Трагедія спільного пастку: Моделювання спільного ресурсу

Hardin's pasture

In a 1968 essay in the journal Science, ecologist Garrett Hardin gave the modern name to an ancient problem: a shared pasture, open to any herder who wants to graze cattle on it. Each herder gains the full benefit of adding one more animal, but the cost of the pasture becoming a little more overgrazed is shared among every herder who uses it. Since the private gain to any individual outweighs their own small share of the collective cost, Hardin argued that each herder is individually incentivized to keep adding animals — right up until the pasture is stripped bare and everyone, including the herder who "won" the most, loses.

Turning the parable into equations

The parable becomes a testable model once the resource’s own regrowth is written down explicitly. A common choice is logistic growth: the resource replenishes fastest at intermediate stock levels, and levels off as it approaches a carrying capacity K, exactly the pattern real fish, timber and grazing-land populations tend to follow:

dR/dt = r * R * (1 - R/K) - H R = current resource stock r = intrinsic regrowth rate K = carrying capacity (max sustainable stock) H = total harvest rate, summed across every user At H = r*K/4, regrowth is at its fastest possible rate — this is the "maximum sustainable yield": the highest harvest the resource can absorb forever without shrinking. Push the total harvest rate H above that maximum sustainable yield, even briefly, and the resource stock R starts a decline that regrowth cannot reverse — the classic collapse trajectory. Below it, the resource can, in principle, support indefinite harvesting forever.

dR/dt = r * R * (1 - R/K)  -  H

R = current resource stock
r = intrinsic regrowth rate
K = carrying capacity (max sustainable stock)
H = total harvest rate, summed across every user

At H = r*K/4, regrowth is at its fastest possible rate —
this is the "maximum sustainable yield": the highest harvest
the resource can absorb forever without shrinking.

Cooperators and free-riders

The interesting dynamics appear once the population of harvesters is split into two strategies. Cooperators voluntarily cap their own harvest at a sustainable share; free-riders harvest as much as they can regardless of the resource's condition. As long as free-riders remain a small minority, cooperators' restraint can keep the total harvest below the sustainable threshold and the resource survives — but every free-rider captures a larger individual payoff than a neighbouring cooperator, so imitation-based strategy updating (copying whichever strategy is currently paying off better) tends to reward defection and erode cooperation over time, exactly the incentive structure Hardin described, now made explicit and quantifiable.

Відповідь Остром

Оригінальна стаття Хардін розглядав трагедію як майже неминучу без приватизації ресурсу або запровадження зовнішнього державного регулювання. Політичний економіст Елінор Осстром протягом десятиліть документувала реальні спільноти, що керували спільними ресурсами – швейцарські альпійські пасовища, японські лісині села, іспанські зрошувальні системи – завдяки власним місцевим інституціям: чітко визначені межі використання ресурсу, прогресивні санкції за надмірне використання та низькобюджетні механізми вирішення конфліктів. Її робота, яка отримала Нобелівську премію з економічних наук у 2009 році, показала, що трагедія є реальним ризиком за відкритого та безрегульованого доступу – не неминучим законом спільних ресурсів загалом.

The harvest quota as a policy lever

In the model, the single most direct intervention is a hard cap on total harvest, set at or below the maximum sustainable yield — the resource-management equivalent of a fishing quota or a grazing permit system. It does not require changing anyone's underlying strategy or motivation; it simply enforces, by external constraint, the outcome that voluntary cooperation alone might fail to sustain once free-riders are present in large enough numbers.

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