Quadratic Voting

Quadratic Voting is a governance mechanism where participants allocate a budget of voice credits to express their preferences across multiple proposals. The key innovation is that the cost of casting votes for a single option increases quadratically with the number of votes cast. For example, 1 vote costs 1 credit, 2 votes cost 4 credits (2²), 3 votes cost 9 credits (3²), and so on. This pricing structure makes it prohibitively expensive for any single participant to dominate an outcome, as the cost of marginal influence rises sharply. It is mathematically designed to approximate the optimal allocation of a public good by aligning individual expenditure with the social welfare created.

The mechanism's core principle is to measure the intensity of preference, not just binary support or opposition. A voter who feels strongly about a proposal can spend a large portion of their credit budget on it, but doing so significantly reduces their ability to influence other decisions. This creates a more nuanced and efficient aggregation of preferences compared to simple one-person-one-vote systems. QV is particularly suited for on-chain governance in decentralized autonomous organizations (DAOs) and funding mechanisms like quadratic funding, where it helps allocate communal resources to projects based on the breadth of community support rather than the depth of funding from a few large contributors.

In practice, implementing QV requires careful design to prevent Sybil attacks, where a user creates multiple identities to gain more voice credits. Common mitigations include identity verification (e.g., proof-of-personhood systems) or correlating voting power with a scarce resource like token holdings, though the latter can dilute the egalitarian ideal. Pioneered by economist Glen Weyl and legal scholar Eric Posner, QV represents a significant advancement in mechanism design, aiming to create fairer and more efficient outcomes in democratic and market-mimicking processes within blockchain ecosystems and beyond.

Quadratic Voting (QV) is a collective decision-making mechanism designed to capture the intensity of voter preferences by allowing individuals to allocate a budget of voice credits across multiple proposals. Unlike one-person-one-vote systems, QV uses a cost function where the cost to cast multiple votes for a single option increases quadratically. For example, casting one vote costs 1 credit, two votes cost 4 credits (2²), three votes cost 9 credits (3²), and so on. This pricing structure makes it exponentially more expensive to concentrate all influence on a single choice, encouraging voters to distribute their limited budget to signal their strongest preferences more accurately.

The core innovation is the quadratic cost curve, which creates a mathematical link between a voter's expressed preference strength and the economic cost of expressing it. This mechanism is theorized to maximize overall social welfare by aggregating preferences in a way that one-person-one-vote or simple token-weighted voting cannot. It effectively mitigates the tyranny of the majority by giving minority groups with strong preferences a financially efficient way to outvote an apathetic majority. The system requires a fungible budget of voice credits distributed equally to participants, which are spent and not staked, ensuring each participant starts with equal potential influence.

In practice, implementing QV involves several key steps: issuing credits, conducting the vote where participants pay the quadratic cost for their chosen vote allocations, and tallying results. A common formula for the cost is cost = (number of votes)². This has been applied in contexts ranging from community treasury grants (like Gitcoin Grants) to corporate decision-making and political theory experiments. The primary computational challenge is preventing Sybil attacks, where an attacker creates many identities to gain more credits; this is typically addressed through robust identity verification or proof-of-personhood systems.

Compared to other systems, QV offers distinct trade-offs. It is more expressive than plurality voting and more resistant to wealth-based dominance than token-weighted voting (often called "plutocracy"). However, it is more complex to implement and explain than simpler models. Its effectiveness relies heavily on voters understanding the cost mechanism to strategize effectively. When functioning ideally, QV produces outcomes that reflect not just how many people support an option, but how strongly they support it relative to other choices, leading to more nuanced and legitimate collective decisions.

The core mathematical formula governing Quadratic Voting is Cost = (Number of Votes)². This means that to cast n votes for a single proposal, a voter must spend n² voice credits from their budget. For example, casting 1 vote costs 1 credit, 2 votes cost 4 credits, 3 votes cost 9 credits, and so on. This quadratic cost function is the defining characteristic that makes QV fundamentally different from simple one-person-one-vote systems, as it imposes a rapidly escalating price for expressing a strong preference, forcing voters to make trade-offs.

The mechanism's power lies in its ability to efficiently aggregate preferences. Theoretically, under certain assumptions, QV leads to decisions that maximize the sum of the utilities of the participants. This is because the marginal cost of an additional vote equals the number of votes already cast, which, in equilibrium, should reflect the voter's marginal utility for that outcome. In practice, each voter solves an optimization problem: given their fixed budget of credits (e.g., 100), they allocate votes across various proposals to maximize their total expressed utility, subject to the quadratic cost constraint.

A critical implementation detail is the use of a square root function for tallying. The vote tally for a proposal is not the sum of raw votes, but the sum of the square roots of each voter's allocated votes: Tally = Σ √(Votes_i). This mathematical transformation is the inverse of the cost function and ensures that a single voter casting 9 votes (costing 81 credits) contributes only 3 units (√9) to the final tally, while nine different voters each casting 1 vote (each costing 1 credit) contribute 9 units (9 * √1) collectively. This prevents wealthy or highly motivated individuals from dominating the outcome.

The quadratic formula creates a natural market for preference intensity. It allows a voter with a moderate preference on many issues to spread their influence, while a voter with a very strong preference on a single issue can concentrate their budget there, albeit at a high marginal cost. This system surfaces not just what people want, but how much they want it relative to other choices, leading to more nuanced and socially optimal outcomes than binary voting in funding public goods, protocol parameter settings, or grant allocations.

The term Quadratic Voting originates from the mathematical relationship at its core: the cost of acquiring votes increases with the square of the number of votes cast. This mechanism was formally proposed by economists Steven Lalley and E. Glen Weyl in their 2017 paper, "Quadratic Voting and the Public Good." It evolved from earlier concepts like quadratic funding and the Vickrey–Clarke–Groves (VCG) mechanism, aiming to capture the intensity of individual preferences in collective decision-making more efficiently than one-person-one-vote systems.

The theory posits that by making additional votes progressively more expensive, Quadratic Voting forces participants to economically express how strongly they care about an outcome. This creates a market for votes where marginal cost equals marginal benefit, theoretically leading to decisions that maximize aggregate welfare. Its initial applications were proposed for corporate governance and public policy referenda, where discerning the strength of minority opinions is crucial for optimal outcomes.

The migration of Quadratic Voting to blockchain and decentralized autonomous organizations (DAOs) began around 2018-2019, pioneered by projects like Gitcoin Grants. The blockchain's transparent ledger and native token systems provided an ideal, tamper-proof environment to implement the complex payment and voting logic. Here, it is often used for retroactive public goods funding, allowing communities to allocate resources in a way that magnifies the preferences of a broad contributor base rather than a few large whales.

A key historical implementation is the Gitcoin Grants matching rounds, which use a variant called Quadratic Funding. In this model, a matching pool is distributed to projects proportionally to the square of the sum of the square roots of individual contributions. This elegantly demonstrates the core principle: many small donations signal stronger community support than a single large one, and the funding algorithm reflects this non-linearly.

The evolution continues with adaptations like Conviction Voting and Quadratic Delegation, which modify the basic formula for continuous funding or delegated representation. As a governance primitive, Quadratic Voting's history is a case study in translating dense economic theory into a practical, algorithmic tool for decentralized collectives, fundamentally altering how on-chain communities express preference intensity and allocate shared capital.