What is Quadratic Voting?

A way of voting in which everyone gets the same budget of points to spread across the options they care about. One vote costs 1 point, two votes cost 4, three cost 9 — the square of the number of votes — so you can show how strongly you feel, but piling everything onto one option gets expensive.

In an ordinary vote, someone who is "slightly in favor" and someone whose life depends on the outcome each get exactly one vote. The strength of their feelings never shows up in the result. Quadratic Voting (QV) was designed to let a vote reflect how much a person actually cares. "Quadratic" simply means "squared".

In short

In QV, everyone receives the same number of points. Each person spreads their points across the options they like. But the cost rises with the square of the number of votes: one vote on an option costs 1 point, two votes cost 4 points, three votes cost 9 points [1][2].

So you can put extra votes on the thing that matters most to you. But if you pour everything into a single option, you get few votes for your points.

For example

Taiwan's Presidential Hackathon (2019). The Taiwanese government used QV for the vote in which participants and citizens helped select the winning teams. That is how chapter 2-2 of the book ⿻ Plurality puts it; chapter 7-0 of the same book says that QV is used for public participation "in selecting the top 20 teams". The two chapters describe what QV selects differently [4][5]. At a planning meeting, Audrey Tang, then Taiwan's Digital Minister, explained it like this: with 100 points, spending all 100 on one project gets you only 10 votes. But putting 7 votes on each of two projects costs 49 + 49 = 98 points, with 2 points left over [1]. In a meeting later that year, Tang said each person actually received 99 points in the real vote [3]. The book ⿻ Plurality says that the method "allowed a wide range of participants to be at least partial winners, by making it likely everyone would have supported some winner and that if someone felt very strongly in favor of one project they could give it a significant boost" [4].

The Colorado state legislature, USA (2019). According to the Chinese-language Wikipedia, Democratic members of the Colorado House of Representatives tried QV in April 2019. Each member had 100 points and could, for example, put up to 10 votes on one bill, or 5 votes on each of four bills. The bill that received the most votes was an equal-pay bill (60 votes), and nobody spent all their points on a single bill [6]. The book Plurality, on the other hand, says QV is used "frequently in the Colorado State Legislature to prioritize spending" [11]. But the concrete example Wikipedia gives is only this April 2019 trial, and no record of how often it has been used confirms the book's "frequently". In a December 2019 interview, Tang said that "the Colorado state budget, is now also determined by the quadratic voting", but did not give a number of times or years [7].

The video game Civilization VI. In the "World Congress" of this strategy game's expansion pack, "every civilization gets a single vote for free, but additional votes cost increased diplomatic favor, at an increasing rate." The book presents this as a variant of the quadratic voting procedure that "one of us invented" (the Japanese edition names Glen Weyl) [8].

How it works

A small worked example

Say you have 16 points [6].

One vote on each of 16 options → 1 point × 16 = 16 points
Four votes on a single option → 4 × 4 = 16 points

Both cost the same 16 points. Concentrating gets you 4 votes; spreading thinly gets you 16 votes in total. You can show strong feeling through more votes, but that leaves fewer points for everything else. In his commentary on the Japanese edition, Ken Suzuki gives the example of 5 votes costing 25 points (5 × 5 = 25) [2].

Why "squared"?

Vitalik Buterin, the founder of Ethereum, restates the idea in an introductory article: the n-th vote costs n dollars [9].

The first vote costs 1, the second 2, the third 3, and so on — each additional vote costs a bit more. So people keep buying votes until the price of the next vote catches up with how much the issue matters to them. If one person cares about an issue twice as much as another, they end up buying exactly twice as many votes. In other words, the number of votes is exactly proportional to the strength of feeling.

Add up the cost of all n votes and you get roughly n² ÷ 2. That, Buterin explains, is where the name "quadratic" comes from [9]. This differs from the rule at the start of this article (n votes cost n² points) only by a factor of 1/2: in both, the total cost is proportional to the square of the number of votes, and the price of the next vote rises in proportion to the number of votes (under the n² rule, the n-th vote costs n² − (n−1)² = 2n − 1 points). The abstract of a paper by Weyl and a co-author likewise says that only a squared cost produces this property, and that it leads to the best outcome for society as a whole [10].

The book's explanation: square roots

Chapter 5-6 of Plurality explains it from another angle [11]. Votes from people who have nothing to do with each other tend to cancel out on average, so their influence grows only with the square root of their number. Votes held together by one person, by contrast, count in full. In the book's words, "10,000 uncorrelated votes will weigh as heavily as only 100 correlated ones" (100 × 100 = 10,000).

So, the book argues, it is fair to give each person a weight equal to the square root of how strongly they care. In QV the number of votes is the square root of the points spent (16 points buys 4 votes), which gives exactly this shape [11].

In a 2019 meeting, Tang put it this way too: spending 10,000 Taiwan dollars gets you only 100 votes, but 100 people chipping in 1 dollar each also make 100 votes — so people with money and people with many connections stand on equal footing [3].

Money or points?

QV can also be designed so that votes are bought with money. In that version, the money collected is spent on public purposes and returned to the voters [6]. The points used at Taiwan's hackathon, however, were not money [1]. Tang also mentions another advantage: on an issue you don't understand well, you can abstain and save your points [3].

Where it came from

Similar squared mechanisms had been proposed since the 1970s. A 2018 paper by Buterin and others summarizes the history this way: Weyl presented the idea anew in the form of voting in 2012, and it was later given a mathematical foundation [12]. QV was also a central proposal in Radical Markets, the earlier book by Weyl and the legal scholar Eric Posner [13].

What is debated

It breaks if one person can pretend to be many. Buterin points out that if accounts (identities) are easy to multiply, people can keep buying the cheap first vote with a fresh identity each time, and the system collapses. Identity doesn't need to be perfect, he says, but it must be robust enough that cheating doesn't pay [9]. → Proof of personhood
Buying, selling and coordinating votes. If votes can be sold, the richest win again. So, Buterin argues, voting needs to be secret enough that even the voter cannot prove to others how they voted [9][14].
Doesn't paying with money favor the rich? Buterin notes that because of the squared cost, 100 times the money buys only 10 times the influence, and that giving everyone the same points narrows the gap even further [9].
Who sets the agenda? Buterin points out that QV has no mechanism for deciding what gets put to a vote in the first place [9]. A Japanese reading note likewise criticizes that the power to decide how issues are divided up goes unexamined [15]. The book itself admits the limit: voting "expresses and determines preferences about decisions already posed by other social processes" [16].
Could it strengthen bloc voting? A note by Hirokazu Nishio cites the point that QV gives more weight to 100 organized people casting one vote each than to one person casting 100 votes, and so could amplify organized blocs [17].
It can't stop a stampede. A Japanese blog post criticizes QV because it takes strength of feeling as its input: amid enthusiasm with almost no opposition, it cannot stop things from running out of control [18].
Second thoughts from its inventor. In 2020, Weyl looked back and said that Radical Markets, including QV, carried a way of thinking that saw people as individuals cut off from society and designed mechanisms from the top down [19].
Not suited to every situation. Nishio writes that QV is a poor fit for programs where the people running the program coach the selected projects for about half a year: a mentor whose favorite lost is unlikely to support the winner with the same enthusiasm [20].

Where it fits in Plurality

In chapter 5-6, "⿻ Voting", Plurality presents QV, alongside Liquid Democracy, as one of two "exciting, though incomplete, approaches" [11].

But the book does not stop at QV. The square-root rule, it says, is "only optimal when voters are perfectly internally unified and perfectly externally uncorrelated/uncoordinated" [11]. So the book moves toward counting votes in a way that takes people's connections into account. Buterin describes this shift: in Radical Markets the problem was intensity of preference, while Plurality tackles a different problem — how to count votes that are connected to each other [13].

Chapter 5-7 applies the same idea to allocating money: Quadratic Funding [21]. And because QV assumes that one person cannot become many, it links to the discussion of Identity and Personhood.

Sources

Discussion of the Presidential Hackathon voting method (2019-02-18, transcript of Audrey Tang, in Chinese) — https://sayit.archive.tw/2019-02-18-%E7%B8%BD%E7%B5%B1%E7%9B%83%E9%BB%91%E5%AE%A2%E6%9D%BE%E7%A5%A8%E9%81%B8%E6%96%B9%E5%BC%8F%E8%A8%8E%E8%AB%96 ［CC0］
Ken Suzuki, commentary on the Japanese edition of Plurality (in Japanese) — https://note.com/digitaldemocracy/n/nb8b59a6b775d
Meeting of Audrey Tang with Chang Jie-ping and Kao Chung-chien (2019-09-23, transcript in Chinese) — https://sayit.archive.tw/2019-09-23-%E5%BC%B5%E6%BD%94%E5%B9%B3%E5%8F%8A%E9%AB%98%E9%87%8D%E5%BB%BA%E6%8B%9C%E6%9C%83 ［CC0］
Plurality, 2-2 "The Life of a Digital Democracy" — https://www.plurality.net/v/chapters/2-2/eng/ ［CC0］
Plurality, 7-0 "Policy", the paragraph on the Presidential Hackathon — https://www.plurality.net/v/chapters/7-0/eng/ ［CC0］
"平方投票法" (Quadratic voting), Chinese-language Wikipedia — https://zh.wikipedia.org/zh-tw/%E5%B9%B3%E6%96%B9%E6%8A%95%E7%A5%A8%E6%B3%95 ［CC BY-SA 4.0］
Audrey Tang, interview with Adrien Simorre (2019-12-27, transcript) — https://github.com/audreyt/transcript/blob/main/2019-12-27-Interview-with-Adrien-Simorre.md ［CC0］
Plurality, 5-6 "⿻ Voting", opening — https://www.plurality.net/v/chapters/5-6/eng/ ［CC0］
Vitalik Buterin, "Quadratic Payments: A Primer" (2019) — https://vitalik.eth.limo/general/2019/12/07/quadratic.html
Lalley and Weyl, "Quadratic voting: How mechanism design can radicalize democracy", abstract (2018) — https://www.microsoft.com/en-us/research/publication/quadratic-voting-how-mechanism-design-can-radicalize-democracy/
Plurality, 5-6 "⿻ Voting", section "⿻ Voting tomorrow" — https://www.plurality.net/v/chapters/5-6/eng/ ［CC0］
Buterin, Hitzig and Weyl, "A Flexible Design for Funding Public Goods" (2018) — https://arxiv.org/abs/1809.06421
Vitalik Buterin, "Plurality philosophy in an incredibly oversized nutshell" (2024-08-21) — https://vitalik.eth.limo/general/2024/08/21/plurality.html
Vitalik Buterin, "On Collusion" (2019) — https://vitalik.eth.limo/general/2019/04/03/collusion.html
syumi_data, reading note on Li Shunzhi's Techno-Despotism and the Road to the Commons (note, 2026-01-03, in Japanese) — https://note.com/syumi_data1230/n/n5d4a2c2115e0
Plurality, 5-6 "⿻ Voting", section "Limits of ⿻ voting" — https://www.plurality.net/v/chapters/5-6/eng/ ［CC0］
Hirokazu Nishio, "Quadratic Voting and bloc votes" (Cosense, in Japanese) — https://scrapbox.io/nishio/Quadratic_Voting%E3%81%A8%E7%B5%84%E7%B9%94%E7%A5%A8
Project Aequitas, "Three people who tried to update democracy, and their common bug" (note, 2026-07-11, in Japanese) — https://note.com/glossy_lemur5728/n/n75e891a7f46b
E. Glen Weyl, "Why I Am not a Market Radical" (2020) — https://www.radicalxchange.org/updates/blog/why-i-am-not-a-market-radical/
Hirokazu Nishio, "QV is not suited to selecting projects for mentored programs" (Cosense, in Japanese) — https://scrapbox.io/nishio/%E4%BC%B4%E8%B5%B0%E5%9E%8B%E3%83%97%E3%83%AD%E3%82%B0%E3%83%A9%E3%83%A0%E3%81%AE%E6%8E%A1%E6%8A%9E%E3%81%ABQV%E3%81%AF%E5%90%91%E3%81%84%E3%81%A6%E3%81%84%E3%81%AA%E3%81%84
Plurality, 5-7 "Social Markets" — https://www.plurality.net/v/chapters/5-7/eng/ ［CC0］

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