1
The pool and the share
Each person contributes a fixed sum or nothing. The pool is multiplied and divided equally among all, contributors or not, so a contributor gets back only the multiplier over the group size of what was given.
Lesson · Games and the Social Contract · Many players
Public Goods and Free-Riding
In a public-goods game each of ten people may put a sum into a pool that is multiplied and shared equally. If the multiplier is smaller than the group, each contribution returns less to its giver than it cost, and not contributing dominates. The lab prints what one free-rider gains, what universal free-riding loses, and where the dilemma stops.
a good for all returns a share to me
multiplier below the group: free-riding wins
one free-rider gains, all free-riding loses
multiplier above the group: giving wins
A share of a pool I fill
Core ideas
What this lesson establishes
The structure is the commons turned around: there a cost fell on others, here a benefit does.
2
A free-rider takes the share and keeps the sum
The person who does not contribute still receives an equal share of the pool. Compared with contributing, that is the same share less the contribution, and the difference is the free-rider's gain.
3
The ratio decides
What matters is the multiplier against the number of people. Below the group size contributing loses on every contribution, at it contributing breaks even, and above it contributing pays whatever others do.
The material
Who pays for the lighthouse
One pool, ten people, and the arithmetic of a share.
Ten people are each given 10. Each may put it into a common pool or keep it. The pool is tripled and divided equally among all ten, whether they contributed or not. A contribution of 10 therefore grows to 30 in the pool, and the contributor gets back one tenth of that, 3, as does every one of the nine others.
As in the commons, count the others, and let k be how many of the other nine contribute. If I contribute, the pool holds k + 1 contributions and I keep none of my 10; if I do not, it holds k and I keep mine. In both cases I receive a tenth of three times the pool.
C(k) = 3·(k + 1) − 10 = 3·k − 7
D(k) = 3·k
The difference is D(k) − C(k) = 7 for every k. Not contributing pays 7 more than contributing whatever the others do, so not contributing dominates, and the lab's tile for dominance says so. Each contribution is worth 30 to the group and 3 to the giver, who has paid 10 for it.
Definition · Free-rider
A free-rider receives the benefit of a good without bearing a share of its cost. In the game, a free-rider is a person who does not contribute and takes a share of the pool.
What one free-rider gains, and what all of them lose
If nine people contribute and I do not, I get D(9) = 27. If I had contributed too I should have had C(9) = 20. A lone free-rider gains 7. Now let everyone do as I did. Nobody contributes, the pool is empty, and each gets 0 where universal contribution would have given 20. Each person loses 20. The lab prints both: the gain alone and the loss when everyone does it.
The best total is when all ten contribute: each receives 20, and the sum is 200. Against that, the only equilibrium has no one contributing, and the sum is 0. The good would benefit everyone, and that is not enough to get anyone to pay for it.
Where the dilemma stops
It stops when the share is large enough. Two people with the same tripling have a share of one half, so a contribution of 10 returns 15 to its giver. Now contributing pays 5 more than not contributing, whatever the other does, and contributing dominates. With ten people and a multiplier of exactly ten, a contribution of 10 becomes 100 in the pool and a tenth of that, 10, returns to its giver: it returns exactly what it cost, and the lab finds neither choice dominant and every number of contributors an equilibrium.
So the rule is the comparison of the multiplier with the group. Below the group size the dilemma holds, and the larger the group, the smaller each person's share of a given multiplier. Public goods are hard to supply for large groups, easy for small ones, and the lab computes where the line falls. What it does not do is say that real people behave like the table, and experiments in which they give something are not refuted by it, for a person who values giving has a different table.
Interactive
The many-player dilemma
payoffs as exact expressions in k, and what follows from them
Method
Setting up a public-goods game
Five moves from the rules to the verdict.
1
State the rules in three numbers
How many people, how much each may give, and by what factor the pool is multiplied before it is shared equally.
2
Write what a contributor gets with k of the others contributing
The multiplier times the contribution times k + 1, over the group size, less the contribution.
3
Write what a non-contributor gets
The same share of a pool with k contributions in it, and no cost.
4
Subtract
If the difference is positive at every k, not contributing dominates. If it is negative at every k, contributing does.
5
Compute the universalisation numbers
What a lone free-rider gains, what everyone loses if all free-ride, and the total when all contribute.
Shares and free-riders
Score 0 / 0Choose the answer you can justify.
Ten people, tripled, a contribution of ten
Each contribution of 10 becomes 30 in the pool, and the pool is shared among ten.
C(k) = 3(k + 1) - 10 = 3k - 7
D(k) = 3k
D(k) - C(k) = 7 for every k: not giving dominates
All contribute: C(9) = 20 each, 200 in all
No one contributes: 0 each
One free-rider among nine givers: D(9) = 27, gain 7
A free-rider gains 7 and the group, if all do likewise, loses 20 a head. Every step of the reasoning is the reasoning of a person who is correct about their own payoff.
Common mistakes
1
Thinking that if everyone benefits from the good, everyone will pay for itA contribution of 10 is worth 30 to the group and returns 3 to the giver, so the benefit to everyone does not reach the person who decides. Not contributing dominates by 7. The good is worth having, 20 a head against 0, and the table still has nobody paying.
2
Confusing the free-rider's gain with the group's lossOne free-rider among nine givers gains 7; everyone free-riding costs each person 20. A person may reason correctly from the first number and the group ends up with the second.
3
Thinking any shared good is a dilemmaThe dilemma depends on the multiplier being below the group size. With two people and a multiplier of 3 contributing dominates, and with ten people and a multiplier of ten the two choices tie. The lab finds the line by changing the multiplier.
Check your understanding
✓
Finish when you can say whether a public-goods game is a dilemma and by how much.
Given the number of people, the contribution and the multiplier, write both payoff lines, state the dominance verdict, compute what a lone free-rider gains and what universal free-riding costs each person, and find the break-even multiplier.
Note: The commons and the public good are one dilemma with the sign turned around: a cost that falls on others there, a benefit that falls on others here. Real public goods are often provided by compulsion, which is a change to the payoffs of the kind the previous lessons studied. As in the commons, the last tile belongs to “The Evolution of Cooperation” and only repeats the starting share while the generations stay at zero.