Liberal Radicalism: Can Quadratic Voting Be the Perfect Voting System?

— — Review of Vitalik’s Public Goods Voting Paper

This article was originally published in Chinese and has been translated from its original source.

Recently, the founder of Ethereum, Vitalik Buterin, published a paper titled Liberal Radicalism: Formal Rules for a Society Neutral among Communities, together with Harvard scholar Zoe Hitzig and Microsoft Principal Researcher E.Glen Weyl. The 41-page paper has become a hot topic after being published. It mainly discusses how to establish a reasonable voting system for charitable and public fundraising projects. The Nebulas Research Institute studied the paper and have provided feedback, with detailed explanations and analysis.


The paper starts with the research problem, that is, the free rider problem in public goods. In economics, a public good is a good that is both non-excludable and non-rivalrous, meaning individuals cannot be effectively excluded from use, and where use by one individual does not reduce availability to others. Goods such as national defense and basic research are examples of public goods. (In microeconomics, there are another three types of goods: private goods, club goods and public resources). The term “free rider” refers to a person who gets benefits from a public good but avoids paying for it.

What’s the relationship between the above two terms?

Here is an example to help answer this question.

A small town planning to install traffic lights at several crossroads to reduce traffic accidents. The funding of this project rely on government taxation, however some people may still enjoy the improved traffic facilities even if they do not pay taxes. These people are so-called free riders who enjoy the benefits of a public good without paying for it.

What is the consequence of this phenomenon? When the public realizes they can enjoy these benefits without paying, they may refuse to pay. Hence it becomes difficult to raise funds for improving traffic facilities. Finally, the plan may fail because of insufficient funding and the wellbeing of everyone is either the same or worse, because everyone’s traffic safety cannot be guaranteed.

The free rider problem exists in every social system, and there are three ways to solve this problem:

1. Taxation and democratic voting: In democratic countries, voting and taxation can be used to determine self-help items and amounts.

2. Converting them to private goods by imposing technologies: for example, digital rights management for information or walls and fee collectors at parks, that allow individuals to be excluded. Those who do not pay cannot enjoy the convenience.

3. Constraints from morality and religion.

All of the above-mentioned solutions have various flaws. They do not solve the free rider problems at the root, and almost all of them, more or less, depend on centralized regulation and coercion, or irrationally assume about individual behavior. However, this paper can solve the free rider problem by effectively designing the incentive mechanism without relying on the above mentioned strong assumptions.

In a social group of N citizens, the utility function of every citizen is expressed as:

Among that

is the funding value of public good P, which can be understood as the necessary fundraising needed for P to take effect.

represents the benefits that citizen i gets from good P.

is the cost that citizen i pays for.

In the paper,

was explained as taxes on users.

is the cost that citizen i pays for.

In the paper,

was explained as taxes on users.


The utility function of the whole social group, or social welfare, can be expressed as:

For the sake of brevity,

is used to represent

that is, the benefit that public good P brings to all people. The paper assumes that when the social welfare of each public item P reaches the maximum, the total social welfare reaches the maximum (efficiency).

So dealing with the derivation of

when VP’ = 1, the social welfare reaches the maximum and the marginal value of the public good P at that time is 1.

Then the paper gives a most common fund-raising strategy,

that is, the funding value of the item P comes from the contribution of all people. At this time, the utility function of citizen i to the public item P is:

Under this social system, if one pursues the maximization of personal interests, one needs to satisfy VP’ = N, which contradicts the previous condition VP’ = 1 for the maximization of social welfare.

At this point, the paper reveals a core problem through theoretical analysis: under the traditional fund-raising system, there is a conflict between the maximization of personal interests and the maximization of social welfare. Later, this paper also pointed out that the democratic system of “one person one vote” has similar problems.

Quadratic Voting (QV)

On this basis, this paper introduces the Quadratic Voting model, which was actually first put forward by Glen Weyl, the co-author of this paper. The biggest difference between QV and the traditional voting or fund-raising system is the calculation of raising level:

In this case, the utility function of citizen i for public goods P is:

Through simple mathematical derivation, it can be concluded that VP’ = 1 can be met when all citizens get the maximum benefits, which is consistent with the condition of social welfare maximization. This consistency fundamentally guarantees the native incentive to citizens and has nothing to do with the specific revenue function VIP of each citizen. In short, if QV model is adopted, whether one is a free rider or a true devotee, he will not harm the overall interests by doing what he likes most, and hence the problem of free rider problem will also be solved.

This is the excellent point of QV model. At the same time Vitalik explained that QV model is the only function satisfying these properties!

After so much analysis, you may be more concerned about the difference between QV model and the actual application. In fact, the process of QV model has not changed much in most scenarios. For example, when raising funds, participants only need to invest costs according to their own wishes. During voting, unlike the one-person-one-vote system, QV will allow every participant to cast more votes, but the voting cost shows superlinear increase with the number of votes.

In the second half of the paper, Vitalik explained the good features of QV, including the effective prevention of community division and support for privatization, etc.


Despite its advantages, quadratic voting (QV) suffers some drawbacks, i.e. the lack of resistance against cheating. One would get higher utility by splitting its voting stake, which is called Sybil attack. QV accommodates the needs of minority as well as the Sybil attacks(There are more details about Sybil attacks in the Nebulas Yellow Paper). For example, in the voting scene, casting N votes is less profitable than hiring N people with each one casting one vote.

The authors of the paper propose to build an effective system of identity verification. However, such system is difficult to realize even in the real society, let alone building a centralized identity verification system on blockchain. And it will lose one most important feature of blockchain: anonymity.

Moreover, we think this paper is extensible in following aspects. For example, the analysis on each public good p is independent in the paper, i.e. each citizen could pay any amount of cost for each good. However, when the public goods are correlated, for example, citizen i has a budget Ci, if

the conclusion in the paper doesn’t hold anymore. Studies on this situation might be a good future work.

In addition, the conclusion of the paper relies on the assumption of the concavity and smoothness of utility function. When this assumption doesn’t hold, the paper proposes solution in engineering without mathematically robust model.


This paper of Vitalik puts forward a QV model for fund-raising of public projects, which guarantees the consistency of the maximization of personal interests and social welfare, and fundamentally solved the free rider problem. However, it cannot solve the problem of Sybil attack, and its applicability is also limited in practical uses. Therefore, we think that if it is applied with effective oversight and avoid Sybil attack, QV is indeed a good model.

This article was originally published in Chinese and has been translated from its original source.

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