoers are spread evenly and walk to whoever is nearest. Each vendor wants the biggest catchment. Every round, each moves to its best response given where everyone else stands.
With two vendors this is the famous result: both slide to the centre and stop. It is why competing shops cluster, why rival petrol stations sit on the same corner, and why two parties converge on the median voter. Hotelling called it the principle of minimum differentiation.
Add a third and something else happens.
One vendor moves per round, in turn, to the best spot available given where everyone else is standing. They choose among twenty-one storefronts, marked on the strip — not among the real numbers, which is why a move is a visible jump rather than a shuffle.
Two vendors settle in about nine moves. Four settle paired at the quartiles. Three run until you stop watching.
This is not an artefact of the starting positions or of the update order. There is genuinely no arrangement of three vendors from which nobody wants to move — the middle vendor is always squeezed and always has somewhere better to be, and whichever way it jumps it creates a new squeezed middle. Eaton and Lipsey proved it in 1975. You can also just count:
The Shared Resource game has a stable state that is exactly reachable and one sentence long, and the question is whether a population finds it. Here there is nothing to find. Steering, if it means anything, has to mean something other than arriving — holding a cycle inside a bound, maybe, or slowing it down.
Answering "no" is as informative as answering "yes", and either way it is arithmetic — market shares are computed, not judged. Markets, routing and pricing all live closer to the boardwalk than to a game with a tidy optimum.
Hotelling (1929), Stability in Competition; Eaton & Lipsey (1975), The Principle of Minimum Differentiation Reconsidered. MIT; exported data CC0.