Quadratic voting

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Inside the Colorado State Capitol Building, where the quadratic voting process took place

Quadratic voting is a collective decision-making procedure which involves individuals allocating votes to express the degree of their preferences, rather than just the direction of their preferences.[1] By doing so, quadratic voting seeks to address issues of the Condorcet paradox and majority rule. Quadratic voting works by allowing users to "pay" for additional votes on a given matter to express their support for given issues more strongly, resulting in voting outcomes that are aligned with the highest willingness to pay outcome, rather than just the outcome preferred by the majority regardless of the intensity of individual preferences. The payment for votes may be through either artificial or real currencies (e.g. with tokens distributed equally among voting members or with real money).[2][1] Quadratic voting is a variant of cumulative voting. It differs from cumulative voting by altering "the cost" and "the vote" relation from linear to quadratic.

Quadratic voting is based upon market principles, where each voter is given a budget of vote credits that they have the personal decisions and delegation to spend in order to influence the outcome of a range of decisions. If a participant has a strong support for or against a specific decision, additional votes could be allocated to proportionally demonstrate the voter's support. A vote pricing rule determines the cost of additional votes, with each vote becoming increasingly more expensive. By increasing voter credit costs, this demonstrates an individual's support and interests toward the particular decision.[3] If money is used, it is eventually cycled back to the voters based upon per capita. Both E. Glen Weyl and Steven Lalley published research in 2017 in which they claim to demonstrate that this decision-making policy expedites efficiency as the number of voters increases.[4] The simplified formula on how quadratic voting functions is[5]

cost to the voter = (number of votes)2.
Vote pricing example
of votes
"Vote credit"
1 1
2 4
3 9
4 16
5 25

The quadratic nature of the voting suggests that a voter can use their votes more efficiently by spreading them across many issues. For example, a voter with a budget of 16 vote credits can apply 1 vote credit to each of the 16 issues. However, if the individual has a stronger passion or sentiment on an issue, they could allocate 4 votes, at the cost of 16 credits, to the singular issue, effectively using up their entire budget. This mechanism towards voting demonstrates[according to whom?] that there is a large incentive to buy and sell votes, or to trade votes. Using this anonymous ballot system provides identity protection from vote buying or trading, since these exchanges cannot be verified by the buyer or trader.[6]

Properties of quadratic voting[edit]


The quadratic cost function has the unique property that people purchase votes directly proportionally to the strength of their preferences. As a result, the total number of votes for a given issue is the sum of the strength of the preferences of the people who voted. This is because the marginal cost of each additional vote increases linearly with the number of votes. If the marginal cost increases less than linearly with respect to the number of votes, then someone who values a vote twice as much will tend to purchase more than twice as many votes, and the system will be predisposed to dominance by special interest groups with strong, concentrated interests. One-dollar-one-vote is the limit of this behavior, wherein the marginal cost of a vote is constant. On the other hand, if the cost function increase more quickly than a linear rise, then the system will be predisposed to a tyranny of the majority, with the limit of this behavior being one-person-one-vote.[citation needed]

By contrast, majority rule based on individual person voting has the potential to lead to focus on only the most popular policies, so smaller policies would not be placed on as much significance. The larger proportion of voters who vote for a policy even with lesser passion compared to the minority proportion of voters who have higher preferences in a less popular topic can lead to a reduction of aggregate welfare. In addition, the complicating structures of contemporary democracy with institutional self-checking (i.e., federalism, separation of powers) will continue to expand its policies, so quadratic voting is responsible for correcting any significant changes of one-person-one-vote policies.[7]


Robustness of a voting system can be defined as how sensitive a voting scheme is to non-ideal behavior from either voters or outside influence. The robustness of QV with respect to various non-idealities has been studied, including collusion among voters, outside attacks on the voting process, and irrationality of the voters. Collusion is possible in most voting schemes to one extent or another, and what is key is the sensitivity of the voting scheme to collusion. It has been shown that QV exhibits similar sensitivity to collusion as one-person-one-vote systems, and is much less sensitive to collusion than the VCG or Groves and Ledyard mechanisms.[8] Proposals have been put forward to make QV more robust with respect to both collusion and outside attacks.[9] The effects of voter irrationality and misconceptions on QV results have been examined critically by QV by a number of authors. QV has been shown to be less sensitive to 'underdog effects' than one-person-one-vote.[8] When the election is not close, QV has also been shown to be efficient in the face of a number of deviations from perfectly rational behavior, including voters believing vote totals are signals in and of themselves, voters using their votes to express themselves personally, and voter belief that their votes are more pivotal than they actually are. Although such irrational behavior can cause inefficiency in closer elections, the efficiency gains through preference expression are often sufficient to make QV net beneficial compared to one-person-one-vote systems.[8] Some distortionary behaviors can occur for QV in small populations due to people stoking issues to get more return for themselves,[10] but this issue has not been shown to be a practical issue for larger populations. Due to QV allowing people to express preferences continuously, it has been proposed that QV may be more sensitive than 1p1v to social movements that instill misconceptions or otherwise alter voters' behavior away from rationality in a coordinated manner.[11]

History of quadratic voting[edit]

One of the earliest known models idealizing quadratic voting was proposed by 3 scientists: William Vickrey, Edward H. Clarke, and Theodore Groves. Together they theorized the Vickery-Clarke-Groves mechanism (VCG mechanism). The purpose of this mechanism was to find the balance between being a transparent, easy-to-understand function that the market could understand in addition to being able to calculate and charge the specific price of any resource. This balance could then theoretically act as motivation for users to not only honestly declare their utilities, but also charge them the correct price.[12] This theory was easily able to be applied into a voting system that could allow people to cast votes while presenting the intensity of their preference. However, much like the majority of the other voting systems proposed during this time, it proved to be too difficult to understand,[13] vulnerable to cheating, weak equilibria, and other impractical deficiencies.[14] As this concept continued developing, E. Glen Weyl, a Microsoft researcher, applied the concept to democratic politics and corporate governance and coining the phrase Quadratic Voting.[1][additional citation(s) needed]

Ideation in democratic politics[edit]

The main motivation of Weyl to create a quadratic voting model was to combat against the "tyranny of the majority" outcome that is a direct result of the majority-rule model. He believed the two main problems of the majority-rule model are that it doesn't always advance the public good and it weakens democracy.[15] The stable majority has always been systematically benefited at the direct expense of minorities.[16] On the other hand, even hypothetically if the majority wasn't to be concentrated in a single group, tyranny of the majority would still exist because a social group will still be exploited. Therefore, Weyl concluded that this majority rule system will always cause social harm.[15] He also believed another reason is that the majority rule system weakens democracy. Historically, to discourage political participation of minorities, the majority doesn't hesitate to set legal or physical barriers. As a result, this success of a temporary election is causing democratic institutions to weaken around the world.[15]

To combat this, Weyl developed the quadratic voting model and its application to democratic politics. The model theoretically optimizes social welfare by allowing everyone the chance to vote equally on a proposal as well as giving the minority the opportunity to buy more votes to level out the playing field.[15]

Ideation in corporate governance[edit]

Quadratic voting in corporate governance is aimed to optimize corporate values through the use of a more fair voting system. Common issues with shareholder voting includes blocking out policies that may benefit the corporate value but don't benefit their shareholder value or having the majority commonly outvote the minority.[17] This poor corporate governance could easily contribute to detrimental financial crises.[18]

With quadratic voting, not only are shareholders stripped of their voting rights, but instead corporate employees can buy as many votes as they want and participate in electoral process. Using the quadratic voting model, one vote would be $1, while two votes would be $4, and so on. The collected money gets transferred to the treasury where it gets distributed to the shareholders. To combat voter fraud, the votes are confidential and collusion is illegal. With this, not only is the majority shareholders' power against the minority stripped, but with the participation of everyone, it ensures that the policies are made for the corporate's best interest instead of the shareholders' best interest.[17]

Criticisms of quadratic voting mechanisms[edit]


The most common objection to QV using real currency is that although it efficiently selects the outcome for which the population has the highest willingness to pay, willingness to pay is not directly proportional to the utility gained by the voting population. More specifically, the wealthy can afford to buy more votes relative to the rest of the population.[3][19][Note 1] This would distort voting outcomes to favor the wealthy in situations where voting is polarized on the basis of wealth. While the wealthy having undue influence on voting processes is not a unique feature of QV as a voting process, the direct involvement of money in the QV process has caused many to have concerns about this method.

Several proposals have been put forward to counter this concern, with the most popular being QV with an artificial currency. Usually, the artificial currency is distributed on a uniform basis, thus giving every individual an equal say, but allowing individuals to more flexibly align their voting behavior with their preferences. While many have objected to QV with real currency, there has been fairly broad-based approval of QV with an artificial currency.[19][20][8]

Other proposed methods for ameliorating objections to the use of money in real currency QV are:

  • To reduce or eliminate the unequal representation due to wealth, QV could be coupled with a scheme that returns incomes from the QV process to the less-wealthy. One such scheme is proposed to by Weyl and Posner,[2]
  • For situations where issues are polarized based on wealth, one-person-one-vote may be a better alternative, depending on how gains in efficiency from preference expression balance with distortions due to wealth polarization. The use of QV vs one-person-one-vote could be determined on an issue-by-issue basis,[8]
  • Votes could be made more expensive to wealthy voters either for all issues, or for issues which are polarized on the basis of wealth.[8]

Quadratic voting has several advantages compared to the current systems of voting in democratic nations. However, it is not a perfect system. It has certain drawbacks in its framework that make it vulnerable to cheating and collusion. This lack of resistance against cheating could make it vulnerable to the Sybil attack, for example.[21] Collusion is also a potential threat to the system, where one might hire multiple people to cast votes for a certain issue. Quadratic voting is also weak against polarization by the wealthy, who can buy and cast more votes than the average voter.

Sybil attacks[edit]

One of the biggest weaknesses of quadratic voting is the lack of moderation when dealing with cheating.[22] The specific term used for cheating with quadratic voting is Sybil attacks. These attacks use sybils, or fake or duplicate identities, to influence community-oriented decisions to push them in their favor. Since a single vote has the potential to tilt a majoritarian group decision, prevention of sybil attacks is an important priority in ensuring the security of quadratic voting.[23] With one of its priorities being an open, peer-to-peer network, an anti-sybil identification software is a requirement to implement widespread quadratic voting.[1]

Some future possible avenues of inquiries include to investigate more intersectional proof of personhood systems that aren't directly blockchain-based.[24] For example, extending the web of trust by having a protocol that verifies proof of identities using social interactions would allow a community of users to assign corresponding levels of trusts to different candidates in relation with others. However, this would require a fully decentralized system.[24] This web-of-trust protocol could even expand to allowing candidates to provide proof of personhood by physical attendance, which could lead to trusted clusters that grow into communities.[25]


United States[edit]

Many areas have been proposed for quadratic voting, including corporate governance in the private sector,[26] allocating budgets, cost-benefit analyses for public goods,[27] more accurate polling and sentiment data,[28] and elections and other democratic decisions.[7]

Quadratic voting was conducted in an experiment by the Democratic caucus of the Colorado House of Representatives in April 2019. Lawmakers used it to decide on their legislative priorities for the coming two years, selecting among 107 possible bills. Each member was given 100 virtual tokens that would allow them to put either 10 votes on one bill (as 100 virtual tokens represented 10 votes for one bill) or 5 votes each (25 virtual tokens) on 4 different bills. In the end, the winner was Senate Bill 85, the Equal Pay for Equal Work Act, with a total of 60 votes.[29] From this demonstration of quadratic voting, no representative spent all 100 tokens on a single bill, and there was delineation between the discussion topics that were the favorites and also-rans. The computer interface and systematic structure was contributed by Democracy Earth, which is an open-source liquid democracy platform to foster governmental transparency.[30]


Taiwan has had 2 applications of quadratic voting so far.[when?] The first event was hosted by RadicalxChange in Taipei, where quadratic voting was used to vote in the Taiwanese presidential Hackathon.[31] The Hackathon projects revolved around 'Cooperative Plurality' – the concept of discovering the richness of diversity that is repressed through human cooperation.[32] Judges were given 99 points with 1 vote costing 1 point and 2 votes costing 4 points and so on. This stopped the follow-up effect and group influenced decision that happened with judges in previous years.[31] This event was considered a successful application of quadratic voting.

Another application is Taiwan's government-run e-democracy platform Join. This platform utilizes the quadratic voting system to encourage public participation in budget matters.[33] Citizens having 99 points to assign to their preferred policies using the standard quadratic voting model.[34] With over 4 million active participants, anyone can start an e-petition for a certain policy. When it surpasses 5,000 signatures, corresponding government sectors will address the questioned issue by holding a collaborative meeting. So far, Taiwan has held 40 collaborative meetings spanning topics of tax filing, medical resource distribution, or environmental maintenance in national parks.[33]


In Leipzig, Germany, Volt Germany – a pan-European party – held its second party congress and used quadratic voting to determine the most valued topics in their party manifesto among its members.[35] Partner with Deora, Leapdao, a technology start-up company, launched its quadratic voting software consisting of a "burner wallet". Since there was limited time and it was a closed environment, the "burner wallet" with a QR code acted as a private key that allowed congress to access their pre-funded wallet and a list of all the proposals on the voting platform.[36] The event was considered a success because it successfully generated a priority list that ranked the importance of the topics.

Quadratic voting also allowed researchers to analyze voter distributions. For example, the topic of Education showed especially high or emotional value to voters with the majority deciding to cast 4 or 9 voice-credits (2 or 3 votes) and a minority casting 25-49 voice-credits (5-7 votes).[36] On the other hand, the topic of Renewed Economy showed a more typical distribution with a majority of voters either not vote or max out at 9 voice-credits (3 votes). This indicates that there are less emotionally invested voters on this proposal as many of them didn't even spend tokens to vote on it.[36]


In Brazil, the city council of Gramado has used quadratic voting to define priorities for the year and to reach consensus on tax amendments.[37]

Quadratic funding[edit]

Vitalik Buterin in collaboration with Zoë Hitzig and E. Glen Weyl proposed quadratic funding, a way to allocate the distribution of funds (for example, from a government's budget, a philanthropic source, or collected directly from participants) based on quadratic voting, noting that such a mechanism allows for optimal production of public goods without needing to be determined by a centralized legislature. Weyl argues that this fills a gap with traditional free markets – which encourage the production of goods and services for the benefit of individuals, but fail to create outcomes desirable to society as a whole – while still benefiting from the flexibility and diversity free markets have compared to many government programs.[38][39][40]

The Gitcoin Grants initiative is an early adopter of quadratic funding. However, this implementation differs in several ways from the original QF scheme.[41] Led by Kevin Owocki, Scott Moore, and Vivek Singh, the initiative has distributed more than $60,000,000 to over 3,000 open-source software development projects as of 2022.[42]

Global hackathon organizer DoraHacks' developer incentive platform HackerLink has leveraged quadratic funding to help many open Web3 ecosystems like Solana, Filecoin and BSC distribute more than $10,000,000 to 1,500 projects. Schemes have been designed by the DoraHacks team to enhance the integrity of quadratic funding. HackerLink and Gitcoin are considered the largest quadratic funding platforms for funding public goods and open source projects.[43]

See also[edit]


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  2. ^ a b Posner, Eric A.; Weyl, E. Glen (2018). Radical markets: uprooting capitalism and democracy for a just society. ISBN 9780691177502. OCLC 1030268293.
  3. ^ a b Posner, Eric (30 December 2014). "Quadratic voting". ERIC POSNER. Retrieved 9 October 2019.
  4. ^ Weyl, E. Glen (1 July 2017). "The robustness of quadratic voting". Public Choice. 172 (1): 75–107. doi:10.1007/s11127-017-0405-4. ISSN 1573-7101. S2CID 189841584.
  5. ^ Ellenberg, Jordan (16 January 2015). "Saving Democracy With Quadratic Equations". Wall Street Journal. ISSN 0099-9660. Retrieved 19 November 2019.
  6. ^ Hasen, Richard L. (2000). "Vote Buying". California Law Review. 88 (5): 1323–1371. doi:10.2307/3481262. ISSN 0008-1221. JSTOR 3481262.
  7. ^ a b Weyl, Eric Glen; Posner, Eric A. (2014). "Voting Squared: Quadratic Voting in Democratic Politics". University of Chicago Law School Coase-Sandor Institute for Law and Economics.
  8. ^ a b c d e f Posner, Eric A.; Weyl, E. Glen (2017). "Quadratic voting and the public good: Introduction". Public Choice. 172 (1–2): 1–22. doi:10.1007/s11127-017-0404-5. S2CID 46616848.
  9. ^ Park, Sunoo; Rivest, Ronald L. (2017). "Towards secure quadratic voting". Public Choice. 172 (1–2): 151–175. doi:10.1007/s11127-017-0407-2. hdl:1721.1/110335. S2CID 5059331.
  10. ^ Patty, John W.; Penn, Elizabeth Maggie (2017). "Uncertainty, polarization, and proposal incentives under quadratic voting". Public Choice. 172 (1–2): 109–124. doi:10.1007/s11127-017-0406-3. S2CID 157943343.
  11. ^ Kaplow, Louis; Kominers, Scott Duke (2017). "Who will vote quadratically? Voter turnout and votes cast under quadratic voting". Public Choice. 172 (1–2): 125–149. doi:10.1007/s11127-017-0412-5. S2CID 157660093.
  12. ^ Algorithmic game theory. Nisan, Noam. Cambridge: Cambridge University Press. 2007. ISBN 978-0-511-35572-1. OCLC 191726233.{{cite book}}: CS1 maint: others (link)
  13. ^ Ausubel, Lawrence M.; Milgrom, Paul (9 December 2005), Cramton, Peter; Shoham, Yoav; Steinberg, Richard (eds.), "The Lovely but Lonely Vickrey Auction", Combinatorial Auctions, The MIT Press, pp. 17–40, doi:10.7551/mitpress/9780262033428.003.0002, ISBN 978-0-262-03342-8, retrieved 5 October 2020
  14. ^ Rothkopf, Michael H. (April 2007). "Thirteen Reasons Why the Vickrey-Clarke-Groves Process Is Not Practical". Operations Research. 55 (2): 191–197. doi:10.1287/opre.1070.0384. ISSN 0030-364X.
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  21. ^ Nebulas (21 September 2018). "Liberal Radicalism: Can Quadratic Voting Be the Perfect Voting System?". Medium. Retrieved 7 October 2020.
  22. ^ Nebulas (21 September 2018). "Liberal Radicalism: Can Quadratic Voting Be the Perfect Voting System?". Medium. Retrieved 3 November 2020.
  23. ^ Shahaf, Gal; Shapiro, Ehud; Talmon, Nimrod (August 2019). "Sybil-Resilient Reality-Aware Social Choice". Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence. California: International Joint Conferences on Artificial Intelligence Organization: 572–579. doi:10.24963/ijcai.2019/81. ISBN 978-0-9992411-4-1.
  24. ^ a b Siddarth, Divya; Ivliev, Sergey; Siri, Santiago; Berman, Paula (13 October 2020). "Who Watches the Watchmen? A Review of Subjective Approaches for Sybil-resistance in Proof of Personhood Protocols". arXiv:2008.05300 [cs.CR].
  25. ^ Howitt, Aleeza (5 March 2019). "Proposal for a Decentralized Unique Identity Seeding Protocol". UBI Research. Retrieved 10 November 2020.
  26. ^ Posner, Eric A.; Weyl, E. Glen. "Quadratic Voting as Efficient Corporate Governance". University of Chicago Law Review (University of Chicago Coase-Sandor Institute for Law & Economics Research Paper No. 643).
  27. ^ Masur, Jonathan S. (2017). "Quadratic voting as an input to cost-benefit analysis". Public Choice. 172 (1–2): 177–193. doi:10.1007/s11127-017-0408-1. S2CID 55731681.
  28. ^ Quarfoot, David; von Kohorn, Douglas; Slavin, Kevin; Sutherland, Rory; Goldstein, David; Konar, Ellen (2017). "Quadratic voting in the wild: Real people, real votes". Public Choice. 172 (1–2): 283–303. doi:10.1007/s11127-017-0416-1. S2CID 155832950.
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  31. ^ a b Huang, Yahsin (2 October 2019). "Highlights from First RadicalxChange Taipei Meetup". Medium. Retrieved 6 October 2020.
  32. ^ Erichsen, Leon (31 May 2020). "GitxChange: Build the Roots of Cooperative Plurality". RadicalxChange. Retrieved 6 October 2020.
  33. ^ a b "Inside Taiwan's new digital democracy". The Economist. ISSN 0013-0613. Retrieved 6 October 2020.
  34. ^ "The web's a threat to democracy? Think again, Taiwan says". Christian Science Monitor. 8 April 2020. ISSN 0882-7729. Retrieved 6 October 2020.
  35. ^ "The new voting system that could save our democracies". nesta. Retrieved 3 November 2020.
  36. ^ a b c "LeapDAO: delivering scalability as global public utility". leapdao.org. Retrieved 3 November 2020.
  37. ^ "The Mathematic Method that Could Offer a Fairer Way to Vote". The Economist.
  38. ^ Buterin, Vitalik. "Quadratic Payments: A Primer".
  39. ^ Buterin, Vitalik; Hitzig, Zoë; Weyl, E. Glen (2018). "Liberal Radicalism: A Flexible Design For Philanthropic Matching Funds". arXiv:1809.06421. doi:10.1287/mnsc.2019.3337. S2CID 198858039. SSRN 3243656. {{cite journal}}: Cite journal requires |journal= (help)
  40. ^ Weyl, Glen; Wiblen, Rob. "Glen Weyl on uprooting capitalism and democracy for a just society". 80,000 Hours Podcast.
  41. ^ Wildrick Thomas, Matthew (11 March 2021). "How to Fund Open Source".
  42. ^ "Gitcoin Grants Explorer". Gitcoin: Grow Open Source.
  43. ^ "HackerLink:Open Source Curation Market".


  1. ^ More formally, willingness to pay is approximately the utility gain experienced by the individual voting normalized by the marginal utility of money. The marginal utility of money decreasing with increasing wealth, and therefore willingness to pay is inflated for wealthy individuals.