*Note: The original author is Ethereum co-founder Vitalik Buterin. In this article, he introduced the concept and application of Quadratic Voting and Quadratic Payment . He stated that if the goal is Optimizing what people want to see, rather than purely satisfying the rich and centralized institutions, then such a mechanism will definitely be of great benefit to us.*

Special thanks to Karl Floersch and Jinglan Wang for their feedback:

If you have been researching application mechanism design or decentralized governance, you may have heard a few buzzwords recently: quadratic voting , quadratic funding , and quadratic focus on buying ( quadratic attention purchase ).

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These ideas have quickly gained popularity in the past few years, and small-scale testing has also begun: Taiwan ’s presidential hackathon campaign uses a quadratic voting mechanism to vote for projects, and Gitcoin Grants uses a quadratic financing mechanism to fund For public goods in the Ethereum ecosystem , the Colorado Democrats have also experimented with quadratic voting to determine their platform.

For supporters of these voting schemes, this is not just a small improvement over the existing system. Instead, this is an initial attempt at a new social technology that has the potential to upend many public decisions, large and small. The final effect of these comprehensive implementation plans may be as profoundly transformative as the free market in the industrial era. But now, you might be thinking, "Those are big promises. Is there any reason to prove that these new governance technologies can achieve this?"

## I. Private Goods and Private Markets

To understand what is happening, let us first consider the existing social technologies: money and property rights-intangible social technologies are usually hidden behind money. Money and private property are extremely powerful social technologies, which is what classical economists have been saying for more than a century. If Bob is producing apples and Alice wants to buy apples, we can economically simulate the interaction between the two, and the results seem to make sense:

Alice has been buying apples until the marginal value of the next apple she wants to buy is lower than the cost of producing apples, which is almost the most ideal thing that can happen. If the cost of producing apples is greater than their value to Alice, then Alice would not buy it:

These are all formal results, such as "the basic theorem of welfare economics". Now, those of you who have studied economics may scream, what about incomplete competition? Information asymmetry? Economic inequality? Public goods? Externality? Many activities in the real world, including those vital to the progress of human civilization, have benefited (or harmed) many people in complex ways. These activities and their consequences cannot often be subtly decomposed into different transaction sequences between the two parties.

But since then, we have counted on a set of technologies to solve all problems? This is not an argument against cars, but an argument against auto extremism. We need cars, not anything else. Much like how private property and markets handle private goods, can we try to infer which social technology is a good way to promote the production of public goods we need?

## Public Goods and Public Markets

In some key ways, private goods (such as apples) are different from public goods (such as parks, air quality, scientific research, this article …). When we talk about personal goods, the production of multiple people (for example, the same farmer produces apples for Alice and Bob at the same time) can be broken down into (i) farmers producing some apples for Alice, and (ii) farmers producing for Bob Some apples. If Alice wants apples and Bob doesn't, then the farmer gives Alice some apples and collects money from Alice, which has nothing to do with Bob.

Even complex collaborations can be broken down into a series of such interactions. However, when we talk about public goods, this decomposition is impossible. As I write this blog post, both Alice and Bob (and others) can read it. I could put it behind a paywall (for paid reading), but if it's popular enough, it will inevitably appear on third-party websites, and the paywall is annoying and not very effective in any case. In addition, offering an item to ten people is not ten times cheaper than giving it to a hundred people. Instead, the costs are exactly the same. So, I either write articles for everyone or don't write to anyone at all.

So the challenge we face is: how to bring people's preferences together? Some private and public goods are worth producing, others are not. In the case of personal belongings, this problem is easy to solve because we can break it down into a series of decisions. No matter how much everyone is willing to pay, they will pay so much for them. The economics involved here is not particularly complicated. **However, in the case of public goods, you cannot "decompose", so we need to summarize people's preferences in different ways.**

First, let's see what happens if we just build an ordinary old market: as long as someone donates at least $ 1,000 to me, I am willing to write an article (funny fact: I did exactly that in 2011 of). Every dollar donated increases the probability of achieving the goal and publishing the article, which we call "marginal probability" `p`

. At the cost of `k`

dollars, you can increase the probability that an article will be published as `k * p`

(although eventually the probability will decrease as the probability approaches 100%). Suppose we told you that the article being published is worth `V`

dollars, would you donate it? Well, donating a dollar will increase the probability of `p`

, so it will give you an expected value of `p * V`

If `p*V>1`

, you donate, and there are many, but if `p*V<1`

, you don't donate at all.

In less mathematical terms, either you pay enough attention to this article (or you are rich enough yourself) so that you will pay for it, and if you pay less attention to this article, you wo n’t Make any contribution to this.

**Therefore, the only blog posts published will be articles donated by a few people (in my 2011 experiment, this prediction was experimentally verified: in most donations, more than half came from the same donor) .**

Please note that this reasoning applies to any mechanism that involves "buying influence" on matters of public concern. This includes paying for public goods, voting for company shareholders, public advertising, bribing politicians, and more. It's like saying that this little guy's influence is too small (not exactly zero), because there is something like altruism in the real world, but the big man's influence is very large.

We can also consider another mechanism: **one person, one vote** . Suppose you can vote for what I deserve for writing this article, or you can vote for what I deserve, and my compensation is directly proportional to the votes I gave.

We can explain it this way: your first "contribution" takes very little effort, so if you care enough about an article, you will support it, but after that, there is no more room to contribute further In other words, the cost of your second contribution is "infinite".

Now, you may notice that the two figures above don't look right. **The first graph gives too much power to those who care a lot (or rich), and the second graph gives too much power to people who don't care much. This is also a problem.** **We say that the desire of one sheep for life is more important than the desire of two wolves for a delicious dinner** .

But what do we want? **Ultimately, what we want is a plan where your "buy" influence is proportional to the degree you care about** . In the mathematical terms above, we want `k`

to be proportional to `V`

, but the question is: `V`

determines how much you are willing to pay for an influence unit.

If Alice thinks the article is worth $ 100 and she must fund it, then she is willing to pay 1 dollar to increase the chance of publishing the article by 1%, and if Bob thinks the article is worth only $ 50, then he only Willing to pay $ 0.5 in exchange for the same "unit of influence."

So how do we match them? *The answer is smart: your nth influence unit will cost you n * units of dollars* . For example, you can buy the first ticket for $ 0.01, but the second ticket costs $ 0.02, the third ticket costs $ 0.03, and so on.

Suppose you are Alice in the above example. In such a system, she will always buy influence units until the cost of the next unit reaches $ 1, so she will buy 100 units. Bob will also buy until the cost reaches $ 0.5, so he will buy 50 units. Alice's 2x higher valuation has become 2x the influence unit purchase.

Let's draw it as a chart:

Now let's look at the relationship between these three:

Note that only quadratic voting has such a good characteristic that the amount of influence you buy is directly proportional to the degree you care about. The other two mechanisms are either good for privileges or very bad for privileges. Now, you might ask, where does this quadratic curve come from? Well, the marginal cost of the nth vote is n * units of dollars (or 0.01 * n dollars), but the total cost of n votes is approximately `{n^2}{2}`

. Can be viewed in the following geometrical ways:

The total cost is the area of a triangle that you may have learned in a math class: the area is radix * height / 2. Because the radix and height are proportional here, this basically means that the total cost is proportional to the square of the number of votes, so it is "square". But to be honest, it's easier to think that "your nth influence unit costs n * units of dollars."

In the end, you may notice that my actual definition of "a unit of influence" above is ambiguous. This is the result of careful consideration. It may mean different things in different environments, and quadratic payment reflects These different perspectives.

## 3. Quadratic voting

See original paper: https://papers.ssrn.com/sol3/papers.cfm?abstract%5fid=2003531

Let's start with exploring the first application of quadratic payment: quadratic voting. Imagine that an organization is trying to make a decision in two choices that affects all of its members. For example, this could be a company (or a non-profit organization) deciding in which part of the town to set up a new office, or a government deciding whether to implement certain policies, or an Internet forum deciding whether its rules allow discussion Cryptocurrency prices.

This seems to be a perfect target for quadratic voting. The goal is to choose A if everyone prefers A, and B if you prefer B. Simple voting ("one person, one vote") ignores the difference between strong and weak preferences, so simple voting on issues where one side has high value to a few people and the other side has low value to more people It is likely to give the wrong answer. **With the private goods market mechanism, people can buy as many votes as they want for the same price, and the individual with the greatest preference (or wealthiest) has everything** . **With quadratic voting, you can vote n in either direction, and the price is the square of n, which creates a perfect balance between these two extremes** .

Note that in the case of voting, we decide on two options, so different people would prefer A over B, or B over A. So unlike the zero-based chart we saw earlier, voting and preferences here can be either positive or negative (which option is considered positive and which is negative, it doesn't matter because The calculation results are the same.)

As shown above, because the cost of the `n`

vote is `n`

, the number of votes you vote is directly proportional to your evaluation of the unit of influence of the decision, and how much you care about A being selected rather than B, and vice versa. So again we have this clean "preference increase" effect.

We can extend quadratic voting in a number of ways. First, we can allow voting between more than two options. Due to the existence of Arrow's theorem and Duverger's law , traditional voting schemes will inevitably fall into various “strategic voting” problems. When there are more than two choices, the second time The party vote is still optimal.

"For those interested, the intuitive argument is: Suppose there are established candidates A and B, and a new candidate C. Some people favor C> A> B, while others favour C> B> A. In the regular voting, if both sides think that C has no chance, they decide that they might as well vote between A and B so that C cannot get the ballot and C's failure becomes a self-evident prediction. In the second party voting, the former group will vote [A +10, B -10, C +1] and the latter group will vote [A-10, B + 10, C + 1]. Therefore, the votes of A and B will be Cancelled, C's popularity will increase. "

Secondly, we can consider not only voting between discrete options, but also voting on the thermostat settings: anyone can push the thermostat up and down 0.01 degrees n times by paying the cost of n ^ 2.

The party that wants to get cold will win only if the other party is convinced that "C" stands for "hot"

## Third-party financing

See original paper: https://papers.ssrn.com/sol3/papers.cfm?abstract%5fid=3243656

Quadratic voting is optimal when you need to make some fixed number of collective decisions. However, a disadvantage of quadratic voting is that it does not have a built-in mechanism to determine the first result of voting. Proposed voting can be a considerable source of power if it is not handled carefully: malicious actors who control voting rights can repeatedly make decisions with weak majority approval and strong opposition by the majority, and then continue to propose until the minority runs out of voting tokens (If you do mathematical operations, you will find that a few people will burn the token faster than most people). Let us consider an example of a quadratic payment that does not encounter this problem and makes the choice of decision itself endogenous (ie part of the mechanism itself). In this case, the mechanism is dedicated to a specific use case: separate provision of public goods.

Let's take an example, someone wants to create a public item (such as a developer writing an open source software program), and we want to figure out whether this program is worth funding. However, instead of considering only one public good, it is better to establish a mechanism so that anyone can raise funds for a project that they claim is a public good. Anyone can contribute to any project, and a mechanism tracks these contributions, and then at the end of a certain period of time, the mechanism calculates payments for each project. This payment is calculated as follows: for any given project, take the square root of each contributor's contribution, add these values, and then take the square of the result.

That is the following formula:

(\ sum_ {i = 1} ^ n \ sqrt {c_i}) ^ 2

If this formula looks a bit complicated, here is an intuitive graphical representation:

In any case, if there is more than one contributor, the calculated payment is greater than the original total contribution; the difference comes from the central subsidy pool (for example, if 10 people donate $ 1 each, the sum of the square roots is $ 10 , And then square calculation is $ 100, so the subsidy is $ 90).

Please note that if the subsidy pool is not large enough to cover the full amount required for each project, we simply divide the subsidy proportionally by any constant to add the total to the budget of the subsidy pool; you can prove that this solved You can also use the subsidy budget for the tragedy of the commons.

There are two ways to explain this formula intuitively. First of all, we can look at it through the lens of "repair market failure", which is a surgical operation on the tragedy of the commons. With Alice contributing to one project and Bob contributing to the same project, Alice is contributing to something that is of value to both parties. And when deciding how much to contribute, Alice will only consider her own benefits, not Bob, and she may not even know that Bob exists. The quadratic financing mechanism adds a subsidy to compensate for this impact, which determines how much she will “contribute” if Alice also considers the benefits that her contribution will bring to Bob. In addition, we can calculate the subsidies for each pair of people separately (Note: if there are `N`

individuals, there are `N*（N-1）/2`

pairs), and add all these subsidies, and then give Bob a comprehensive subsidy from all pairs . It turns out that this is exactly the formula for quadratic financing.

Secondly, we can study this formula through the quadratic voting lens. We explain second-party financing as a special case of second-party voting, where a project's investor voted for the project and a fictitious participant voted against it: the subsidy pool. Each "project" is a motion to get money from the grant pool to the creator of the project. Everyone who sent `c_i`

funds voted `\sqrt{c_i}`

, so a total of `\sum_{i=1}^n \sqrt{c_i}`

votes in favor of this motion.

To terminate the motion, the subsidy pool will need to vote more than `\sum_{i=1}^n \sqrt{c_i}`

, which will be more than `(\sum_{i=1}^n \sqrt{c_i})^2`

pay more. Therefore, `(\sum_{i=1}^n \sqrt{c_i})^2`

is the largest transfer from the subsidy pool to the subsidy pool without voting to stop the project.

At present, some people are exploring the application of the quadratic financing mechanism. Gitcoin grants' public goods funding in the Ethereum ecosystem is the best example. In my personal opinion, this is to support projects that the community considers valuable. Pretty good distribution.

The white numbers are the total number of original contributions, and the green numbers are additional subsidies.

## Fourth, the second party pays attention to payment

See original post: https://kortina.nyc/essays/speech-is-free-distribution-is-not-a-tax-on-the-purchase-of-human-attention-and-political-power/

An important feature that people love and hate about modern capitalism is advertising. Our cities have advertising:

Source: https://www.flickr.com/photos/argonavigo/36657795264

Advertisements are also printed on the turnstiles of our subway stations:

Source: https://commons.wikimedia.org/wiki/File:NYC,_subway_ad_on_Prince_St.jpg

And American politics is dominated by advertising:

Source: https://upload.wikimedia.org/wikipedia/commons/e/e3/Billboard_Challenging_the_validity_of_Barack_Obama%27s_Birth_Certificate.JPG

Even the river and the sky are full of advertisements. Of course, there seem to be no problems in some places:

But in fact, they just have a different kind of advertisement:

Recently, some people have tried to do some advertising in some cities, or on Twitter. Let's look at the problem systematically and try to find the source of the problem. The answer is actually surprisingly simple: Public advertisements are evil twins produced with public goods. As far as the production of public goods is concerned, one actor bears the expense of producing a certain kind of goods, and this kind of goods will benefit a large number of people. Because these people cannot effectively coordinate their own bills for public goods, we get much less public goods than we need, and we get public goods that are favored by wealthy actors or centralized authorities. **Here, some participants benefited greatly from forcing others to look at an image, and this action hurt a lot of people.** **Because these people are unable to coordinate the purchase of advertising space, we are presented with advertisements that we do not want to see, and they are provided by wealthy people or centralized authorities.**

So, how do we solve the dark image produced by this public good? A bright mirror image of financing through quadratic parties: quadratic fees! Imagine a billboard, anyone can spend $ 1 for a minute of advertising, but if someone wants to advertise multiple times, the price will increase several times: $ 2 in the second minute and $ 3 in the third minute And so on. Note that you can spend money on billboards to extend the life of someone else ’s ad, which only costs you $ 1 in the first minute, even if others have spent money multiple times to extend the life of the ad. We can once again interpret it as a special case of quadratic voting: it is basically the same as the "thermostat voting" example above, but the "thermostat" discussed here refers to the number of seconds the ad stays.

If the goal is to optimize what people want to see, rather than purely satisfy the rich and centralized institutions, then this payment model can be applied to cities, websites, conferences or many other occasions.

## V. Complexity and Warning

Perhaps the biggest challenge we need to consider for this type of second-party payment concept is the practical implementation of identity and bribery / collusion.

Any form of second-party payment requires an identity model, in which individuals cannot easily obtain the identity they want: if they can, then they can continue to obtain a new identity and continue to pay $ 1 to influence their desire If the number of decisions is required, then this mechanism will collapse into a linear bribery.

Note that in the sense of preventing multiple identities, the identity system does not need to be sealed, and there are indeed good reasons why the identity system should probably not try to seal. Instead, it just needs to be powerful enough that the cost of manipulation is unaffordable.

**Collusion is also tricky** . If we can't stop people from selling their votes, these mechanisms will collapse again into a dollar-a-vote mechanism. Not only do we need to vote anonymously and confidentially, but we also need to make the end result provable and public. We need to keep votes secret so that even voters themselves cannot prove to others what they have voted for, which is difficult to do. Secret voting works well in the offline world, but secret voting is a technology from the 19th century, and for us who want to see a large number of quadratic votes and participation in the 21st century, this voting method is efficient too low.

Fortunately, there are some technical means that can help us to combine zero-knowledge proofs, encryption, and other cryptographic techniques to obtain the precise set of privacy and verifiability attributes needed. People have also proposed some technical solutions to verify that the private key is actually personally owned, rather than some hardware or cryptographic system that restricts the use of the private key. However, these technologies are untested and require considerable improvement.

Another challenge is that as a payment-based mechanism, quadratic payments will still benefit the rich. It should be noted that because the voting cost is quadratic, the impact will be suppressed: **people who have more than 100 times the capital have only about 10 times the influence, not 100 times, so the severity of the problem will decrease by 90 %** (For the super-rich, this weakening will be stronger). In other words, it may be desirable to further alleviate this inequality in power.

This can be achieved by naming the quadratic payment separately as a token for which each person has a fixed number of units, or allocating funds that can only be used for quadratic payment applications: this is basically Andrew Yang's "Democratic Dollar" "proposal.

The third challenge is the issue of " **rational neglect** " and " **rational mischief** ". **Decentralized** public decision-making has a weakness, that is, the influence of any one person on the outcome is small, so there is no incentive to ensure that they support the decision that is best for long-term On the contrary, tribal relations etc. may dominate. And any form of secondary payment is rarely helpful in solving this problem.

Of course, second-party voting is more useful than the one-person-one-vote system to alleviate this problem, and these problems may not be as serious as major decisions affecting millions of people for medium-sized public goods, so at the beginning It may not be a big challenge, but it is definitely an issue worthy of facing.

One approach is to combine quadratic voting with ranking elements, while another, possibly longer-term approach, is to combine quadratic voting with another economic technology called a predictive market.

A simple example is a system in which quadratic financing is retrospective, so people vote on which public goods are valuable some time ago (for example, two years ago), and projects postpone by selling these Share of voting results and advance funding. By buying equity shares, people can either fund the project or bet on which project will be considered successful after two years. There is a lot of design space here to experiment.

## 6. Conclusion

As I mentioned at the beginning, quadratic payments cannot solve all problems. They address resource management issues that affect large populations, but they do not address many other types of problems. A particularly important issue is the asymmetry of information and the generally low quality of information. For this reason, I like to use technologies such as prediction markets (eg, electionbettingods.com) to solve the problem of information collection. By combining different mechanisms, many applications can be made most effective.

For me personally, a particularly important reason is what I call "corporate public goods" (public goods that only a few people consider important at present, but more people will value them in the future). In the 19th century, contributing to the abolition of slavery may be an example, and in the 21st century, I cannot give examples that please every reader, because the nature of these commodities determines their importance in the future. It will become common sense, but I will point out that life extension and artificial intelligence risk research are two possible examples.

That said, we don't need to solve everything today. The second-party payment is an idea that has only become popular in recent years. We have only seen small-scale experiments of second-party voting and financing, and the second-party payment application has never seen it! There is still a long way to go, but if we can get these mechanisms to land, they will definitely be of great benefit to us!