The commoner who added the extra animal has increased the size of his or her herd from two to three cattle, a 50% increase in numbers. However, the value of that herd, which is the value of that commoner's share of the productivity of the common, has only increased by 36%, as determined previously. This is because all of the cattle are now in less than optimal condition.

When the common land was stocked at the optimum rate of ten head of cattle, the cattle were all in 100% condition. But now there are eleven animals, and the productivity of the common land (based on the amount of grass it sustainably produces) cannot be increased; there is still only feed for ten cattle. This means that each animal only gets 10/11ths (91%) of its optimum feed, and will only be in 91% condition. So, in terms of productivity, the three animals are only 91% of three animals, which is 2.73 animals, which is the 36% increase over the original two animals.

The two animals of each of the other commoners are also only in 91% condition, and so in terms of productivity they are only 91% of two animals, which is 1.82 animals.

In terms of productivity, the total number of animals on the common is 2.73 + 1.82 + 1.82 + 1.82 + 1.82, which adds up to ten animals (allowing for rounding errors), which is the optimum stock of the common land, as was nominated at the beginning. So there is no overall gain in productivity, just a redistribution of the share in favour of the commoner that added the extra animal.

Actually, there is an overall loss in productivity as the land is used at greater than its optimum stocking rate (initially, while there is still plenty of grass, the cattle won't stop eating when they get to 91% of their full capacity); so the land will be stocked at greater than its optimum stocking rate, and will be degraded.

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Tragedy of the Commons with quantities

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