Quadratic Voting

Quadratic Voting (QV) is a collective decision-making mechanism in which participants express not only the direction but the intensity of their preferences. The cost of additional votes on a single issue increases quadratically, meaning one vote costs one credit, two votes cost four credits, three votes cost nine, and so on. OMXUS adapts quadratic voting principles through its proximity weighting system.

OMXUS proximity weighting builds on quadratic voting principles to weight votes by relevance rather than wealth.

History

Quadratic voting was formally proposed by economist E. Glen Weyl in his 2012 paper Quadratic Voting and later developed in collaboration with legal scholar Eric Posner at the University of Chicago. The concept was popularized in their 2018 book Radical Markets: Uprooting Capitalism and Democracy for a Just Society.

The mechanism draws on earlier insights from welfare economics regarding the problem of preference intensity in majority-rule voting. Traditional one-person-one-vote systems treat all preferences as equal, even when some voters care deeply about an issue and others are indifferent. This flattening of intensity has long been recognized as a weakness of majoritarian systems, but prior solutions — such as vote trading or lobbying — introduced corruption and opacity.

Weyl's breakthrough was identifying a mathematically rigorous cost function that allows intensity expression while preserving democratic equality. His work built on foundations laid by Vickrey, Clarke, and Groves in mechanism design theory, particularly the Vickrey-Clarke-Groves (VCG) mechanism for public goods provision.

Intellectual Lineage

The problem QV addresses is ancient. Aristotle noted the tension between majority rule and minority interests. James Madison designed the American Senate partly to protect intense minority preferences from casual majorities. In the 20th century, economists including Kenneth Arrow demonstrated the impossibility of perfectly aggregating preferences (Arrow's Impossibility Theorem), while public choice theorists like James Buchanan explored the failures of majority voting.

QV does not claim to solve Arrow's theorem. Instead, it operates in a different framework: rather than ranking candidates, voters allocate continuous resources across issues. This sidesteps the ordinal constraints that generate Arrow's impossibility result.

How It Works

The Mathematics

Each voter receives a budget of voice credits. To cast n votes on a single issue, a voter must spend n² credits:

Votes Cast	Credits Spent	Marginal Cost	Cumulative % of 100-Credit Budget
1	1	1	1%
2	4	3	4%
3	9	5	9%
4	16	7	16%
5	25	9	25%
6	36	11	36%
7	49	13	49%
8	64	15	64%
9	81	17	81%
10	100	19	100%

This quadratic cost function ensures that expressing strong preferences is possible but increasingly expensive, forcing voters to allocate their influence across issues according to genuine priority.

Formally, the cost function is:

<math>C(v) = v^2</math>

where v is the number of votes cast. The marginal cost of the nth vote is 2n - 1, meaning each additional vote costs more than the last. A voter with budget B maximizes utility by distributing votes across issues such that the marginal value of influence equals the marginal cost on every issue.

Why Quadratic?

The quadratic function is not arbitrary. It is the unique cost function that, under standard economic assumptions, produces outcomes that maximize aggregate welfare. Linear costs would allow wealthy voters to dominate; exponential costs would suppress all intensity signals. The quadratic sits precisely between these extremes.

The intuition can be understood through two boundary cases:

Cost Function	Behavior	Problem
Linear (cost = votes)	Voters buy as many votes as they can afford	Plutocracy; wealth determines outcomes
Exponential (cost = 2votes)	Almost no one casts more than one vote	Collapses back to 1p1v; intensity lost
Quadratic (cost = votes2)	Moderate intensity expression at increasing cost	Optimal welfare under standard assumptions

Weyl proved that the quadratic is the unique function where, as the number of voters grows large, the outcome converges to the utilitarian social welfare maximum. This result holds under independent private values and is robust to a range of behavioral assumptions.

Credit Distribution

A critical design choice is how voice credits are distributed. Three primary approaches exist:

1. Equal periodic allocation — Every participant receives the same credit budget per voting cycle. This is the most egalitarian approach and the one most commonly proposed.
2. Earned credits — Credits are earned through participation, contribution, or other prosocial activity. This rewards engagement but risks creating new inequalities.
3. Purchased credits — Credits are bought with money. This is theoretically elegant (willingness to pay reflects intensity) but reintroduces wealth effects.

OMXUS rejects purchased credits entirely, using proximity-based weighting instead of credit-based intensity expression.

Glen Weyl and RadicalxChange

Glen Weyl founded the RadicalxChange Foundation in 2018 to advance the ideas from Radical Markets into practical implementation. The foundation operates as a global community of activists, academics, artists, and technologists working on mechanisms for a more equitable society.

RadicalxChange has promoted several related mechanisms alongside QV:

• Quadratic Funding (QF) — A mechanism for public goods funding where individual contributions are matched by a central fund proportional to the square root of each contribution. Used extensively by Gitcoin for Ethereum ecosystem grants.
• Common Ownership Self-Assessed Tax (COST) — A property tax regime where owners self-assess asset values, pay tax on those values, and must sell to anyone willing to pay the declared price.
• Data Dignity — Frameworks for compensating individuals for the data they produce.

Advantages Over Traditional Voting

One-Person-One-Vote (1p1v)

Traditional voting treats a voter who is deeply affected by a policy identically to one who is indifferent. QV corrects this by allowing intensity expression while maintaining democratic equality through equal credit budgets.

Consider a community voting on whether to build a factory. Under 1p1v, 51 mildly supportive voters defeat 49 deeply opposed residents who will live next to the factory. Under QV, the 49 can express their intense opposition by spending more credits, potentially changing the outcome to reflect actual aggregate welfare.

Plurality and Ranked Choice

These systems capture preference ordering but not magnitude. A voter who slightly prefers option A over B is treated the same as one for whom the distinction is existential. Ranked-choice voting (RCV) improves on plurality by eliminating spoiler effects, but it still cannot distinguish between mild and intense preferences.

Feature	Plurality	Ranked Choice	Quadratic Voting
Captures preference direction	Yes	Yes	Yes
Captures preference ordering	No	Yes	Yes (implicitly)
Captures preference intensity	No	No	Yes
Handles multiple issues simultaneously	No	No	Yes
Resistant to strategic voting	Low	Medium	High
Welfare-maximizing (theoretical)	No	No	Yes (asymptotically)

Lobbying and Log-Rolling

In representative systems, preference intensity is expressed through lobbying, campaign donations, and vote trading (log-rolling). These are informal, opaque, and heavily biased toward the wealthy. QV provides a transparent, equitable channel for the same information. It makes explicit what lobbying does implicitly — and does so without corruption.

The Colorado Experiment

In April 2019, the Colorado House Democratic Caucus became the first legislative body in the world to use quadratic voting for real governance decisions. Members used QV to prioritize legislative bills for the upcoming session.

Design

Each of the 41 Democratic representatives received 100 tokens to distribute across 107 bills. They could place multiple tokens on bills they cared most about, with quadratic costs applying. The process replaced the traditional system where party leadership unilaterally decided priorities.

Results

• Representatives spread their tokens across an average of 12 bills each
• The results surfaced consensus priorities that differed from what leadership had assumed
• Several bills that would have been deprioritized under the old system received strong QV support and were ultimately passed
• Participants reported that the process felt more fair and representative than the traditional approach

Lessons

The experiment demonstrated that QV is practically implementable in real political contexts and that it surfaces information invisible to traditional methods. It also revealed challenges: some participants found the system confusing at first, and a few attempted strategic coordination (though the quadratic costs limited its effectiveness).

Limitations and Critiques

Collusion

Groups can coordinate to concentrate votes, undermining the mechanism. If 10 people each cast 1 vote (cost: 10 credits total), they achieve the same influence as 1 person casting 10 votes (cost: 100 credits). This makes coordinated groups more cost-effective. Weyl proposes various anti-collusion strategies, including correlation discounting and identity verification.

Complexity

QV requires voters to understand credit allocation, which is less intuitive than simple majority voting. User interface design becomes critical. Experiments show that comprehension improves rapidly with experience, but first-time users often underperform relative to their true preferences.

Wealth Effects

If credits are purchased rather than equally distributed, QV can amplify existing inequality. This is the strongest critique of market-based QV implementations. Equal distribution of credits eliminates this concern but introduces questions about the optimal budget size and refresh frequency.

Cultural Assumptions

QV assumes voters can and will accurately assess and express their preference intensities numerically. This assumption may not hold uniformly across cultures, education levels, or cognitive styles. Some critics argue that QV advantages analytically-minded voters.

Connection to OMXUS

OMXUS does not implement pure quadratic voting but draws on the same foundational insight: not all votes should count equally on all issues. Rather than using purchased credits to express intensity, OMXUS uses proximity weighting to adjust vote influence based on:

• Geographic proximity — Closer to the affected area means greater weight
• Social proximity — Connected to affected people via the web of trust
• Domain proximity — Expertise or direct experience with the issue
• Temporal proximity — More recent experience with the issue carries more weight

This approach preserves the quadratic voting insight (intensity matters) while eliminating the wealth-dependence problem. Proximity is earned through lived experience, not purchased.

Dimension	Quadratic Voting	OMXUS Proximity Weighting
Intensity signal	Voice credits spent	Proximity to affected area/people
Source of weight	Budget allocation (potentially purchased)	Lived experience and relationship
Collusion resistance	Correlation discounting	Sybil-resistant identity via Web of Trust
Welfare maximization	Asymptotically optimal	Targets affected-party welfare
Complexity for voter	Must calculate credit allocation	Automatic based on verified relationships

The OMXUS approach can be understood as a revealed preference version of QV: rather than asking people to declare how much they care (via credit spending), the system observes how much they are affected (via proximity). This eliminates both the wealth problem and the cognitive complexity problem.

Real-World Applications

Application	Year	Context	Outcome
Colorado House Democrats	2019	Bill prioritization	Surfaced hidden consensus; several reprioritized bills passed
RadicalxChange Foundation	2018-present	Community governance experiments	Ongoing refinement of QV protocols
Gitcoin Grants	2019-present	Quadratic funding for open-source software	Over $50M distributed via QF mechanism
Taiwan (g0v / PDIS)	2016-present	Digital democracy experiments	QV principles incorporated into pol.is platform
Denver Democratic Party	2020	Platform prioritization	Used QV to rank policy positions
European Commission	2021	Internal pilot for priority setting	Tested QV for cross-departmental resource allocation

Quadratic Funding

Quadratic Funding (QF) is a closely related mechanism designed by Buterin, Hitzig, and Weyl for funding public goods. In QF, a matching pool amplifies small donations quadratically:

<math>\text{Funding}_p = \left(\sum_i \sqrt{c_{i,p}}\right)^2</math>

where c_i,p is individual i's contribution to project p. This means a project with many small donors receives proportionally more matching funds than one with a few large donors. The mechanism optimally funds public goods under the same assumptions that make QV welfare-optimal for voting.

Gitcoin has deployed QF at scale in the Ethereum ecosystem, distributing over $50 million to public goods projects through multiple rounds since 2019. This represents the largest real-world deployment of any mechanism from the Radical Markets family.

Philosophical Implications

QV raises deep questions about the nature of democracy:

• Is equal voice a right or a resource? — 1p1v treats voice as a right; QV treats it as a resource to be allocated. Both respect equality (equal budgets), but the unit of equality differs.
• Should intensity matter? — Classical liberalism protects minorities through rights; QV protects them through intensity expression. These are complementary but distinct strategies.
• What is the purpose of voting? — If voting is about preference aggregation (finding what society wants), QV is superior. If voting is about equal dignity (affirming each person's standing), 1p1v may be more symbolically appropriate.

OMXUS resolves this tension by using proximity weighting, which preserves equal dignity (everyone has a token, everyone can vote) while incorporating intensity information (those more affected have more weight). See Principles and Freedom-Preserving Governance for more on how OMXUS navigates these tradeoffs.

See Also

• Voting
• Direct Democracy
• Principles
• Freedom-Preserving Governance
• Web of Trust
• Sybil Resistance
• Two Monkey Theory
• Main Page

References

• Posner, E. A., & Weyl, E. G. (2018). Radical Markets: Uprooting Capitalism and Democracy for a Just Society. Princeton University Press.
• Weyl, E. G. (2017). "The Robustness of Quadratic Voting." Public Choice, 172(1-2), 75-107.
• Lalley, S. P., & Weyl, E. G. (2018). "Quadratic Voting: How Mechanism Design Can Radicalize Democracy." AEA Papers and Proceedings, 108, 33-37.
• Quarfoot, D., et al. (2017). "Quadratic Voting in the Wild." Public Choice, 172(1-2), 283-303.
• Buterin, V., Hitzig, Z., & Weyl, E. G. (2019). "A Flexible Design for Funding Public Goods." Management Science, 65(11), 5171-5187.
• Arrow, K. J. (1951). Social Choice and Individual Values. Yale University Press.
• Posner, E. A., & Weyl, E. G. (2015). "Voting Squared: Quadratic Voting in Democratic Politics." Vanderbilt Law Review, 68(2), 441-499.