Tragedy of the Shared Pasture

Everyone grazes sheep on one shared pasture — watch why no grass survives.

The pasture belongs to everyone; the sheep belong to each herder. The more sheep there are, the less grass each one eats. Still, each 🧑 herder counts only the profit from their own animals. So the total overshoots the right flock size, and the pasture's profit evaporates. The right size is in the formula below.

For example, a class 📦 box holds 30 🍬 candies. Take just one a day and they last a whole month. If everyone grabs handfuls, the box is empty in three days. One simple rule — at most two each — keeps the candies alive.

v(S)=a−bSv(S) = a - bS
S∗=a−c2bS^* = \frac{a - c}{2b}

Part of these learning paths

Examples & Exercises

A 🐑 sheep is worth v(S)=10−Sv(S) = 10 - S and costs 22 to raise.

If 33 herders share the pasture, how many sheep graze in total, and how far is that from the social optimum?

33 herders, no limit — the reality marker has passed the optimum.
Show proof

  1. The social optimum is where total profit S(10−S−2)S(10 - S - 2) peaks: S=4S = 4 sheep, earning 4×4=164 \times 4 = 16.

  2. But each herder counts only their own sheep.

    Acting alike, the three push the total to 3×83+1=6\frac{3 \times 8}{3 + 1} = 6 sheep.

    Total profit falls to 6×2=126 \times 2 = 12 — nobody did anything wrong, yet 44 evaporated into the grass.

Now there are 44 herders.

What happens with no limit — and what if each herder is capped at 11 sheep?

44 herders, capped at 11 sheep each — the total sticks to the optimum of 44.
Show proof

  1. Unlimited, the total climbs to 4×84+1=6.4\frac{4 \times 8}{4 + 1} = 6.4 sheep and total profit sinks to 6.4×1.6=10.246.4 \times 1.6 = 10.24. More herders means a smaller slice each — and a bigger waste.

  2. Capping each herder at 11 sheep makes the total 4×1=44 \times 1 = 4 — exactly the social optimum.

    Total profit recovers to 1616.

    One sharing rule is all it took to stop the tragedy.

1. Intro to Microeconomics