The tragedy of the shared field

Every herder who adds one more cow to the shared pasture is right to — it's the rational move for them. Add up every individually correct decision, though, and the pasture ends up worth less than if a single owner had run it. Nobody has to be greedy. The arithmetic does it on its own.

What you're seeing

A pasture can support some grazing before it's crowded. Push past that and every cow on it produces less, because there's less grass to go around. The curve on the chart is the total value the pasture produces at every possible level of grazing — low near zero (too little use, hardly any cows), highest in the middle (well-used but not crowded), and back to zero once it's overgrazed entirely.

A single owner would stop right at the peak — the level worth the most. But split the pasture among N herders who each decide for themselves, and something else happens. Each one keeps adding cows exactly until it stops paying them personally to add another. That point comes well past the peak, because part of the cost of each extra cow — the little bit of extra crowding it adds — is a cost every other herder eats too, and no one has any reason to count a cost that lands on somebody else's herd. The dot doesn't sit at the peak. It slides down the back slope, and it slides further every time one more herder joins.

Watch what that costs. At one owner the dot sits exactly on the peak — full value, nothing wasted. Add herders and it drops fast at first, then more slowly, but it never turns around and it never levels off: as the crowd of herders grows without bound, the total actually grazed climbs toward exactly double the efficient level, and the value captured shrinks toward nothing — not "somewhat less than ideal," but arbitrarily close to zero. Already by the sixty herders at the top of this dial, well under a tenth of the possible value is left, with no floor in sight.

Now press the toggle. Give each herder their own private plot, with the identical crowding curve — only now the crowding is theirs alone, never shared. The dot stops moving. It sits on the peak no matter how many herders you add, because there's no longer a cost that lands on anyone but the person making the choice. That's the whole difference: it was never "too many decision-makers" or "diminishing returns." It's sharing the resource itself.

The rule, exactly. N identical herders each choose a usage x_i ≥ 0. The value per unit falls with the total usage X = Σx_i: v(X) = a − bX, and each unit costs k. Herder i's payoff is x_i(v(X) − k). Best-responding to the others and solving the symmetric fixed point gives X_Nash = N(a−k) / (b(N+1)), while a planner maximizing total welfare W(X) = X(v(X) − k) chooses X_social = (a−k) / (2b). Their ratio, 2N/(N+1), and the welfare actually retained, r(2−r) with r that ratio, both depend on N alone — not on a, b, or k. Verified in node (improve/verify/84-commons.js, 85 checks): a damped best-response fixed-point solver — seeded from zero usage, never from the formula — converges to the same X_Nash; an independent ternary-search maximizer finds the same X_social; a true N-player simulation started from random, unequal usage still converges to the identical symmetric outcome; and the welfare-retained law is confirmed to run to exactly zero as N grows without bound. A private-plots negative control — the identical crowding curve, but never shared — gives a ratio of exactly 1 at every N tested, both by formula and by an independent best-response iteration with no externality term to ignore at all. This is the classic non-cooperative baseline, not a claim that every real commons fails. Real ones — fisheries, grazing lands, groundwater basins — are very often managed successfully through communication, monitoring and shared norms; Elinor Ostrom's work documenting exactly this won a Nobel prize. What's modeled here is deliberately the pessimistic case: what happens with no coordination between the users at all.

← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 84.