Quadratic voting (QV) and quadratic funding (QF) are a pair of mechanisms designed to fix the central defect of one-token-one-vote: that it collapses into one-dollar-one-vote, letting a handful of large holders decide everything. Both work by making influence concave in money — each additional unit of voting power costs quadratically more than the last, so a voter's effective say grows only with the square root of what they spend. The intent is a middle path between plain majority rule, which ignores how strongly anyone cares, and plutocratic weighted voting, which lets the richest participant simply buy the outcome.

The catch — and the reason QV/QF remains niche in on-chain DAO governance despite its elegance — is that it only works if you can stop one person from posing as many. That makes it entirely dependent on an identity layer, the hardest unsolved problem in permissionless governance.

The core idea: a quadratic price for influence

Under quadratic voting, a participant is given a budget of voice credits and may buy as many votes on an issue as they like — but the cost is the square of the number of votes. One vote costs 1 credit, two votes cost 4, three cost 9, ten cost 100. The formalization is due to Steven Lalley and Glen Weyl in "Quadratic Voting: How Mechanism Design Can Radicalize Democracy" (2018), building on the argument in Eric Posner and Glen Weyl's book Radical Markets.

Why square the cost? Because it makes the marginal cost of each additional vote grow linearly, and under standard assumptions that aligns each voter's spending with the true intensity of their preference: someone who cares twice as much will rationally buy twice as many votes, not four times as many. The result is a rule that captures how much people care, not just which side they are on — while still charging steeply enough that no single deep-pocketed actor can dominate. A voter with 100× the credits of another ends up with only 10× the votes.

Quadratic funding: QV applied to public goods

Quadratic funding generalizes the same square-root logic to allocating a shared matching pool across many projects. Introduced by Vitalik Buterin, Zoë Hitzig and Glen Weyl in "A Flexible Design for Funding Public Goods" (2018, published in Management Science), the mechanism sets each project's total funding proportional to the square of the sum of the square roots of its individual contributions:

match ∝ ( Σ √c_i )²

The consequence is that the number of contributors matters far more than the size of any one contribution. A project funded by 100 people giving $1 each receives a much larger match than one funded by a single $100 donor, because 100 × √1 = 100, whereas √100 = 10. QF therefore surfaces what a community broadly wants rather than what its wealthiest member wants — the theoretically optimal way to fund public goods when contributions are voluntary. RadicalxChange calls the family of designs "plural funding."

The Achilles' heel: identity and collusion

Concave voting only rewards breadth if each "person" really is one person. The moment an actor can split a large holding across many wallets, the defense inverts: a whale who divides $100 into a hundred $1 identities is treated as a hundred small, independent backers and reaps the full breadth bonus. This is the classic Sybil attack, and it is fatal to a naïve implementation.

QF is also acutely vulnerable to collusion: two colluding donors (or two fake accounts owned by one person) can each contribute their full balance to the same grants and extract an outsized match. Buterin's own review of Gitcoin's early rounds introduced the standard patch — pairwise-bounded QF, which caps the matching any single pair of contributors can trigger and discounts groups that donate to the same cluster of grants, penalizing coordinated behaviour while preserving the breadth signal from genuinely independent backers. Even so, every real deployment leans on some identity or proof-of-personhood layer, and none of them is watertight.

Where it is used

Gitcoin Grants is the flagship QF deployment — it has channeled well over $60M into open-source and public-goods projects across dozens of matching rounds, and remains the reference implementation of pairwise-bounded QF at scale (see Gitcoin's own explainer and its DeSci rounds). Beyond crypto, the Colorado State Legislature ran the highest-profile QV pilot: in 2019 the Democratic caucus used quadratic voting to prioritize dozens of competing appropriations bills, producing a clearer signal of intensity than a show of hands could (Colorado Sun; RadicalxChange case study) — though a 2024 court ruling that the anonymous ballots violated the state's open-meetings law halted the practice, a reminder that the mechanism's secrecy assumptions can collide with governance transparency.

The limits — and a 2026 impossibility result

The deepest critique is not that QV/QF is hard to Sybil-proof in practice, but that no concave weighting can be Sybil-proof in principle. A May 2026 analysis, "Concave is the New Linear: The Impossibility of Anti-Plutocratic DAO Governance" (Bennett, Vander Vos, Le & Belenkiy), proves that whenever a wallet of any size yields nonzero voting power, an attacker who splits tokens across enough wallets recovers voting power that grows at least linearly in their holdings — i.e. a concave rule degrades back to plain token-weighting under attack. Testing the five largest DAOs (ENS, Compound, Uniswap, Arbitrum, ZKsync), the authors measured amplification factors above 1,100× under quadratic voting, with attack costs far below the value at stake in real governance votes.

The lesson for DAO designers is stark: without a hard, unforgeable identity layer, an "anti-plutocratic" curve buys little — it adds complexity and a false sense of fairness while a determined whale simply pays the (small) cost of splitting. This is why token-weighting, for all its plutocratic flaws, remains the on-chain default, and why the frontier of the debate is now identity and measurable concentration rather than the shape of the voting curve itself.

How Caper approaches this

A caper does not use a quadratic or otherwise concave vote curve — and that is a deliberate choice, not an omission. Its canonical weight is linear in a member's token balance, multiplied by the non-transferable "vote tokens" they have earned by participating: one factor is stake, the other is demonstrated engagement. Because the 2026 impossibility result shows a concave curve collapses to linear under Sybil-splitting anyway, chasing anti-plutocratic fairness through the curve would add complexity for a defense that does not hold.

Instead, a caper bounds the power of concentrated wealth two other ways. First, influence and exit are the same formula: the share of the treasury a member can claim by burning their stake is computed identically to their voting weight, so a large holder who steers the DAO against its members cannot also extract more than their pro-rata share on the way out — and the exit door is always open. Second, weight rewards sustained participation through soulbound vote tokens rather than one-shot buying power, so idle capital and freshly-acquired tokens count for less than committed, engaged members. It is an honest trade: a caper accepts that on-chain votes are wealth-weighted, and makes that weight accountable rather than pretending a curve can erase it.

References

• Steven P. Lalley & E. Glen Weyl, Quadratic Voting: How Mechanism Design Can Radicalize Democracy, AEA Papers & Proceedings 108 (2018).
• Vitalik Buterin, Zoë Hitzig & E. Glen Weyl, A Flexible Design for Funding Public Goods, Management Science 65(11) (2019).
• Vitalik Buterin, Review of Gitcoin Grants CLR Round 3 (2019) — origin of pairwise-bounded QF.
• Bennett, Vander Vos, Le & Belenkiy, Concave is the New Linear: The Impossibility of Anti-Plutocratic DAO Governance (2026).
• RadicalxChange, Plural Funding and Quadratic Voting in Colorado.
• Gitcoin, Quadratic Funding.