Democracy has a math problem.
One person, one vote sounds fair until you realize it ignores intensity. The person mildly in favor counts the same as the person whose life depends on the outcome. This seems wrong, but the obvious fix—letting people cast more votes if they care more—immediately advantages those with more votes to cast. Weight votes by resources and you've built plutocracy with extra steps.
Quadratic mechanisms attempt to thread this needle. They let people express intensity of preference while making concentrated power expensive. The math is elegant. The implementation is treacherous.
The Core Insight
Linear systems scale linearly. If one vote costs one credit, ten votes cost ten credits. Someone with a hundred times your resources gets a hundred times your voice.
Quadratic systems scale quadratically. Your first vote costs one credit. Your second costs four. Your third costs nine. Your tenth costs a hundred. Influence becomes expensive fast.
Under quadratic costs, someone with a hundred times your resources doesn't get a hundred times your voice—they get ten times. The square root compresses advantages. Wealth still matters, but it matters less.
This compression has a specific mathematical property: it approximates optimal preference aggregation under certain assumptions. When people pay quadratically for influence, the resulting decisions tend to maximize aggregate welfare better than linear voting. The mechanism doesn't just feel fairer—it produces better outcomes by some defensible definition of "better."
Quadratic Voting in Practice
The simplest implementation gives everyone equal voice credits per decision period. You allocate those credits across proposals, paying the square of your allocation for each.
Suppose you have 100 credits and three proposals on the ballot. You could spread them evenly—allocating about 5-6 votes to each, costing around 25-36 credits per proposal. Or you could concentrate on one thing you care deeply about—allocating 10 votes to it (costing all 100 credits) and nothing to the others.
The system forces tradeoffs. Expressing strong preference on one issue means having no voice on others. This constraint makes expressed preferences meaningful. You can't care maximally about everything.
Colorado's state legislature experimented with quadratic voting for budget prioritization. Taiwan has used it in public consultations. Various blockchain governance systems have implemented versions. Results are mixed but instructive.
Quadratic Funding: The Public Goods Version
Quadratic voting allocates decision-making power. Quadratic funding allocates money.
The setup: a matching pool exists, funded by some central source. Individuals make contributions to projects they support. The matching pool then amplifies those contributions based on the number of contributors, not the amount contributed.
Specifically, the match is proportional to the square of the sum of square roots of contributions. This sounds complicated, but the effect is simple: many small contributions trigger more matching than few large contributions totaling the same amount.
If a thousand people each give $1, that project gets far more matching funds than if one person gives $1,000—even though the direct contributions are identical. The mechanism privileges breadth of support over depth of pockets.
Gitcoin Grants has run quadratic funding for public goods in the Ethereum ecosystem, distributing millions of dollars. The mechanism genuinely shifts resources toward projects with broad grassroots support rather than wealthy backers.
The Identity Problem Returns
Here's where elegant mathematics crashes into messy reality.
Quadratic mechanisms assume one identity per participant. The entire cost curve is built around making concentrated influence expensive. But concentration is only meaningful if we're measuring it correctly.
If I can create ten accounts, I can split my resources across them. Instead of paying 100 credits for 10 votes from one account, I pay 10 credits (1 vote each) across ten accounts. The quadratic cost curve becomes linear. The entire defense collapses.
For quadratic funding, Sybil attacks are even more devastating. The matching formula explicitly rewards number of contributors. Fake accounts don't just evade cost curves—they actively exploit the mechanism's core logic. One person with a hundred sock puppets triggers matching as if they had grassroots support from a hundred real humans.
Every implementation of quadratic mechanisms must solve the identity problem or fail. There are no exceptions. The math only works if the participants are real.
Identity Approaches and Their Failures
Various attempts to establish identity for quadratic systems have been tried:
Government ID verification - Effective against casual Sybils, but excludes people without ID, creates privacy risks, and remains vulnerable to ID fraud. Also philosophically uncomfortable for systems trying to improve on existing democratic infrastructure.
Social verification - Networks of vouching, where existing participants verify new ones. Creates onboarding friction and can be gamed through collusion. Vouching networks become attack surfaces.
Biometric verification - Iris scans, fingerprints, face recognition. Technically robust but creates honeypots of sensitive data, excludes people with accessibility needs, and faces accuracy problems across demographic groups. Also dystopian.
Proof of personhood protocols - Emerging systems like Worldcoin (iris scanning) or Proof of Humanity (video verification plus vouching). Each has tradeoffs between security, accessibility, and privacy. None are clearly adequate yet.
Financial identity - Bank accounts, credit histories, existing KYC verification. Excludes the unbanked, advantages those already integrated into financial systems, and doesn't necessarily prove uniqueness (people can have multiple bank accounts).
Behavioral analysis - Machine learning to identify coordinated inauthentic behavior. Can catch obvious Sybils but faces adversarial adaptation. Sophisticated attackers modify behavior to evade detection.
No approach solves the problem cleanly. Every identity system either excludes legitimate participants, creates unacceptable privacy risks, or remains gameable by motivated attackers. Usually all three.
The Collusion Variant
Even with perfect identity, quadratic mechanisms face a related attack: collusion.
If ten real people agree to coordinate—voting together, contributing together, splitting rewards—they can achieve outcomes impossible for ten individuals acting independently. Quadratic mechanisms assume independent preferences. Coordinated groups violate that assumption.
This is distinct from Sybil attacks. No fake identities are involved. Real people are really coordinating. The mechanism can't distinguish legitimate coalition (people who genuinely share preferences discovering each other) from illegitimate collusion (people who would prefer different outcomes agreeing to game the system together).
Some implementations try to detect and penalize correlated behavior. But correlation isn't causation. Punishing people who vote similarly punishes communities with genuinely shared values. The treatment is worse than the disease.
Partial Credit: What Quadratic Mechanisms Actually Achieve
Despite these problems, quadratic mechanisms provide real value when properly bounded.
They compress wealth advantages even imperfectly. If your Sybil detection catches 90% of fake accounts, the remaining 10% still face quadratic costs. Imperfect identity verification still helps—it just doesn't provide the complete defense the math promises.
They shift strategic calculations. Even in adversarial environments, knowing that concentrated spending is expensive changes behavior. Attackers might choose different targets or methods. The mechanism shapes the landscape even when it doesn't fully secure it.
They provide signal about preference intensity. Linear voting tells you how many people support something. Quadratic voting tells you how much people care. That information is valuable even if the mechanism is gameable—it's just not trustworthy enough for high-stakes binding decisions.
They work better in bounded contexts. Among verified members of a small organization—employees of a company, students at a university, members of a cooperative—identity is solved by existing institutional structures. Quadratic mechanisms can operate on that foundation without building identity infrastructure from scratch.
Quadratic Voting vs. Quadratic Funding
These mechanisms have different vulnerability profiles despite sharing mathematical foundations.
Quadratic voting is more robust because it's zero-sum within each decision. My increased influence on a proposal comes at the cost of my influence elsewhere. Sybil attacks let me evade costs, but the total influence in the system is still bounded by how many voice credits exist.
Quadratic funding is more vulnerable because it's positive-sum. The matching pool amplifies contributions. Sybil attacks don't just evade costs—they actively extract matching funds that would otherwise go to legitimate projects. The attacker steals from the pool, not just from other participants.
This means quadratic voting can tolerate some identity leakage while quadratic funding requires much tighter identity guarantees. The same level of Sybil resistance that makes quadratic voting "good enough" might make quadratic funding "actively exploited."
Implementation Lessons From Gitcoin
Gitcoin's quadratic funding rounds offer a live laboratory for these dynamics.
Early rounds were heavily gamed. Airdrop farmers created accounts to receive contribution matching, then extracted value without genuine interest in funded projects. Coordinated groups inflated each other's matching. The mechanism distributed funds, but not always to projects with real community support.
Successive rounds implemented increasingly sophisticated defenses: passport scores combining multiple identity signals, machine learning fraud detection, community flagging of suspicious projects, caps on matching per contributor. Each defense closed some attacks and opened others.
The current state is functional but imperfect. Gitcoin acknowledges that matching calculations are approximations given identity uncertainty. They've reduced fraud to manageable levels rather than eliminating it. The mechanism produces better outcomes than simple plutocratic funding but doesn't achieve the theoretical optimum.
This is probably the realistic ceiling for quadratic funding in open systems: "better than alternatives" rather than "mathematically optimal."
The Cost Curve Zoo
Quadratic is just one option in a family of cost curves.
Linear (1 credit = 1 vote) - No compression. Pure plutocracy. But also no identity requirements—Sybil attacks don't help because there's no cost curve to evade.
Quadratic (n votes cost n² credits) - Square root compression. Moderate wealth advantages, moderate Sybil vulnerability.
Cubic or higher (n votes cost n³ credits) - Stronger compression, but stronger Sybil incentives. The more you compress concentrated influence, the more you reward splitting across fake identities.
Logarithmic (diminishing returns) - Heavy compression of large holdings, but potentially too much—might disincentivize legitimate large contributions when those are actually valuable.
Capped linear (linear up to a limit, then zero) - Maximum influence per participant. Simple, but the cap creates arbitrary cliffs and still requires identity for enforcement.
Different contexts warrant different curves. The quadratic is popular because it has nice mathematical properties, but it's not uniquely correct. Understanding why you're choosing a particular cost function matters more than defaulting to quadratic because it's trendy.
When To Use Quadratic Mechanisms
They fit well when:
- Identity is reasonably solved by existing structures (membership organizations, verified communities)
- You want to measure preference intensity, not just preference direction
- Stakes are low enough that imperfect security is acceptable
- You're aggregating input for decisions made by humans, not automating decisions directly
- Broad participation is more important than preventing all gaming
They fit poorly when:
- Identity is unsolved and adversaries are sophisticated
- Stakes are high enough that gaming incentives are serious
- You need binding outcomes that can't be revisited if exploitation is discovered
- The population is anonymous or pseudonymous by design
- Wealth compression is your only goal (simpler mechanisms might suffice)
The Honest Summary
Quadratic mechanisms represent genuine innovation in collective decision-making. The mathematical insight is real. The compression of wealth advantages is valuable. The expression of preference intensity adds information that simpler mechanisms lose.
They are not robust against adversarial conditions without solved identity. Every deployment must grapple with Sybil attacks, and no current solution is fully adequate. Using quadratic mechanisms in high-stakes anonymous contexts is asking to be exploited.
The appropriate stance is enthusiasm tempered by humility. Use quadratic approaches where their assumptions hold. Don't pretend those assumptions hold where they don't. Combine them with other defenses rather than treating them as complete solutions.
Mechanism design is not a search for the one correct answer. It's navigation through tradeoffs that can't be escaped, only chosen deliberately.