Cooperation in the Commons

In 1833, British economist William Forster Lloyd presented a setting in which selfish individual economic behavior might lead to collapse. He posited the scenario of a plot of land held in common by a set of farmers on which each could graze his or her cows. If each farmer grazes as many cows as possible following self-interest, the grass would deplete and the “commons” would have no value at all. This same phenomenon could apply to fishing, hunting, aquifer use, and so on. That in turn suggests the need for governmental regulatory regimes involving licensing and the like.

To see how to avoid such a value and environmental collapse without resorting to legal sanctions, let’s start with two partners, Bob and Alice, who own a common piece of land in equal shares. Based on historical records, they determine the land can support a total of 100 grazing cows.

They devise a mutual tax scheme. Each of Bob and Alice pays for the privilege of grazing a cow (up to 100 cows in total) on the commons into a kitty. Any such cow is eventually sold for the benefit of the cow’s owner. The kitty is divided equally between Bob and Alice. The idea is to prevent overgrazing while maximizing outcomes.

Now, it turns out Bob has higher fixed costs, but lower per cow costs than Alice. Specifically, Bob pays $5 per cow from birth until sale and a fixed cost regardless of the number of cows he raises of $10. If Bob has b cows, that comes to: $10+(b×$5). Alice pays a fixed cost of $7 plus $7 per cow from birth until sale, that is, $7+(a×$7). The sale price of the cow is $12.

Can Alice and Bob keep the commons productive while maximizing their profits?

To determine who grazes how many cows so that a+b = 100, they start with the plan in which each grazes 50 cows for a kitty price of 0. Then either one can propose a kitty price. The other can choose to pay that kitty price for as many cows as he or she wants. The kitty price proposer must graze any left so that a+b = 100.

Warm-Up: Suppose Alice sets the kitty price to $4 and Bob decides to graze 70 cows, leaving 30 cows for Alice. What will be the net profit of each?

Solution: Bob will pay ($4×70) to the kitty and ($5×70)+$10 to graze the cows. So, Bob’s total expenses will be ($4×70)+($5×70)+$10 = $640. Bob will receive ($12×70) for the sale of 70 cows and half the kitty ($4×100/2). So, Bob will have revenues of $840 + $200 or $1,040 in total. Thus, Bob will have a net profit of $400. By similar reasoning, Alice will have a net profit of ($12×30)+($4×100/2)−(($4×30)+($7×30)+$7) = $223.

Warm-Up: Suppose they stick with the kitty price of $0 and each grazes 50 cows. What is the net profit for each of them?

Solution: Bob will have a net profit of $600−$260 = $340. Alice will have a net profit of $600−$357 = $243.

Question Friendly Farmers: Will either Bob or Alice have some interest in changing the kitty price from the starting point (of zero for the kitty price and 50 cows each) and then allowing the other to choose the number of cows to buy at that kitty price? Assume Bob and Alice are friends, so if some decision does not change the profit for one (X) but benefits the other (Y), then X will make that decision.

Solution: Because Bob has lower costs per cow, it is overall more efficient for Bob to graze 100 and for Alice to graze none. However, if Alice sets the kitty price to below $5, then Bob will graze 100 cows and Alice will receive less than the $243 she would have received under the $0 kitty price, 50/50 arrangement. If Alice sets the kitty price above $7, then Bob will not graze any cows, forcing Alice to graze 100 and she will again receive less than $243. However, by setting the kitty price to exactly $7 and assuming Bob’s friendliness, Bob will graze 100 cows, make the same profit ($340) as he does under the 50 cows each arrangement and Alice will have a greater net profit of $343 (she still has fixed costs even if she grazes no cows).

Question Unfriendly Farmers: Suppose Bob and Alice have an argument, so Bob and Alice are no longer friends. This means if some decision does not change the profit for one (X) but hurts the other (Y), then X will make that decision. However, each wants to maximize his or her profit above all so if a decision helps X and helps Y, then X will make that decision. Should Alice change the kitty price from $0 (where each grazes 50 cows and Bob makes a profit of $340 and Alice of $243) to something else in such a way that they will both benefit?

If each farmer grazes as many cows as possible following self-interest, the grass would deplete and the “commons” would have no value at all.

Solution: If Alice sets the kitty price to 7, then Bob will spite Alice by grazing 0 cows forcing Alice to graze 100 and reduce her profit to $143. Bob will make $340 whether he grazes 0 cows or 100 cows. However, if Alice sets the kitty price to $6, then Bob will graze all 100 cows and he will make a profit of $390 and Alice a profit of $293. This benefits both of them.

Bob and Alice have a bad year, so they each sell 5/6 of their shares in the commons to an investor. As a result, the investor gets 10/12 of the share of the kitty and each of Bob and Alice will get 1/12. The investor however does not plan on raising any cattle. As part of the sales contract, the investor sets a minimum kitty price of $4.

Question Investor: Assuming Bob and Alice are friendly again and assuming they start with 50 cows each at a kitty price of $4, does either have any interest in changing the kitty price upward to $5 or more knowing the other will then choose the number of cows to purchase at that kitty price?

Solution: At the $4 kitty price, 50/50 cow arrangement, Bob will have a net profit of $173.3 and Alice $76.3. This is lower than before the investor, because Bob and Alice will get a smaller share of the kitty. If Bob moves the kitty price to $5, then Alice makes $34.7 whether she buys 0 cows or 100. Assuming she still feels friendly to Bob, though, she might let him still buy all 100 cows and then he will have a profit of $231.7.

The problem with the solution to the last question is that Alice may not feel friendly when Bob lowers her profit. A better solution for Bob and Alice would be for Bob to buy 100 cows and pay Alice for this privilege.

Question Side Payment: In the same scenario with the investor, but where Bob and Alice are friends, is there some other allocation of cows along with a side payment that can benefit both Bob and Alice compared with the 50/50 arrangement without changing the out come for the investor? In which range is the side payment?

Some farmers are mutually friendly and the rest are mutually unfriendly. (Mutual friendliness is not transitive.)

Solution: Yes. If Bob buys all 100 cows, then the profits for Bob become $323.3 and, for Alice, $26.3. However if Bob makes some side payment to Alice of more than $50 and less than $150, then both Bob and Alice benefit relative to the 50/50 arrangement (staying with the $4 kitty). For example, if Bob makes a side payment of $100, then Bob would end with $223.3 (versus $173.3 in the 50/50 arrangement) and Alice $126.3 (versus $76.3 in the 50/50 arrangement) while the investor still receives 5/6 of the $4 kitty price per cow.

Upstart: Assume f farmers and i investors, each with equal shares in the kitty. The investors raise no cattle. Each farmer has a fixed cost plus a marginal cost to raise each cow. Some farmers are mutually friendly and the rest are mutually unfriendly. (Mutual friendliness is not transitive.) A total of g cows can be grazed. Suppose the kitty price is fixed at r. Initially, each of the f farmers is allocated g/f cows at the kitty price of r. Assume the farmers can change allocations among themselves all without changing the kitty price. Only mutually friendly farmers can make side payments to one another. Find an algorithm to determine how many cows each farmer will graze.