Which economic idea says a resource everyone may use and nobody owns will be exhausted?

The tragedy of the commons, from a 1968 essay by Garrett Hardin: each herdsman gains the full benefit of adding one more animal to shared pasture while the cost of overgrazing is spread across everyone, so each rationally adds animals until the pasture fails.

The example is historically wrong, which matters. English commons were not open to all; they were governed by detailed local rules about who could graze how many animals and when, and they worked for centuries. Elinor Ostrom spent a career documenting real commons - Swiss alpine pasture, Japanese forests, Spanish irrigation, Maine lobster grounds - that had been managed sustainably by their users without either privatisation or state control, and won a Nobel for it in 2009, the first woman to take the economics prize.

Her design principles are the useful residue: clear boundaries, rules made by the people affected, graduated sanctions, cheap conflict resolution, and recognition from higher authorities. The tragedy is a real failure mode - it describes ocean fisheries and the atmosphere well - but it is a description of what happens when governance is absent, not an argument that governance is impossible. Hardin's own conclusions ran towards coercive population control and he later wrote openly racist things, which is now generally noted alongside the essay.