In a recent paper, Vitalik Buterin, Zoë Hitzig, and E. Glen Weyl propose what they refer to as a Liberal Radicalism mechanism for handling the provision of public goods.

The difference between a public good and a private goods has to do with rivalry and excitability. Private goods are rivalrous and excludable while public goods are non-rivalrous and non-excludable. A good is rivalrous if one person consuming it inhibits another person’s ability to consume it. For example, a sandwich is highly rivalrous because if someone eats it then no one else gets to eat it but information is non-rival because one person learning something doesn’t inhibit someone else’s ability to learn that thing. A good is excludable if there are effective ways of preventing people from consuming it. For example, most of your personal belongings are hopefully excludable in the sense that you can effectively control who gets to use them but a fireworks show is relatively non-excludable because it’s not really feasible for the people who put on the display to selectively choose who gets to see it. Of course, all goods really exist on a spectrum for both of these traits so there is no line in the sand at which something becomes a public good but, in general, the more something behaves like a public good the harder it is for traditional markets to handle.

This has to do with the fact that these markets rely on property rights. The less rivalrous something is, the more the benefits of it will come to people besides the person who paid for it and thus less of the good’s actual benefit will be taken into account by the market. The less excludable a good is, the harder it will be to define and enforce property rights in the first place. The combination of these two effects leads to what is commonly known as the free rider problem where individuals are incentivized to free-ride off of others contributions to public goods rather than contributing themselves which leads to the goods being under-provided overall.

There are variety of methods that have been used to try to handle this problem such as developing new technologies to make the good excludable (e.g. copyrights for information), having the good be provided or subsidized by the state, assurance contracts (in which people agree to contribute conditional on others contributing), or using moral, cultural, or social motives to compel individuals to contribute. One naive approach you might try if you’re a government or an influential philanthropist is to ask people how much they value the good and then look at the difference between the total amount that people value it and the current amount that people are paying for it and supply the difference. The problem with this is that people won’t be properly incentivized to tell you how much they really value the good. In fact, in such a situation people would most likely be incentivized to say they valued the good as much as possible since there’s no cost to giving a higher stated value.

One attempt to try to deal with this is matching programs where organizations agree to match individual contributions at some ratio such as 1:1 (this is usually only done up to some amount for each individual). While it seems reasonable to try to infer some information about how much people value the good from their contributions and compensate based on that, there is still arbitrariness in choosing the matching ratio and cap and there is no way to guarantee that the optimal amount is being compensated. Ideally, we would like to set up the system so that we can infer how much individuals value the good based on how much they contribute so we can know what the socially optimal amount is. The new mechanism being proposed in the paper attempts to do exactly this.

They do this by having a mechanism where each individual makes a contribution and then the organization implementing it makes an additional contribution so that the total amount contributed will equal the following:

C_{T }= (C_{1}^{1/2 }+ C_{2}^{1/2} + C_{3}^{1/2} + … + C_{N}^{1/2})^{2}

(Where C_{T} is the total contribution and C_{i} is the contribution of the i^{th} individual out of N total people)

A couple of things to note initially about this system:

- The total contribution will always be greater than or equal to the sum of the individual contributions
- If there is only one individual then this behaves like a private good because the total contribution is equal to their individual contribution
- The additional contribution made by the organization will be monotonic in all of the individual contributions (which means someone will never cause the total to decrease by contributing more themselves)
- If each individual multiples their contribution by X then the total will be multiplied by X as well (this means the mechanism will be unaffected by irrelevant factors such as the currency used, how goods are split or combined, and how often the mechanism is run)

These all seem like desirable properties but does this mechanism really guarantee that it will pick the socially optimal allocation? It achieves this in the sense that the Nash Equilibrium strategy for each individual involves them paying amounts such that the total amount of the public good that is provided is the amount at which the sum of everyone’s marginal benefit is equal to the marginal cost of providing the good.

To see why let’s look at an example.

Let’s imagine we have a good x that is perfectly non-rival and costs $60 to produce per unit along with two individuals whose true willingness to pay can be described using the following functions:

B_{1}(x) = 50x – (x^{2})/4 B_{2}(x) = 50x – x^{2}

(So, for example, the second individual would be indifferent between losing $50 and gaining 1 unit of the good). We can get the marginal benefit each person receives for an additional unit of the good by taking the derivatives of these functions:

MB_{1}(x) = 50 – x/2 MB_{2}(x) = 50 – 2x

Because we are assuming that the good is perfectly non-rival, one of the individuals using it does not impair the other’s ability to use it at all so the total marginal benefit will simply be the sum of the individuals’ marginal benefits:

MB_{T}(x) = 100 – 5x/2

We said that each unit costs $60, so the marginal costs will just be $60:

C(x) = 60x MC(x) = 60

If we set the total marginal benefit of the good equal to the marginal cost we will get x=16. This is significantly greater than what the market equilibrium quantity would be (which in this case is actually x=0!). But now let’s think about what would happen under this new mechanism. Since the equation for the mechanism determines how much of the good both people get we can get new functions for their benefit by substituting the equation for the mechanism into their old benefit functions:

B_{1}(x) = 50(x_{1}^{1/2 }+ x_{2}^{1/2})^{2} – (x_{1}^{1/2 }+ x_{2}^{1/2})^{4}/4

B_{2}(x) = 50(x_{1}^{1/2 }+ x_{2}^{1/2})^{2} – (x_{1}^{1/2 }+ x_{2}^{1/2})^{4}

Getting the marginal benefits and setting them equal to the marginal cost again, we get the following two equations:

50(C_{1}^{1/2} + C_{2}^{1/2})/C_{1}^{1/2} – 0.5(C_{1}^{1/2} + C_{2}^{1/2})^{3}/C_{1}^{1/2} = 60

50(C_{1}^{1/2} + C_{2}^{1/2})/C_{2}^{1/2} – 2(C_{1}^{1/2} + C_{2}^{1/2})^{3}/C_{2}^{1/2} = 60

Each of these equations tells us how each individual can best respond to the other’s contribution, so by solving the system of equations we can find how much they will contribute when they are best responding to the each other’s contributions (i.e. the Nash equilibrium). Doing this we find that C_{1} = 196/25 and C_{2} = 36/25. Plugging these back into the equation for the mechanism we find that the total quantity will be 16, which is exactly the optimal amount!

Up until this point I’ve been assuming that the good in question was perfectly non-rival and this was why the LR mechanism was achieving the efficient allocation. However, as mentioned previously, rivalry comes on a spectrum and most things that fall into the fuzzy category of public goods are not perfectly non-rivalrous. Luckily, this can be accommodated by making the mechanism provide an amount of the good that is a weighted average of what the LR mechanism and a traditional market would provide:

C_{T }= α(C_{1}^{1/2 }+ C_{2}^{1/2} + C_{3}^{1/2} + … + C_{N}^{1/2})^{2 }+ (1 – α)(C_{1} + C_{2} + C_{3} + … + C_{N}) where (0 ≤ α ≤ 1)

They also offer some alternative versions of the mechanism to deal with budget caps and negative contributions in the paper. There are some rather standard concerns someone might have about such a mechanism including:

- Is social welfare (in the sense of the sum of people’s willingness to pay) what we should be optimizing?
- Will real people using this mechanism actually tend to move towards the Nash equilibrium strategy?
- How can we ensure that people aren’t able to collude and how robust is the mechanism to collusion?

Additionally, it’s worth noting that if this mechanism were being used by a state and funded through taxation then it would need to be the case that increasing the amount the government is contributing doesn’t directly lead to the people in the mechanism being taxed more as this would change the incentives. I’ll be excited to see it tested out (starting out either in the lab or small-scale real world settings presumably) and I’m happy to take it as a reminder that there is still progress to be made in designing mechanisms to improve cooperation and collective outcomes.