Quadratic voting
Quadratic voting (QV) is a collective decision-making procedure in which participants receive a fixed endowment of credits to acquire votes on binary propositions, with the cost of the nth vote for a given option equal to n credits, yielding a total quadratic expenditure that incentivizes expressing the relative intensity of preferences across issues.[1][2] Originally proposed by economist E. Glen Weyl in a 2012 working paper titled "Quadratic Vote Buying," the mechanism draws on mechanism design principles to address limitations of traditional one-person-one-vote systems, which fail to capture how strongly individuals care about outcomes.[2][3] Theoretical analysis by Weyl and mathematician Steven P. Lalley demonstrates that, under standard assumptions including large electorates and convex vote costs, QV converges to efficient aggregation of preferences in Bayesian Nash equilibrium, revealing truthful valuations without requiring interpersonal utility comparisons.[2][3] Eric Posner and Weyl further elaborated QV's potential for democratic politics, corporate governance, and public goods provision in their 2014 paper "Voting Squared" and 2018 book Radical Markets, arguing it mitigates the tyranny of the majority by balancing voice with stake.[1][4] Empirical evaluations, including experiments with quadratic voting for survey research (QVSR), show it outperforms Likert-scale methods in eliciting preferences that better predict policy support intensities, though strategic behaviors and budget constraints can introduce deviations from ideal efficiency in finite settings.[5][6] While QV has seen niche applications in decentralized autonomous organizations and experimental platforms, its scalability faces challenges from collusion risks and the need for verifiable credit distribution, limiting adoption beyond proofs-of-concept.[4][7]Core Mechanism
Definition and Basic Operation
Quadratic voting is a decision-making procedure designed to elicit the intensity of participants' preferences over multiple options by imposing a quadratic cost on the number of votes purchased. Participants receive an endowment of voice credits, which serve as currency to buy votes on propositions, typically binary yes/no issues. The cost to acquire the k-th vote on any issue is k credits, resulting in a total expenditure of approximately k2/2 credits for k votes, though implementations may normalize this to k2 for simplicity.[8][9] This quadratic pricing structure increases the marginal cost of additional votes, incentivizing voters to concentrate credits on issues where their preferences are strongest while limiting influence on many weakly held views. Votes are tallied linearly across all participants, with the aggregate determining outcomes such as approval or ranking. For instance, in a system with N propositions and K credits per voter, strategic allocation reveals relative utilities, as the convex cost function aligns individual spending with preference gradients under equilibrium assumptions.[3] Basic operation often involves one-time or periodic credit distributions, with credits non-transferable between voters to prevent collusion, though variants allow refunds or multi-round adjustments. Propositions can span policy choices, resource allocation, or prioritization, where the mechanism contrasts with one-person-one-vote systems by amplifying minority intensities without unbounded voice. Empirical pilots, such as those in corporate governance or community forums, demonstrate feasibility, with software handling credit deductions and vote aggregation in real-time.[10][11]Mathematical Formulation and Variants
In the canonical formulation of quadratic voting (QV) for a binary collective decision, as developed by Lalley and Weyl, a population of N voters each endowed with one unit of voice credits faces a choice between two alternatives, say A or B. Each voter i privately values alternative A over B by v_i, drawn independently from a symmetric distribution around zero. Voter i purchases q_i \geq 0 votes in favor of their preferred alternative, incurring a cost of q_i^2 credits paid to a central clearinghouse. The alternative receiving the greater total votes (\sum q_j) is selected; ties are resolved randomly.[3][12] Under the assumption of quasilinear utility (u_i = v_i \cdot \mathbf{1}_{\{A \text{ wins}\}} - q_i^2), the game possesses a unique symmetric Bayes-Nash equilibrium in large electorates. Voters act as price-takers, purchasing votes up to the point where the marginal cost equals the expected marginal benefit, yielding total votes approximately equal to the expected absolute aggregate value \mathbb{E}\left[\sum |v_j|\right]. This price converges to the competitive equilibrium price \lambda = \mathbb{E}[|v|] as N \to \infty, rendering the mechanism asymptotically efficient and strategyproof in expectation, as deviations yield negligible pivotality.[12][8] Variants extend this framework. In multi-issue QV, voters allocate a fixed budget across M independent binary decisions, purchasing q_{i,m} votes for issue m at cost q_{i,m}^2, with \sum_m q_{i,m}^2 \leq 1. Equilibrium strategies reveal relative intensities, with credits flowing to issues of highest private value density, preserving approximate efficiency under separability.[7] Fixed-budget QV imposes a global expenditure cap per voter while allowing variable vote supply, contrasting with the endowment-based clearinghouse model; Posner and Weyl analyze its application to corporate governance, where quadratic costs mitigate logrolling incentives.[4] Multiple-alternatives QV generalizes to K > 2 options via pairwise or bundled bidding, where voters buy votes against a status quo or across contests, maintaining quadratic marginal costs to approximate utilitarian welfare; Eguia et al. prove convergence to efficient randomized social choice rules in large populations.[13]Theoretical Foundations
Claims of Efficiency
Proponents claim that quadratic voting achieves greater efficiency in aggregating preferences by enabling voters to express the intensity of their utilities through differentially priced votes, where the cost of each additional vote increases quadratically. This structure discourages over-voting on low-intensity issues and concentrates influence on matters of high personal stakes, theoretically leading to outcomes that more closely maximize aggregate social welfare than linear voting systems, which ignore preference strengths.[14] In theoretical models, the quadratic cost function aligns individual incentives with truthful revelation of valuations, as voters optimally allocate limited credits to proposals proportional to the square root of their utility differences, resulting in collective signals that approximate efficient resource allocation. For instance, Posner and Weyl argue that in corporate governance contexts, this mechanism outperforms traditional share-weighted voting by allowing dispersed shareholders with intense preferences—such as on executive compensation or mergers—to override apathetic majorities, thereby enhancing firm value maximization. Empirical simulations and equilibrium analyses support that such efficiency holds under assumptions of rational, budget-constrained agents without collusion.[4][2] Further claims posit that quadratic voting's efficiency extends to public decision-making, where it mitigates the inefficiencies of majority rule in handling heterogeneous intensities, such as protecting minority interests with high stakes. Lalley and Weyl's analysis demonstrates that in large populations, the mechanism's Nash equilibrium converges rapidly to the socially optimal decision, leveraging asymptotic properties akin to market clearing for public goods. This convergence relies on voters' strategic behavior, which, unlike in linear systems, can enhance rather than undermine efficiency by amplifying signals from informed participants. However, these efficiency claims presuppose sufficient credits, no externalities in vote purchases, and voters' ability to accurately assess their utilities, conditions that may not hold in practice.[15][16]Assertions of Robustness
Proponents of quadratic voting assert that the mechanism exhibits approximate strategy-proofness in large electorates, where voters' optimal strategy converges to revealing their true preference intensities. In a model where voters' utilities are independently drawn from a known distribution and the number of participants approaches infinity, Lalley and Weyl demonstrate that the unique symmetric Bayes-Nash equilibrium involves each voter purchasing a number of votes equal to the absolute value of their private utility for the binary decision, rendering deviations from truth-telling unprofitable in expectation.[17] This equilibrium arises because the quadratic pricing—where the cost of the v-th vote equals v credits—approximates a competitive market equilibrium, aligning individual incentives with efficient aggregation of intensities without requiring dominant-strategy incentives.[12] The system is further claimed to be robust to collusion by small subgroups, as the quadratic cost structure limits the amplifying effect of coordinated behavior. Spencer analyzes an equilibrium model of QV with colluding minorities, using heuristic approximations to quantify that a subgroup comprising a fraction \epsilon of the electorate can at best distort the outcome by an amount on the order of \sqrt{\epsilon}, rather than linearly in \epsilon as in one-person-one-vote systems. For instance, in electorates of size n exceeding 10,000, collusions involving fewer than \sqrt{n} participants yield negligible influence, as pooling fixed per-person voice credits incurs rapidly escalating marginal costs that deter over-investment beyond the group's proportionate stake. This property stems from the convexity of the cost function, which penalizes disproportionate vote concentration more severely than linear mechanisms. Additional robustness assertions highlight resistance to certain forms of vote-buying or external manipulation, provided voice credits are non-transferable and tied to verified identities. Unlike cash-based vote trading, the internal currency of credits prevents monetary corruption from directly scaling influence, as each participant's budget remains capped and quadratically constrained.[8] However, these claims hold under idealized assumptions of independent utilities and no large-scale coordination; vulnerabilities emerge if a majority colludes or if sybil attacks inflate participant counts, though proponents argue real-world identity verification mitigates the latter. Empirical analogs in controlled experiments, such as those comparing QV to plurality voting, support reduced strategic abstention, with participants expressing intensities more consistently across issues.[18]Equilibrium Analysis
In quadratic voting mechanisms for binary decisions, equilibrium analysis models voter behavior as a Bayesian game where each of N agents possesses a private valuation u_i drawn independently from a smooth distribution F with bounded support and positive density, representing the utility from the preferred outcome. Voters allocate a budget to purchase v_i votes at quadratic cost v_i^2, aiming to maximize expected utility \Psi(S + v_i) - v_i^2, where S aggregates opponents' net votes and \Psi is a smooth, increasing function capturing outcome probabilities.[12] Pure-strategy Nash equilibria exist and are monotone increasing in valuations for N > 1, under standard conditions including compactly supported \Psi with derivative bounded away from extremes. In symmetric equilibria, moderate voters purchase votes approximately linearly in their valuations, scaled by a factor converging to zero as $1/\sqrt{N}, reflecting strategic attenuation due to pivotal probabilities shrinking with electorate size.[12] For balanced electorates where the mean valuation \mu = 0, equilibria are continuous, and strategies approximate myopic revelation adjusted for aggregate uncertainty, yielding asymptotic efficiency: expected inefficiency, measured as welfare loss relative to full information optima, vanishes as N \to \infty for any bounded distribution F. When \mu \neq 0, equilibria exhibit discontinuities in the tails, where extremists purchase disproportionately many votes—scaling linearly with N—to insure against moderate opponents, yet efficiency still holds asymptotically, with inefficiency decaying as O(1/N).[12] Multiple equilibria can arise from self-fulfilling discontinuities, particularly in unbalanced settings, allowing inefficient outcomes like uniform abstention or coordination on suboptimal sides if voters share correlated beliefs; however, symmetric equilibria are conjectured unique except at isolated points, and robustness analyses indicate low vulnerability to collusion or fraud in large populations, with empirical inefficiencies rarely exceeding 10%. These properties stem from quadratic costs internalizing externalities akin to market pricing, though deviations from exact truth-telling persist due to interdependent pivotalities.[12]Historical Development
Precursors and Early Ideas
The concept of quadratic voting emerged from broader traditions in mechanism design theory, which seeks to create incentive-compatible rules for eliciting truthful preferences in collective decisions. Foundational work includes William Vickrey's 1961 proposal for second-price auctions, where bidders reveal true valuations by paying the second-highest bid if they win, establishing principles of truthful revelation without strategic misrepresentation. This approach influenced subsequent developments in social choice, emphasizing costs or payments to align individual incentives with efficient outcomes.[19] Further precursors lie in the Vickrey-Clarke-Groves (VCG) mechanism, articulated by Edward Clarke in 1971 and Theodore Groves in 1973, which generalizes Vickrey's ideas to public goods provision. In VCG, agents report valuations, and payments equal the externality imposed on others, often resulting in quadratic-like terms when utilities exhibit diminishing marginal returns or in multi-agent settings. These mechanisms aimed to achieve Pareto efficiency by making truth-telling a dominant strategy, though they faced practical challenges like high informational demands and budget imbalances—issues quadratic voting later addresses through symmetric, credit-based quadratic costs. Early mechanism design thus provided the theoretical scaffolding for incorporating preference intensities via convex pricing, contrasting with traditional voting's equal-weight aggregation that disregards varying stakes.[20] The immediate early ideas for quadratic voting proper trace to E. Glen Weyl's working paper circulated in February 2012, initially titled "Quadratic Vote Buying." This explored voters purchasing additional votes at a quadratic cost in voice credits, arguing it incentivizes proportional expression of intensity while preventing dominance by intense minorities through escalating marginal costs.[8] Weyl demonstrated equilibrium existence and efficiency properties for convex (including quadratic) cost functions, building directly on mechanism design to mitigate issues like the tyranny of the minority in one-person-one-vote systems. This formulation predated broader dissemination, serving as the conceptual seed later refined for democratic and governance applications.[2]Formal Introduction and Key Publications
Quadratic voting was formally introduced by economist E. Glen Weyl in a working paper circulated in February 2012, initially titled "Quadratic Vote Buying," which proposed the mechanism as a method for voters to purchase additional votes at a quadratic cost to better reflect preference intensities in collective decision-making.[2] The paper formalized the core idea using game-theoretic equilibrium analysis, demonstrating that under quadratic pricing, voters reveal truthful marginal valuations, leading to efficient aggregation akin to market outcomes.[2] Weyl's framework addressed limitations of one-person-one-vote systems by allowing differential vote expenditures proportional to squared quantities, with credits distributed equally or based on stakes, and emphasized applications beyond politics, such as corporate governance.[4] Subsequent key publications expanded and refined the concept for democratic contexts. In 2015, Eric A. Posner and Weyl published "Voting Squared: Quadratic Voting in Democratic Politics" in the Vanderbilt Law Review, adapting the mechanism for legislative and electoral use, where voters receive voice credits to buy votes on bills or candidates, with costs rising quadratically to prevent majority dominance over intense minorities.[21] This work argued for quadratic voting's superiority in balancing fairness and efficiency, supported by simulations showing reduced strategic abstention compared to linear voting.[21] Posner and Nicholas O. Stephanopoulos further developed electoral applications in their 2016 paper "Quadratic Election Law," proposing district-level implementations to mitigate gerrymandering and enhance representation of varied intensities.[22] Theoretical advancements continued with Steven P. Lalley and Weyl's 2018 contribution, "Quadratic Voting: How Mechanism Design Can Radicalize Democracy," presented at the American Economic Association, which provided mechanism design proofs for incentive compatibility under incomplete information and large electorates, confirming convergence to efficient social welfare maximization.[3] These publications collectively established quadratic voting's foundations, influencing subsequent empirical trials and variants, though early works like Weyl's focused more on abstract efficiency than practical robustness critiques.[23]Theoretical Refinements
Following the initial formulation of quadratic voting by Posner and Weyl in 2015, subsequent theoretical work focused on establishing the existence and properties of equilibria. In a 2019 analysis, Lalley and Weyl demonstrated that quadratic voting admits approximate Bayesian Nash equilibria where voters reveal their true intensities truthfully in expectation, particularly as the electorate size grows large; this result relies on voters having quasilinear preferences and facing proposals with binary outcomes, with strategic deviations becoming negligible due to the quadratic cost structure aggregating signals efficiently.[24] This refinement addressed early concerns about manipulability by showing convergence to the utilitarian optimum under mild conditions, though it assumes common priors on proposal values.[24] Weyl further refined the theory in 2017 by examining robustness to deviations from ideal assumptions, such as partial collusion among voters or non-quasilinear utilities. Using heuristic approximations, he proved that quadratic voting maintains near-optimal efficiency even when up to a constant fraction of voters collude or when preferences exhibit moderate risk aversion, as the mechanism's signal aggregation dampens strategic noise more effectively than linear voting.[25] These results hold for electorates exceeding hundreds of participants, with welfare losses bounded by factors independent of scale, contrasting with one-person-one-vote systems that amplify minority suppression under similar perturbations. Extensions to multi-issue and multi-alternative settings emerged around 2019, with Lalley and Weyl analyzing quadratic voting over multiple binary proposals under fixed budgets. They established that truthful equilibria persist when issues are independent, but interdependence introduces approximation guarantees rather than exact truthfulness, with efficiency approaching the optimum as budgets allow flexible allocation across issues.[13] Posner and Stephanopoulos complemented this in 2017 by formalizing fixed-budget variants, proving incentive compatibility for discrete vote allocations while preserving intensity expression, though at the cost of minor distortions in large-budget limits.[26] These developments highlighted trade-offs, such as reduced robustness to correlated preferences in multi-dimensional spaces, prompting further equilibrium refinements under asymmetric information.[27]Practical Implementations
Political and Governmental Trials
One of the earliest governmental trials of quadratic voting occurred in the Democratic caucus of the Colorado House of Representatives in April 2019, involving 41 members who each received 100 voice credits to allocate across approximately 60 to 100 appropriations bills.[28] Participants purchased votes quadratically, with costs escalating as the square of votes cast for any option, in an anonymous, non-binding poll to signal funding priorities amid a $40 million allocation for over $120 million in requests.[29] The process highlighted strong support for initiatives like Senate Bill 85 on equal pay, which garnered 60 votes, while producing a "long tail" of lower-priority items, enabling clearer prioritization than traditional methods.[28] Subsequent iterations expanded the trial's scope in Colorado. In June 2020, executive branch interagency groups used quadratic voting to rank goals, contributing to the creation of a new Behavioral Health Administration, though some outcomes were not adopted by the governor's office.[30] From 2021 to 2023, both Democratic and Republican caucuses in the state House and Senate employed the mechanism—often via RadicalxChange tools or spreadsheets—for legislative polls on over 80 bills and appropriations, describing it as a tool for nuanced preference revelation in resource-constrained settings.[30] In Spring 2023, quadratic voting was applied to participatory budgeting in New York City's Harlem District 9 by former Council Member Kristin Richardson Jordan, using an online platform with voter-roll verification and QR code authentication for residents.[31] Participants allocated credits across spending proposals, with guides and videos aiding engagement; the top outcome funded "The Baxter" affordable housing project with $1 million after receiving 136 votes, alongside an 80% satisfaction rate among voters, though 18% did not exhaust their budgets.[31] Nashville implemented quadratic voting for its 2023 county budget process through an online tool, as part of Mayor John Cooper's participatory budgeting pilot, which had previously engaged 500 residents in generating 400 project ideas for potential $20 million expansion.[32] Supported by Metro Council member Burkley Allen and enabled by state legislation from Governor Bill Lee recognizing decentralized decision-making, the trial aimed to prioritize broadly supported needs using voice credits.[32] Jersey City, under Mayor Steven Fulop, incorporated similar mechanisms into participatory budgeting for community projects like playgrounds and arts funding, allocating $900,000 across 89 recipients in 2020, though specifics on quadratic cost structures were integrated into broader engagement efforts.[32] These trials have remained small-scale and advisory, focusing on budget prioritization rather than binding decisions, with implementations often relying on third-party software amid concerns over software reliability in early adopters like Colorado's Senate Democrats.[29]Corporate and Private Sector Uses
In corporate governance, quadratic voting has been proposed as a mechanism to enhance shareholder decision-making by allowing investors to allocate votes quadratically based on the intensity of their preferences, addressing limitations of traditional one-share-one-vote systems where dispersed ownership leads to apathy and managerial opportunism. Eric Posner and E. Glen Weyl argued in a 2014 paper that quadratic voting could achieve efficient outcomes in shareholder votes on issues like mergers or executive compensation by enabling minority shareholders with strong views to amplify their influence without requiring proportional ownership, thus reducing agency costs identified since Berle and Means's 1932 analysis.[4][33] Theoretical models suggest quadratic voting promotes collective efficiency in corporate decisions, as shareholders rationally allocate limited voice credits to maximize utility, converging on outcomes aligned with the median intensity-weighted preference under large electorates. A 2024 study comparing quadratic voting to majority voting in shareholder contexts found that, assuming collective rationality, both systems yield efficient firm decisions, but quadratic voting better captures preference intensities, potentially improving resolutions on complex proposals like bylaw amendments.[34] However, implementation remains largely theoretical, with no widespread adoption in public companies as of 2025, due to regulatory hurdles under securities laws and challenges in verifying vote credits without collusion risks. In private sector applications beyond public firms, quadratic voting has been explored for internal decision processes, such as product prioritization or resource allocation in tech firms and investment portfolios. For instance, analyses have applied it to portfolio planning, where stakeholders use quadratic credits to signal strong convictions on asset selections, theoretically outperforming linear voting by weighting passion over mere headcount. Blockchain-based private organizations, including decentralized autonomous organizations (DAOs) in venture funding, have piloted quadratic voting for governance proposals, enabling token holders to express vote intensities on protocol upgrades, though empirical scalability issues persist.[35] These uses highlight quadratic voting's potential to democratize private decision-making while preserving efficiency, but real-world deployments are limited to experimental or niche settings, often integrated with cryptographic tools for anonymity and refund mechanisms.Digital and Decentralized Applications
Quadratic voting has been implemented in digital platforms to facilitate preference aggregation in online communities and organizations, often through web-based interfaces that allow users to allocate voice credits via quadratic costs. For instance, software tools developed by organizations like RadicalxChange enable experimental quadratic voting in digital settings, where participants purchase votes using predefined credits, with costs scaling quadratically to reflect intensity without requiring physical assembly.[11] These digital applications extend to non-blockchain environments, such as corporate decision-making tools, but gain prominence in decentralized contexts for their compatibility with pseudonymous participation. In decentralized autonomous organizations (DAOs) on blockchain networks, quadratic voting addresses governance challenges posed by token concentration, where large holders (whales) could dominate linear voting systems. By treating token stakes or allocated credits as quadratic budgets, it empowers minority voices and reduces plutocratic tendencies, as the marginal cost of additional votes rises quadratically.[36] This mechanism has been proposed and partially adopted in proof-of-stake (PoS) blockchains, where validators' stakes serve as vote budgets, promoting broader consensus on protocol upgrades or fund allocations.[37] Empirical analysis in DAOs shows quadratic voting variants improving decentralization metrics, though adoption remains limited due to computational overhead in smart contract execution.[38] Specific decentralized implementations include the Quadratic Voting Plugin in Realms, a Solana-based DAO governance platform launched in early 2025, which tempers the influence of high-stake voters by enforcing quadratic pricing on vote allocations for proposals.[39] Similarly, protocols like QV-net, introduced in June 2025, enable self-tallying quadratic voting on blockchains with maximal ballot secrecy, allowing public verification of results post-voting without revealing individual preferences during the process.[40] These systems leverage smart contracts for automated enforcement, as seen in experimental DAO elections where quadratic rules mitigate collusion risks compared to one-token-one-vote models.[41] However, challenges persist, including scalability on high-throughput chains and resistance to sybil attacks via identity verification layers.[42]Related Concepts
Quadratic Funding Mechanics
Quadratic funding allocates a fixed matching pool M to multiple public goods projects based on private donations, prioritizing projects with broad donor support over those funded by few large contributors.[43] The mechanism computes a subsidy for each project j by first determining a "voice" value v_j = \sum_{i \in D_j} \sqrt{c_{ij}}, where D_j is the set of donors to project j and c_{ij} is the contribution from donor i to j.[44] The target total funding for the project is then t_j = v_j^2, representing the quadratic aggregation of supporter intensities.[43] The raw subsidy is s_j = \max(0, t_j - p_j), where p_j = \sum_{i \in D_j} c_{ij} is the total private funding received by j. To fit within M, subsidies are scaled proportionally: final subsidy f_j = s_j \cdot \frac{M}{\sum_k s_k} if \sum_k s_k > M, or f_j = s_j otherwise.[44] This formula derives from a model where the public funder seeks to maximize social welfare under assumptions of additive utility across projects and unit-elastic private marginal valuations, effectively treating small donations as signals of diverse support.[43] For instance, if 100 donors each contribute $1 to a project, v_j = 100 \times 1 = 100, so t_j = 10,000; with p_j = 100, the raw subsidy is $9,900, which would be scaled based on competing projects and M. In contrast, a single $100 donation yields v_j = 10, t_j = 100, and raw subsidy $0, illustrating amplification of diffuse participation.[44] The approach assumes verifiable identities to prevent sybil attacks and infinitesimal contributions for continuous approximation, though practical variants cap contributions or use discrete adjustments.[43] Variations include pairwise quadratic funding, which bounds interactions between donor pairs to mitigate collusion, computing subsidies as M \times \frac{\sum_{i < k \in D_j} \min(\sqrt{c_{ij}}, \sqrt{c_{kj}})^2}{\sum_l \sum_{i < k \in D_l} \min(\sqrt{c_{il}}, \sqrt{c_{kl}})^2} or approximations thereof.[44] Minimum matching requirements can be imposed, where projects must exceed a private funding threshold to qualify, ensuring genuine interest before subsidy allocation.[45] These mechanics extend quadratic voting principles to funding by subsidizing contributions as if the pool were simulating collective decision-making with quadratic costs.[43]Distinctions from Quadratic Voting
Quadratic funding mechanisms, though inspired by similar quadratic principles, diverge from quadratic voting in their core objectives and operational frameworks. Quadratic voting aims to facilitate collective decision-making on predefined options, such as policy choices or platform planks, by enabling voters to express preference intensities through a budget of voice credits where the cost of casting the k-th vote on an option scales quadratically (k2 total credits for k votes), while the outcome tally aggregates votes linearly to balance minority passions against majority sentiments.[8] In contrast, quadratic funding targets the provision of public goods, like open-source software, by subsidizing projects with matching grants calculated as the square of the sum of square roots of individual donor contributions, thereby prioritizing broad-based support from many small donors over concentrated funding from few large ones to mitigate free-rider problems.[46] Mechanistically, quadratic voting imposes quadratic costs on voters' limited, equalized credit endowments to discourage frivolous spending and reveal true valuations across competing discrete alternatives, often within a fixed ballot or agenda.[2] Quadratic funding, however, leverages donors' voluntary monetary inputs without personal credit budgets, drawing from an external subsidy pool to amplify collective contributions dynamically; this allows for continuous, endogenous project emergence rather than voter-imposed selection among static options.[44] The former thus serves binary or ranked-choice scenarios in governance, as trialed in the Colorado Democratic Party's 2018 platform development, while the latter supports ongoing resource allocation, exemplified by Gitcoin's grants rounds starting in 2019 for Ethereum ecosystem projects.[11]| Aspect | Quadratic Voting | Quadratic Funding |
|---|---|---|
| Primary Purpose | Aggregating intensities for decisions on discrete options (e.g., approve/reject policies) | Allocating subsidies for continuous public goods provision (e.g., funding multiple initiatives) |
| Input Mechanism | Fixed credits per voter; quadratic cost to express k votes | Donor-chosen contributions; subsidy as (∑√ci)2 where ci are inputs |
| Output Type | Winner-take-all or ranked outcomes based on linear vote sums | Proportional matching grants favoring donor diversity |
| Constraints | Predefined agenda; equal starting budgets | No fixed options; relies on subsidy pool availability |
| Key Theoretical Basis | Optimizes utilitarian welfare under preference intensity (Lalley-Weyl, 2015)[8] | Addresses under-provision of public goods via matching (Buterin-Hitzig-Weyl, 2018)[46] |