The phrase comes from a 1968 essay in the journal Science called "The Tragedy of the Commons", by an ecologist with an excellent first name, Garrett Hardin. His example was a pasture shared by herders, each adding one more animal, and his conclusion was gloomy: shared things get ruined unless somebody owns them or a government polices them. The political scientist Elinor Ostrom spent decades checking, in the field, whether that was true. She studied Swiss mountain meadows, Spanish irrigation canals, forests in Nepal, and lobster grounds in Maine, and found that communities very often avoid the tragedy with rules they design, monitor, and enforce themselves, with nobody owning the resource and no government doing it for them. In 2009 she won the Nobel Prize in economics, the first woman to do so. The coin pot with punishment comes from experiments by the economists Ernst Fehr and Simon Gächter, published in 2000 and 2002. With punishment allowed, people paid to punish free riders even when it cost them and they would never meet those players again, and contributions rose over the rounds instead of fading away.
The pond that everyone owns
Four fishers share a pond. The fish grow back, but not fast enough for everyone to take as much as they want. This is the Cookie Game with four players and a future, and the pond is what crumbles. You will play it, watch it collapse, work out exactly how much it can give, and then build the rules that keep it alive.
Here is a pond. It belongs to nobody, which means it belongs to everybody: you and Robo 1, Robo 2, and Robo 3 all fish in it. It starts with 60 fish. Every season, each of the four of you decides how many fish to catch, from 0 to 10. Then the fish that are left have babies, the pond regrows, and a new season starts.
The regrowing is the whole point, so here is the rule exactly. If F fish are left after the catch, the pond adds
new fish before the next season. Try it: 60 fish left gives 0.5 × 60 × 0.4 = 12 new fish. 20 fish left gives 8. A pond with 100 fish is full and adds nothing. A pond with 0 fish adds nothing, forever. If the four of you ask for more fish than the pond has, you split what is there in proportion to what you asked for.
Play twenty seasons. Choose your catch with the slider and press Next season, or let it run. The three robots come in flavors: Greedy always takes 10, Careful takes 3, Copycat takes whatever you took last season, and Mixed gives you one of each. Your goal: after 20 seasons the pond still has at least 40 fish in it, and you have caught at least 70. With Careful robots that is possible. Taking 3 every season gets you 60; the last 10 takes some thinking.
The pond
Four fishers, 20 seasons. The pond regrows after every catch.Why it collapses
Start with the arithmetic. The pond can only give back what it grows, and the most it ever grows in one season is 12.5 fish, which happens when exactly 50 are left after the catch (the Math corner shows why). So four fishers can take about 12 fish a season between them, forever. Three each works: the pond sits in the sixties, everyone catches 60 fish in 20 seasons, and season 21 looks just like season 1. Four each is 16 a season, more than the pond can ever grow, so it shrinks a little every season, then a lot, and is empty by season 9. Ten each empties it in two seasons. After that, everyone catches nothing, forever.
Now the game theory, because the arithmetic is not the interesting part. Put yourself in one fisher's boots. Whatever the other three do, taking one more fish this season gives you one more fish, right now, for certain. The cost is a slightly smaller pond next season, and that cost is split four ways and arrives later. One whole fish for you now, against about a quarter of a fish for you later. So taking more is a dominant strategy, exactly like Grab in the Cookie Game of Chapter 2, and every fisher can see the same arithmetic. If the pond game is played for a single season, everyone taking 10 is its only Nash equilibrium (Chapter 3). Played for 20 seasons with a last season everybody knows about, thinking backwards (Chapter 5) unravels it the same way it unraveled the repeated Cookie Game in Chapter 11: grab at the end, so grab before the end, all the way back to season 1.
This is the tragedy of the commons. A commons is anything shared that nobody owns: a pond, a pasture, the air, the fish in the sea. The tragedy is that everyone's best move adds up to a disaster nobody wanted. Nobody has to be evil for it to happen. Try the calculator: pick a catch, imagine all four fishers making it every season, and watch where the pond goes.
What if everyone did that?
All four fishers take the same amount, every season, for 20 seasons.The public goods game
Here is the same trap with coins instead of fish, stripped down so you can see the gears. Four players, and each of you gets 10 coins. Each of you secretly puts some of your coins into a shared pot, from 0 to 10. The pot is then multiplied by 1.6 and split equally among all four players, whether they put anything in or not.
Check the two extremes first. If all four put in everything, the pot is 40, it becomes 64, and everyone walks away with 16 coins. Everyone is richer. If nobody puts in anything, everyone keeps 10. Now the gears. A coin you put in the pot becomes 1.6 coins, split four ways, so 1.6 ÷ 4 = 0.4 of a coin comes back to you. The other 1.2 coins go to the other three players. Keeping the coin gives you the whole coin. So every coin you contribute costs you 0.6, no matter what anyone else does. Contributing is dominated. The best thing for the group is for everyone to contribute everything, and the best thing for each player is to contribute nothing, and both of those are true at the same time. Grown-ups call this a public goods game, and someone who contributes nothing while collecting a share of the pot is a free rider.
Play 10 rounds. Everyone starts each round with a fresh 10 coins, and what you end each round with piles up in your total. The robots come as Giver (puts in all 10), Free rider (puts in 0), Conditional (puts in whatever the other three averaged last round, starting at 5), or Mixed, one of each. Before you play, answer the question on the board.
The coin pot
10 coins each. The pot is multiplied by 1.6 and split four ways.If you put 1 coin into the pot, how much of it comes back to you?
Rules that work
So far this chapter is bad news. Here is the good news: the players are allowed to change the game. Two fixes, both waiting for you in the boards above.
A quota with a fine. The fishers agree that nobody catches more than q fish a season, and anyone who does pays a fine. A fine is just a number in the payoff grid, and it only works if it is big enough. A Greedy robot facing a quota of 3 wants 7 extra fish. If the fine is 5 fish, it cheats and comes out 2 ahead. If the fine is 8, it obeys. The robots do exactly this arithmetic: a robot breaks the quota when the extra fish are worth more than the fine. Switch on the quota in the pond, set three Greedy robots loose, and find the smallest fine that keeps the pond alive.
Punishment. In the coin game there is no warden, so let the players do it themselves. After each round, any player can pay 1 coin to take 3 coins away from someone. That is expensive for the punisher, which is exactly why it is interesting: punishing is itself a contribution to a public good (a well-behaved group) that everyone else enjoys for free. Turn on punishment. A robot that gave at least the average will punish anyone who gave 3 or more coins less than the average, and a punished Free rider matches the others for three rounds before it tests you again. Watch the Conditional robots. Without punishment, a free rider drags the average down and they follow it. With punishment on, the free rider is made to give, the average climbs, and the Conditional robots climb with it.
These are not toy fixes. Quotas, fines, and neighbors who will call you out are exactly the tools real communities use to keep a commons alive. Notice what they have in common: they change the payoffs, so that the selfish move and the good-for-everyone move become the same move. That is the whole trick. You do not have to make people nicer. You have to make the game better.
Logistic growth. The pond follows growth = r × F × (1 − F / K) with r = 0.5 and K = 100. r is how fast the fish breed when there is plenty of room, and K is the most fish the pond can hold (biologists call it the carrying capacity). The (1 − F / K) part is the fraction of the pond that is still empty. At F = 60 that fraction is 0.4, so growth = 0.5 × 60 × 0.4 = 12. At F = 20: 0.5 × 20 × 0.8 = 8. At F = 80: 0.5 × 80 × 0.2 = 8. Small ponds grow slowly because there are few parents; nearly full ponds grow slowly because there is no room. The pond grows fastest when it is half full, and not at all when it is empty or completely full.
Exactly in the middle. Write the growth as 0.5 × F × (100 − F) / 100. The two numbers F and 100 − F always add up to 100, and the product of two numbers with a fixed sum is biggest when they are equal: (50 + d) × (50 − d) = 2500 − d², which is largest when d = 0. So growth peaks at F = 50, where it equals 0.5 × 50 × 0.5 = 12.5. In symbols the peak is r × K / 4. Fishery scientists call it the maximum sustainable yield: the most you can take, season after season, without the pond shrinking. Ask for more than that and no pond size can keep up.
The pot. With a multiplier m and n players, each coin you contribute returns m / n to you. Here that is 1.6 / 4 = 0.4. Whenever m / n < 1, keeping the coin beats contributing it no matter what the others do, so contributing is dominated. Whenever m > 1, the group as a whole gains from every coin. Both are true here at once, and that gap is the whole dilemma. In Chapter 2 it was two players and Grab beating Share; here it is four players and a pot, and the arithmetic has the same shape.
When many people share something that regrows, each person's best move is to take a little more, and the sum of everyone's best moves can destroy the thing. Nobody has to be greedy or foolish for this to happen; it is the Cookie Game with many players and a future. The way out is not to hope for nicer players but to change the payoffs: quotas, fines, and punishment that the players agree on and enforce themselves.
1. Use the growth formula to find how many fish the pond adds when 20, 50, and 80 are left after the catch.
Answer
0.5 × 20 × 0.8 = 8, 0.5 × 50 × 0.5 = 12.5, and 0.5 × 80 × 0.2 = 8. Twenty and eighty give the same growth: one has few parents, the other has little room.
2. Change the coin pot's multiplier from 1.6 to 4.4, still with 4 players. Is contributing still dominated?
Answer
No. Each contributed coin now returns 4.4 / 4 = 1.1 coins to you, more than the 1 coin you would keep. Contributing everything is now the dominant strategy, and there is no dilemma at all. The trap only exists when the multiplier is less than the number of players.
3. Five fishers share the pond and each takes 4 fish a season. Can the pond keep that up?
Answer
No. That is 20 fish a season and the pond never grows more than 12.5. There is no pond size at which 20 fish grow back, so the pond shrinks every season no matter where it starts, and eventually empties.
Find the largest total catch per season that the four of you can keep up forever, and the pond size that supports it. First by experiment: use the calculator, or play the pond with Careful robots, and watch what the pond settles at. Then by the formula.
Answer
The formula says 12.5 fish a season, with exactly 50 fish left after the catch (so 62.5 in the pond before it). With whole fish that means 12 a season, three each. Run the calculator at 3 and the line is still creeping upward at season 20: it is heading for 72 before the catch and 60 after it, because 60 left grows by 0.5 × 60 × 0.4 = 12, exactly replacing the catch.
There is a second, dangerous answer. Forty fish left also grow by 12 (0.5 × 40 × 0.6 = 12). But a pond with 41 left grows by a bit more than 12 and climbs, and a pond with 39 left grows by a bit less and slides. Sixty is a balance point the pond returns to; forty is a cliff edge. Keep a commons well above its cliff.
Stars in this chapter
Earn them by doing the clever thing, not by clicking around.
- Keep the pond at 40 or more for 20 seasons while catching 70 fish yourself
- Work out what one contributed coin gives back, then play 10 rounds of the coin pot
- Tame three Greedy Robos with a quota and a fine