Quadratic voting

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

Quadratic voting is a collective decision-making procedure where individuals allocate votes to express the degree of their preferences, rather than just the direction of their preferences.[1] By doing so, quadratic voting helps enable users to address issues of voting paradox and majority-rule. Quadratic voting works by allowing users to 'pay' for additional votes on a given matter to express their preference 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 in the class of cardinal 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 preference for or against a specific decision, additional votes could be allocated to proportionally demonstrate the voter's preferences. 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 preferences and interests toward the particular decision.[3] This money is eventually cycled back to the voters based upon per capita. Both Weyl and Lalley conducted research 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
Number of votes "Vote credit" cost
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 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 votingEdit

Quadratic voting is a promising voting method because it efficiently mirrors population preferences, is simple and easy to understand, and is robust.


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]


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.[7] Proposals have been put forward to make QV more robust with respect to both collusion and outside attacks.[8] 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.[7] When the election is 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.[7] Some distortionary behaviors can occur for QV in small populations due to people stoking issues to get more return for themselves,[9] but this issue has not been shown to be a practical issue for larger populations. Due 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.[10]

History of quadratic votingEdit

The primary motivation for the creation of QV was that an optimal voting mechanism for decisions involving public goods was created in the 60's and 70's by Vickrey, Clarke, and Groves (VCG).[11][12][13] Despite high initial excitement about this mechanism, including Vickrey receiving a Nobel Prize for the work, the VCG mechanism was not sufficiently robust and practical to be implemented, and the mechanism found almost no practical adoption. Quadratic voting was intended by its inventors to deliver similarly optimal outcomes as the VCG mechanism while being easier for people to use and understand, and being more robust with respect to collusion and other practical considerations. A mechanism closely resembling quadratic voting was first published in 1977 by Groves and Ledyard,[14] with a similar mechanism being proposed by Hylland and Zeckhauser in 1980.[15] More recently, E. Glen Weyl has done research on the mechanism, invented the name, and extensively promoted the mechanism.[16][Note 1] After circulating working papers on the idea starting in 2012, Weyl worked on QV with Steven Lalley and Eric Posner to further refine the formalism of quadratic voting and its applications.[1][7][17][2]

Many experiments at various scales [18][19][20] and simulations,[21] and theoretical analyses[9][1] have been done on quadratic voting since 2012. A major step in the adoption of quadratic voting for political purposes happened in April 2019, when the Democratic caucus of the Colorado House of Representatives used quadratic voting to make budgetary decisions. The experiment in Colorado was generally viewed as successful in that the budget allocations were reported to be reasonable, and the process was smooth.[22][23] Several organizations and communities have formed to promote adoption of quadratic voting concurrent with its continued academic research, including Democracy Earth (an online platform for quadratic voting), Collective Decision Engines (an app to facilitate QV adoption), and RadicalxChange (a community dedicated to decentralized forms of society and governance).

Criticisms of quadratic voting mechanismsEdit

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][24][Note 2] 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.[24][25][7]

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,[7]
  • Votes could be made more expensive to wealthy voters either for all issues, or for issues which are polarized on the basis of wealth.[7]

Other criticisms have been made, including: 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.[17]


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,[20] and elections and other democratic decisions.[17]

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 9 votes on one bill (as 81 virtual tokens represented 9 votes for one bill) and 3 votes on another 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.[23] 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.[28]

Quadratic fundingEdit

Vitalik Buterin in collaboration with 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.[29][30][31]

The Gitcoin Grants initiative is an early adopter of quadratic funding. Led by Kevin Owocki, Scott Moore, and Vivek Singh, the initiative has distributed more than $2,000,000 to open-source software development projects as of early 2020.[32]

Similar projectsEdit

There are various projects that follow the general trends of innovation past the traditional paper ballot voting methodologies, including the ideologies of quadratic voting. Other practices that are currently in development include instant-runoff voting (IRV), which is centered on a ranked preferential voting method. Voters are allowed to rank their candidates by personal preference, and if a candidate wins over half of the votes as first choice, then the candidate wins. This method contains research that suggests IRV is one of the least manipulative voting methods.[33]

The optical scan voting system adds additional security on the paper ballot voting system by using an optical scanner in an electronic voting system. This method enables voters with disabilities or those who are not familiar with the voting system to cast their votes. This also reduces time to process votes, announce results, and allow more users to vote.[34] Most voters use electronic or optical-scan ballots and is recorded through computer memory.[35]

Power voting, also known as e-voting, gained popularity in Estonia with their municipal elections in 2005[36] and Estonian parliamentary election in 2007.[37] This is voting that can be conducted through the Internet, which is reported to be the most cost-efficient method that was found within Estonia's Electoral System.[38]

See alsoEdit


  1. ^ a b c d Lalley, Steven; Weyl, E. Glen (2017-12-24). "Quadratic Voting: How Mechanism Design Can Radicalize Democracy". Rochester, NY. SSRN 2003531. Cite journal requires |journal= (help)
  2. ^ a b c 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 (2014-12-30). "Quadratic voting". ERIC POSNER. Retrieved 2019-10-09.
  4. ^ Weyl, E. Glen (2017-07-01). "The robustness of quadratic voting". Public Choice. 172 (1): 75–107. doi:10.1007/s11127-017-0405-4. ISSN 1573-7101.
  5. ^ Ellenberg, Jordan (2015-01-16). "Saving Democracy With Quadratic Equations". Wall Street Journal. ISSN 0099-9660. Retrieved 2019-11-19.
  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 c d e f g 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.
  8. ^ 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.
  9. ^ a b 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.
  10. ^ 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.
  11. ^ Vickrey, William (1961). "Counterspeculation, Auctions, and Competitive Sealed Tenders". The Journal of Finance. 16 (1): 8–37. doi:10.2307/2977633. JSTOR 2977633.
  12. ^ Clarke, Edward H. (1971). "Multipart pricing of public goods". Public Choice. 11: 17–33. doi:10.1007/BF01726210.
  13. ^ Groves, Theodore (1973). "Incentives in Teams". Econometrica. 41 (4): 617–631. doi:10.2307/1914085. JSTOR 1914085.
  14. ^ Groves, Theodore; Ledyard, John (1977). "Optimal Allocation of Public Goods: A Solution to the "Free Rider" Problem" (PDF). Econometrica. 45 (4): 783. doi:10.2307/1912672. JSTOR 1912672.
  15. ^ Hylland, Aanund; Zeckhauser, Richard (1980). "A mechanism for selecting public goods when preferences must be elicited". Harvard Kennedy School of Government Discussion Paper D, 70. Cite journal requires |journal= (help)
  16. ^ Weyl, E. Glen (2012). "Quadratic Vote Buying". doi:10.2139/ssrn.2003531. Cite journal requires |journal= (help)
  17. ^ a b c 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.
  18. ^ Goeree, Jacob K.; Zhang, Jingjing (2012). "Electoral Engineering: One Man, One Vote Bid". Cite journal requires |journal= (help)
  19. ^ Cárdenas, J. C.; Mantilla, C.; Zárate, R. D (2014). "Purchasing votes without cash: Implementing quadratic voting outside the lab". Cite journal requires |journal= (help)
  20. ^ a b 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.
  21. ^ Weyl, E. Glen (2017). "The robustness of quadratic voting". Public Choice. 172 (1–2): 75–107. doi:10.1007/s11127-017-0405-4.
  22. ^ "Colorado Tried a New Way to Vote: Make People Pay—Quadratically". Wired. ISSN 1059-1028. Retrieved 2019-10-09.
  23. ^ a b "A New Way of Voting That Makes Zealotry Expensive". Bloomberg.com. May 2019. Retrieved 2019-10-09.
  24. ^ a b Laurence, Ben; Sher, Itai (2017). "Ethical considerations on quadratic voting". Public Choice. 172 (1–2): 195–222. doi:10.1007/s11127-017-0413-4.
  25. ^ Ober, Josiah (2017). "Equality, legitimacy, interests, and preferences: Historical notes on Quadratic Voting in a political context". Public Choice. 172 (1–2): 223–232. doi:10.1007/s11127-017-0409-0.
  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. ^ "Quadratic voting as an input to cost-benefit analysis". doi:10.1007/s11127-017-0408-1. Cite journal requires |journal= (help)
  28. ^ "Democracy Earth - Borderless governance". democracy.earth. Retrieved 2019-11-20.
  29. ^ Buterin, Vitalik. "Quadratic Payments: A Primer".
  30. ^ Buterin, Vitalik; Hitzig, Zoë; Weyl, E. Glen. "Liberal Radicalism: A Flexible Design For Philanthropic Matching Funds". arXiv:1809.06421. doi:10.2139/ssrn.3243656. Cite journal requires |journal= (help)
  31. ^ Weyl, Glen; Wiblen, Rob. "Glen Weyl on uprooting capitalism and democracy for a just society". 80,000 Hours Podcast.
  32. ^ "Gitcoin Grants Explorer". Gitcoin: Grow Open Source.
  33. ^ Bartholdi, John. "Single transferable vote resists strategic voting" (PDF). Georgia Tech.
  34. ^ Mahoney, Matt. "Flawless Vote Counts". MIT Technology Review. Retrieved 2019-11-13.
  35. ^ "Most U.S. voters use electronic or optical-scan ballots". Pew Research Center. Retrieved 2019-11-20.
  36. ^ "Estonia forges ahead with e-vote". 2005-10-14. Retrieved 2019-11-13.
  37. ^ "Estonia to hold first national Internet election - CNET News". archive.is. 2012-07-13. Retrieved 2019-11-13.
  38. ^ Krimmer, Robert; Duenas-Cid, David; Krivonosova, Iuliia; Vinkel, Priit; Koitmae, Arne (2018). Krimmer, Robert; Volkamer, Melanie; Cortier, Véronique; Goré, Rajeev; Hapsara, Manik; Serdült, Uwe; Duenas-Cid, David (eds.). "How Much Does an e-Vote Cost? Cost Comparison per Vote in Multichannel Elections in Estonia". Electronic Voting. Lecture Notes in Computer Science. Springer International Publishing. 11143: 117–131. doi:10.1007/978-3-030-00419-4_8. ISBN 9783030004194.


  1. ^ This paper has been revised several times, but was originally published online in 2012.
  2. ^ 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.