The Problem of Providing Public Goods

TODO

The Two Sides of Prisoners’ Dillemas

Let’s consider the town of Perindale. There is a proposal to construct a public garden in downtown Perindale, and the mayor is planning on how to faciliate this. The mayor has two important restrictions:

the mayor cannot exclude anyone from enjoying the garden (it’s supposed to be in a public garden, after all)
the mayor cannot force anyone to help construct to garden
the mayor must ask the same donation amount from each resident

So, how can the mayor best recruit people to construct the garden?

Additionally, for the sake of simplicity, let us assume the following (which can be relaxed later):

There are \( n \) residents of Perindale.
Each resident gains \( X \) utility from the garden being constructed
All the residents are the same with respect to factors influencing the funding and construction of the garden
Constructing the garden costs \( C \) utility.
The garden cannot be constructed partially; if not turned into a garden, the lot will be bought by an out-of-town euntrepenuer and turned into a private helicopter landing pad, from which residents get \( 0 \) utility.
Each resident acts perfectly rationally (i.e. maximizes expected utility based on their beliefs).

Proposal 1: Simple Donations

The first idea that comes to the mayor’s mind is to simply ask Perindale residents to donate \( D \) utility each, where \( D \leq C/(n - 1) \) i.e. the garden would still be constructed if one resident decided not to contribute (we shall consider the case \( D = C/(n-1) \) in the next section). Additionally, \(( D < X \)) i.e. each resident’s donation would be smaller than the amount they value the garden.

Since each resident gains \( X \) utility from a constructed garden and is being asked to donate \( D \) utility, each resident is in the same position with respect to deciding what to do. So, the mayor asks an aribtrary resident, Jo, how they would decide, and then generalizes this result to all residents.

Jo will decide what to do based on their beliefs about the outcome given everyone’s actions. Since all the residents are the same, Jo can expect one of two situations:

Every other resident donates
Every other resident refrains

The table below shows the utility payoffs to Jo and the other residents given Jo’s desicion (leftmost column) and the other residents’ desicion (topmost row).

Jo, others	others donate	others refrain		donate	\(
X - D, (n-1) (X - D) \)	\( -D, 0 \)		refrain	\( X , (n-1) (X - D)
\)	\( 0, 0 \)

Then, Jo decides as follows, there are two cases for what Jo believes and so how they will decide:

Suppose that Jo believes that every other resident will donate. Then Jo decides not to donate since \( X \geq X - D \) i.e. Jo still recieves the utility of the constructed garden without having to pay. Here, Jo is a free rider.
Suppose that Jo believes that every other resident will refrain. Then Jo decides not to donate since \( 0 \geq -D \) their donation will only go towards a partial garden that is worth \( 0 \) utility.

The mayor realies the tragic situation (more specifically, prisoner’s dillemma): in either case, Jo decides not to contribute. And since Jo is an arbitrary resident, this implies that every resident acts like Jo, and thus no resident contributes. The garden will not be constructed by this simple donations proposal.

Altogether for this proposal: refraining is a dominant strategy since there is always an incentive to refrain.

As shown in Jo’s reasoning, there are two sides to this tragedy:

Jo is able to free-ride when the other residents donate
Jo won’t donate when they know the garden won’t be constructed anyway

The following proposals 2 and 3 will cumulatively remove these malincentives: assurance contracts prevent free-riders, and dominant assurance contracts additionally incentivizes pessimistic residents to still donate.

The mayor thanks Jo, and returns to their office to think up the new proposals.

Proposal 2: Assurance Contracts

To tackle the free-rider problem, the mayor needs to write up a contract that removes the incentive for Jo to not contribute if Jo believes that the other residents will contribute. Additionally, the mayor wants to avoid wasting resources. So, they write up the following assurance contract:

I promise to donate \( D \) utility if and only if every other resident also signs this contract. Otherwise, I refraine from donating.

The mayor consults Jo again, asking what they would do with this proposal. Jo has the same two possible situations to expect:

Every other resident promises (signs the contract)
Every other resident refrains

The table below shows the new utility payoffs to Jo and the other residents’ desicions.

Jo, others	others promise	others refrain		promise	\(
X - D, (n - 1) (X - D) \)	\( 0, 0 \)		refrain	\( 0 , 0 \)	\(
0, 0 \)

Then, Jo decides how to act based on their belief:

Suppose that Jo believes that every other resident will promise. Then Jo decides to donate since \( X - D \geq 0 \) i.e. Jo prefers to donate and then enjoy the garden over getting nothing.
Suppose that Jo believes that every other resident will refrain. Then Jo is indifferent between donating and refraining since both given them a payoff of \( 0 \).

The mayor has made some progress! There will never be free riders since it is strictly better to donate than refrain when Jo believes that others will donate. Additionally, Jo is not disincentivized from donating when other refrain. However, Jo is only indifferent here i.e. Jo is not strictly better off when they donate rather than refrain.

Altogether for this proposal: promising is a weakly dominant strategy since there is never a disincentive from promising.

The mayor still hopes to do better. Not wasting utility was a good move, but even one pessimistic believer in the whole town results in no garden. In fact, if even one resident forgets to sign the contract then there will be no garden! That is where the weakness in “weakly dominant strategy” comes in. The mayor thanks Jo again and returns to their office for, hopefully, a final draft.

Proposal 3: Dominant Assurance Contracts

Ideally, the mayor wishes that each resident always be actively incentivized to donate, not just indifferent about it, and even if they are pessimistic about the others. In other words, that donating is a dominant strategy. The mayor thinks for a while, and finally writes the following contract:

I promise to donate \( D \) utility if and only if every other resident also signs this contract. Otherwise, I donate \( 0 \) and the mayor pays me \( F \) utility.

Where \( D \leq C/n \) i.e. the garden is still constructed if just one resident decides not to sign.

The mayor reflects on the contract, it does feel a little risky. If enough of the residents decide not to sign, then the mayor will be losing money by having to pay out to those that did sign, and still no garden will be constructed. Nevertheless, the mayor has a hunch that the details work out in everyone’s favor in the end.

The mayor finds Jo and presents the new contract. Jo follows the same method of consideration from earlier, where the table below shows the new utility payoffs.

Jo, others	others promise	others refrain		promise	\(
X - D, (n - 1) (X - D) \)	\( F, 0 \)		refrain	\( 0 , 0 \)	\(
0, 0 \)

Jo considers the new circumstances, and explains his updated belief-dependent desicions:

Suppose that Jo believes that every other resident will promise. Then Jo decides to donate since \( X - D > 0 \) i.e. Jo prefers to donate and then enjoy the garden over getting nothing.
Suppose that Jo believes that every other resident will refrain. Then Jo decides to donate since \( F > 0 \) i.e. Jo opts in to be paid by the mayor since the garden isn’t going to be constructed anyway.

“Aha!” the mayor exclaims. Jo concedes that this proposed contract, no matter their beliefs, yields a situation where Jo is always incentivized to sign. Since every other resident will also follow this logic, this implies that every resident will sign! And so the garden is garunteed to be fully funded.

Even though the mayor had to include a clause about paying \( F \) to signers when there aren’t enough signers, the mayor can expect this situation never to arise as long as the residents of Perindale are true to their perfect rationality.

Dominant Assurance Contracts

TODO: generalized definition and justification

Complications

All about deciding the optimal value of \( F \).

Buffering

TODO: How to account for a certain number of agents deciding not to sign anyway? TODO: Avoiding sel-fulfilling negative beliefs

Incomplete Information

TODO: How to account for agents having incomplete information about their peers? And probabilistic beliefs

some graphs from Tabarrok

Differentiated Evaluations

TODO: How to account for when agents have different evaluations of the public good

Privatization

TODO: What if a private firm hosts the DAC instead of the mayor? Where the firm earns profits from the surplus donations.

competition between firms will drive down required donations

Glossary

Some simple definitions and explanations of the formal terms used relating to public goods and collective action.

Goods

A non-excludible good is a good that cannot have its enjoyment limited to specific consumers. Examples: national armies, outdoor air, TODO.

A non-rivalrous good is a good for which its enjoyment by some consumers does not decrease the amount it can be enjoyed by other consumers. Examples: intellectual property, blogs, national armies, street lights.

A public good is a good that is non-excludible and non-rivalrous. Examples: national armies, ideas, outdoor air, vistas, public parks,

A free rider problem is a collective action problem of funding a public good when agents have the ability to opt-out of paying and still enjoy the good. A free rider is an agent that decides not to pay towards funding the public good, yet still enjoys the public good to the same extent as other agents that did decide to pay. Examples: funding many public services like parks, armies, etc.

Collective Action Problems

A collective action problem is a problem of coordinating agents to cooperate when there are incentives for each agent to defect, and if too many agents defect then every agent is worst off. Three popular classes of collective action problems are free rider problems, tragedies of the commons, and prisoners’ dillemas.

A tragedy of the commons is a collective action problem of rationing a public good that, while non-rivalrous at reasonable levels of use, becomes rivalrous over a certain volume of use. The tragedy is that each agent has an incentive to use the public good a reasonable amount on their own, but too many agents deciding to do so overuse the good. Examples: rationing use of public parks, lakes, air quality.

A prisoners’ dillema is a collective action problem where cooperating agents are well off, defecting agents are the best off when there are enough other cooperating agents, if too many agents defect then all agents are the worst off, and no communication between agents is allowed. (Note that a free rider problem without communication is a prisoner’ dillema.) The name is inspired by the following story:

Two prisoners, A and B, are on trial for a bank robbery they comitted as partners. They are put into seperate rooms and each offered the same options: (cooperate) recieve the standard sentence of 2 years, or (defect) reveal evidence incriminating your partner to increase their sentence by 3 years, and decrease your sentence by 1 year

For each scenario, the sentences (in years) are arranged below. The entry “x, y” represents that prisoner A gets a sentnce

A , B	cooperate	defect		cooperate	2 , 2	5 , 1
defect	1 , 5	4 , 4

The prisoners have no opportunity to communicate before making their desicions, since they are in separate cells. From the table above, it is clear that the prisoners are best off in total if they both cooperate: they recieve sentencings of 4 years totaled between then.

Yet, consider the ways that each prisoner considers their partner. If one knows that the other will cooperate, then it is best for one to defect and recieve 1 year instead of 2. Alternative, if one knows that the other will defect, then it is best for one to defect and recieve 4 years instead of 5. So, it is inevitable that, whatever either prisoner believes of the other, the best option for each is to defect. This is a problem because even though there exists the best option overall to both cooperate, perfectly-rational prisoners will never obtain it.

Assurance Contracts

An assurance contract is a contract between a population of agents where each agent pledges to contribute towards an action if at least a certain amount is pledged in total. Examples: escrow, certain kinds of crowdfunding.

A dominant assurance contract is an assurance contract between a population of agents where agents are promised a compensation in case when the contract fails to facilitate the target action, in such a way that yields agreeing to the contract as a dominant strategy. Dominant assurance contracts were primarily introduced in Tabarrok’s “[The private provision of public goods via dominant assurance contracts][tabarrok1998].”

Bibliography

Amoveo Editor, 2019. [Amoveo use-case: Crowdfunding via a Dominant Assurance Contract (DAC)][amoveo2019].
Guerra-Pujol F, 2018. [Nozick, Tabarrok, and Dominant Assurance Contracts][guerra-pujol2018]
Michel M, 2019. [Crowdsourcing the Public Good: Dominant Assurance Contracts][michel2019].
Tabarrok A, 1998. [The private provision of public goods via dominant assurance contracts][tabarrok1998].
Tabarrok A, 2013. [A Test of Dominant Assurance Contracts][tabarrok2013].
Tabarrok A, 2017. [Making Markets Work Better: Dominant Assurance Contracts and Some Other Helpful Ideas][tabarrok2017].
Taylor J. Dominant Assurance Contracts with Continuous Pledges

[guerra-pujol2018]: https://priorprobability.com/2018/07/27/nozick-tabarrok-and-dominant-assurance-contracts/ [amoveo2019]: https://medium.com/amoveo/amoveo-use-case-crowdfunding-via-a-dominant-assurance-contract-dac-1be3482e7792 [michel2019]: https://commonwealth.im/edgeware/proposal/discussion/101-crowdsourcing-the-public-good-dominant-assurance-contracts [tabarrok1998]: https://mason.gmu.edu/~atabarro/PrivateProvision.pdf [tabarrok2013]: https://marginalrevolution.com/marginalrevolution/2013/08/a-test-of-dominant-assurance-contracts.html [tabarrok2017]: https://www.cato-unbound.org/2017/06/07/alex-tabarrok/making-markets-work-better-dominant-assurance-contracts-some-other-helpful