Quadratic Mechanisms

Established The prize is a pricing rule that reads how much a group cares, not only which side it is on, and a proved theorem that it does so with an error that shrinks as the group grows. Under quadratic voting a participant pays the square of the votes cast, so the marginal vote costs its own count and a voter buys intensity until price meets value. Lalley and Weyl prove that in the resulting equilibrium the per-decision welfare loss falls toward zero as the electorate grows. Majority rule discards intensity by construction — one voter, one bit — and this is the mechanism that recovers it against a stated bound.

Established The magnitude, in the units the field reports. Where one-person-one-vote returns a single bit per voter, quadratic voting returns a real number: the one legislature that has run it, the Colorado House Democratic caucus in 2019, ranked 107 competing bills against a roughly $40 million discretionary budget and produced one ordering its leaders called a cleaner signal than a show of hands. The sister mechanism, quadratic funding, matches a shared pool on the number of distinct backers a project attracts rather than the dollars; its largest deployment, Gitcoin Grants, reports having moved more than $60 million to over 3,700 projects by that rule (programme’s own figures).

Frontier Who gets the upside is decided by who holds the pool and who sets the agenda. The gain flows to intense minorities that lose under majority rule and to small projects with broad, shallow support — but only if each account is one person. Where identities can be manufactured, the same gain flows instead to whoever can manufacture them most cheaply, and the concave square-root that rewards breadth is precisely the lever a fake-identity attack pulls.

Handwave The condition is population-scale proof that one account is one person, and no deployment in this record has it. Every optimality result assumes distinct honest participants; in Gitcoin’s rounds a fraud-detection team has had to strip thousands of accounts each cycle to hold the distortion to a couple of percent of the matching pool. If robust one-person-one-account verification existed the upside would be bankable; it does not, which is why the bottleneck list in section 5 opens on identity rather than on the pricing rule.

Established Quadratic mechanisms are a family of rules whose defining move is to price the intensity of a preference at the square of its size. The two members that matter are quadratic voting (QV), a rule for collective decisions, and quadratic funding (QF), a rule for allocating a matching pool to public goods. Both descend from the same insight in public-goods theory: for a good whose value is the sum of many private valuations, a participant should face a private marginal cost equal to the social marginal value, and squaring the individual’s expression is the pricing rule that delivers this.

Established Quadratic voting: a voter is given a budget of voice credits and buys votes for or against each question at a cost equal to the square of the votes bought. One vote costs one credit, two votes cost four, three votes cost nine. Because the marginal vote costs its own number, a voter keeps buying votes on a question until the credits spent on the last vote equal the value of shifting the outcome — so the number of votes cast tracks the square root of the credits allocated, and the credits allocated track intensity. Distributing an equal budget of artificial credits, rather than letting people pay in money, is what separates QV as a democratic tool from QV as an auction.

Established Quadratic funding: a project’s share of a matching pool is set by the square of the sum of the square roots of its individual contributions, less the contributions themselves. Ten people giving $1 each generate a matched amount proportional to the square of ten times the square root of one — a hundred — while one person giving $10 generates the square of the square root of ten, which is ten. The rule therefore rewards the number of backers far more steeply than the size of any one gift, and its designers describe it as funding a public good in proportion to the breadth of support rather than the depth of a donor’s pocket.

Frontier The two rules share a single fragility that the overview must name up front. The same square-root concavity that makes breadth valuable makes a split contribution valuable: a donor who divides one gift across many identities converts depth into apparent breadth and captures matching that the rule intended for genuinely broad support. QV faces the mirror problem — a coordinated bloc or a duplicated identity buys intensity it does not feel. The mechanism’s guarantees are therefore conditional on a property outside the mechanism: that the accounts it counts are distinct people.

Established The reason the exponent is two, and not some other number, is the mathematics of the marginal vote. When the cost of expression is squared, the cost of the marginal unit is linear in the units already bought, so a self-interested participant equates that linear marginal cost with their private value and reveals it; any other exponent breaks the alignment between what a person pays at the margin and what the collective outcome is worth to them. The square is not a metaphor for “caring more costs more” but the specific curvature that makes truthful revelation an equilibrium.

Established The theoretical result is real and it is precisely stated. Lalley and Weyl, in the working paper that became the 2018 American Economic Association proceedings piece, prove that for a broad class of value distributions the expected inefficiency of quadratic voting — the gap between the welfare of the QV outcome and the welfare-maximising outcome — is bounded above by a constant that tends to zero as the number of voters grows, and for the symmetric case they give the rate as of order one over the number of voters. This is a genuine optimality theorem: not that QV is good, but that its welfare loss vanishes in large populations at a stated rate.

Frontier The theorem rests on assumptions that do not describe a real election, and the honest position states them. The proof assumes price-taking — each voter treats the cost of a vote as fixed and ignores their effect on the aggregate — independent private values, quasilinear utility (value measured in the same units as the credits), and no collusion. Weyl’s companion robustness analysis argues QV degrades gracefully when several of these are relaxed and describes it as the pricing rule uniquely robust across a range of settings, which is a weaker and more defensible claim than exact optimality. Where price-taking fails — small committees, correlated values, organised blocs — the guarantee is a limit result being applied outside its limit.

Established Quadratic funding carries its own optimality theorem, and the same caveat. Buterin, Hitzig and Weyl prove in Management Science (2019) that the square-of-summed-square-roots rule is the mechanism that, under their standard model of rational contributors with independent values, achieves first-best public-goods provision — the allocation a benevolent planner with full information would choose. The paper is explicit that this holds under that model and that collusion and identity duplication are outside it; the same authors elsewhere spend more words on how to defend the rule than on the theorem, which is the tell.

Frontier The empirical record for quadratic voting is thin, positive on preference recovery, and silent on outcomes. Survey-experiment work — Quarfoot and colleagues’ study of QV run on real respondents — finds that people use the mechanism as intended, spreading credits in a way that tracks stated intensity better than a checkbox does, and that they do not collapse into confusion. What no study shows is that a decision reached by QV is a better decision by any outcome measure, because that comparison has not been run.

Frontier The Colorado case is the only time a legislature has bound a real process to the rule, and it is a natural experiment without a control. In April 2019 the Colorado House Democratic caucus, organised with the RadicalxChange and Democracy Earth foundations, used QV to prioritise 107 spending bills competing for roughly $40 million; each member received a fixed budget of voice credits — reported as 100 by later accounts and as 25 by the contemporaneous Colorado Sun coverage, a discrepancy no primary document this brief could reconcile — and Senate Bill 19-085, Equal Pay for Equal Work, emerged on top with 60 votes in aggregate. Representative Chris Hansen, who ran it, said the caucus “got a better signal with less noise.” Whether the resulting budget was better than a show of hands would have produced is unmeasured and, given a sample of one caucus in one session, unmeasurable from this deployment alone.

Established Quadratic funding has by contrast a large field record, and it is dominated by one operator whose figures are its own. Gitcoin Grants reports cumulative distribution of more than $60 million to over 3,700 projects (programme’s own figures). Individual rounds are better documented: Round 7, in October 2020, drew $274,830 in contributions and distributed $450,000 in matching across 857 projects; Round 9, in April 2021, matched $500,000 on $1.38 million donated by more than 12,000 contributors to 812 projects. These are real transfers to real projects, and they are the strongest evidence that the mechanism is operable at scale.

Established The same rounds are also the strongest evidence that the mechanism cannot run untended. In Round 9 the operator reported that more than 15% of contributions were detected as probably fraudulent, and rather than reweight the round it absorbed the cost — about $33,000, roughly 6.6% of the matching — out of its own treasury. Independent anti-fraud evaluations by BlockScience put the “fraud tax,” the difference in matching caused by removing suspected sybils, at $14,400 (2.1% of a $700,000 pool) in Round 10 and $5,787 (about 0.6% of a $965,000 pool) in Round 11, where 853 of 15,986 contributors (5.3%) were flagged and the true sybil share was estimated at 6.4%, in a range of 3.6% to 9.3%.

Frontier The count of fake identities is large, volatile, and much bigger than its dollar footprint — which is the whole finding. In Round 12 (December 2021), against a $3.3 million pool, BlockScience flagged 8,100 of 28,987 contributors — 27.9% — as potential sybils. That a round can be one-quarter fake accounts while the matching distortion is held to low single-digit percentages is not evidence the problem is small; it is evidence that an active detection function, tuned each round and for years partly closed-source, is doing continuous work the optimality theorem assumes away. BlockScience estimated its own process caught on the order of 83% of sybil users, which means the residual is a policy choice about how much fraud to tolerate, not a solved problem.

Frontier The defences are real, partial, and in tension with the theorem they protect. Two families are deployed: identity verification — phone, BrightID, Idena, later a stamps-based passport — which raises the cost of an extra identity, and pairwise or cluster bounding, which discounts matching when the same donors keep co-funding the same grants. Gitcoin’s current variant, connection-oriented cluster matching, moves this discounting inside the formula by scoring how socially diverse a project’s backers are; the operator reports it improves allocations by around 50% over base quadratic funding and has run more than $5.5 million through it (programme’s own figures, no independent replication). Every one of these defences penalises genuine tight-knit communities along with attackers, which is the cost the theorem cannot see.

Frontier The load-bearing gap in the whole subject is the absence of a causal comparison of allocation quality. No study consulted for this brief routes the same budget through quadratic matching, an expert panel and simple proportional matching and then scores the funded work against each other. Until that exists, every claim that quadratic mechanisms allocate better — as opposed to differently, or more inclusively by construction — is a claim about a theorem’s assumptions, not about an outcome anyone has measured.

Established A sybil does not print matching; it diverts it. The pool is fixed, so an attacker’s manufactured breadth captures matching that would otherwise have reached genuine projects, which is why the harm is measured as a redistribution — the fraud tax — rather than as a theft from the sponsor. It is also why a low fraud tax can sit beside a high fake-account count: many tiny fake contributions each move little matching, and detection strips them before the pool is divided.

Frontier The rounds also surfaced concentration, the failure mode opposite to fragmentation. In Round 7 the leading recipients received on the order of $39,000 to $48,000 each, and the operator worried in print about whether frontrunners were getting too much — while noting those recipients were “uncontroversial and universally beloved” ecosystem mainstays, which he read as the mechanism being more self-correcting than expected. Both readings appear in the same retrospective, and this brief records the tension rather than resolving it.

Frontier Even the remedy is contested inside the operator. In Round 11 the Gitcoin stewards voted not to pay out the calculated fraud-tax adjustment at all, and by Round 12 the metric was being de-emphasised — a reminder that how much fraud to correct for is a governance decision taken round by round, not a fact the formula settles.

Frontier No public deployment has used real money for quadratic voting, and that is a finding about feasibility rather than an oversight. Colorado ran on equal credit budgets; Gitcoin’s funding side spends real dollars but pairs them with square-root matching, not square-root voting. The money-denominated QV that critics attack, and that theory finds plutocratic, exists mainly on the page — worth stating because the strongest objection to the family targets a variant almost no one runs.

Frontier Can proof-of-personhood reach population scale without excluding the people it counts? Every quadratic mechanism collapses to a wealth or effort auction the moment identities are cheap, so the live research question is a verification that is hard to fake and easy to pass. The deployed answers — biometrics, social-graph vouching, government credentials, staked reputation — each trade a false-accept rate against a false-reject rate, and the false rejects are disproportionately the poor, the stateless and the offline, the very groups a breadth-rewarding rule is sold as helping.

Frontier Do the optimality theorems survive their own publicity? Both proofs assume behaviour fixed independently of the algorithm. In the field the algorithm is public and adversaries optimise against it; Vitalik Buterin’s own retrospective states that “making algorithms that prevent undesired manipulation, but continue to do so despite the attackers themselves knowing what these algorithms are, is really hard.” Whether a collusion- and sybil-resistant variant exists whose guarantees hold under common knowledge of the rule is open.

Frontier Is preference intensity even stable enough to price? QV assumes a voter has a cardinal value for an outcome that they can convert into credits. Whether stated intensity is a stable quantity or an artefact of the elicitation — framing, budget size, the number of questions on the ballot — is unsettled, and it bears directly on whether the number QV recovers means anything.

Speculative Can the matching pool itself be made incentive-compatible? A standing question is where the subsidy comes from and whether a self-funding version — taxing the winners to seed the next round, or drawing on a common asset — can avoid the capture that follows when one sponsor controls the pool. No deployed system has closed this loop; the pool is exogenous charity in every case in this record.

Frontier What is the empirical efficiency, as opposed to the theoretical bound? The theorems give an asymptotic rate under idealised assumptions. The frontier question is what the realised welfare loss is in a real committee of realistic size with correlated values and imperfect information — a number that could be one percent or fifty, and that no field measurement yet pins down.

Frontier Does the mechanism’s inclusiveness survive its own defences? Quadratic funding is sold as amplifying the small, broadly-supported project, yet every sybil defence raises the bar to participate, and the burden falls hardest on newcomers with no verification history. Whether a defended quadratic system is more or less inclusive than the unweighted alternative it replaces is an empirical question no deployment has answered.

Frontier Does cheap synthetic-identity generation break the personhood assumption for good? If plausible distinct-seeming participants can be manufactured at near-zero marginal cost, the attacker’s side of the square-root asymmetry gets cheaper every year while verification does not obviously keep pace. Whether the identity layer can stay ahead of synthetic generation is the question that decides whether the family has a future, and it is being settled outside this field.

Established The binding bottleneck is identity, and it is not a detail of implementation but the assumption the mathematics rests on. A quadratic mechanism prices intensity on the premise that the units it counts are distinct people; the square-root concavity that is the source of all its desirable properties is, read from the attacker’s side, an exact instruction for how to profit from duplication. There is today no verification that is simultaneously cheap to pass, hard to fake, privacy-preserving and universal, and until there is, every deployment substitutes a fraud-detection team for the missing assumption.

Frontier Collusion is the second bottleneck and it is harder than sybil resistance because the colluders are real. Sybil detection can in principle key on the fakeness of accounts; bribery and coordination among genuine distinct people leave no such signature. Buterin has documented open vote-buying for grant contributions on community platforms, and the standard analysis holds that quadratic voting is about as sensitive to collusion as ordinary majority voting — better than the classic efficient mechanisms, but not robust. Pairwise bounding attacks the symptom by discounting suspicious co-funding, at the cost of penalising real coalitions.

Frontier Transparency and ballot secrecy pull in opposite directions and the mechanism sits on the fault line. Auditing a quadratic result — checking that no cluster captured the pool — wants the full record of who backed what; democratic legitimacy and coercion-resistance want the ballot secret. Colorado hit this wall in law rather than in code: the very anonymity that made the caucus vote feel safe is what a court found illegal, discussed in section 10.

Established Agenda and pool control is a bottleneck the pricing rule cannot touch. QV prices intensity over a fixed slate of questions and QF matches over a fixed set of eligible projects; whoever sets the slate or curates eligibility holds the outcome upstream of any voter. In Colorado the questions were the caucus’s own bills; in Gitcoin eligibility is a curation decision by the operator. A perfect pricing rule over a captured agenda is a captured process.

Speculative Voice-credit budgeting is an unsolved design problem wearing the appearance of a solved one. How many credits, refreshed how often, carried over or not, priced how against future rounds — these choices determine outcomes as much as the square-root does, and there is no theory that fixes them. The contested Colorado credit figure, 100 or 25, is a small instance of a general fact: the free parameters do the quiet work.

Frontier Participation itself is a bottleneck the theorems assume away. The optimality results assume voters turn out and contributors give; in the field, quadratic mechanisms inherit the same steep participation gradient as every other civic tool, so the breadth the rule rewards is the breadth of whoever showed up, which over-represents the connected and the motivated. A rule that weights by number of backers is only as representative as its enrolment.

Speculative Cluster and pairwise defences may not scale to the populations the mechanism aspires to. Scoring the social diversity of every project’s backers is a computation over a contribution graph, and the more the method leans on off-chain social signals to judge that diversity, the more it depends on data that is itself gameable. The defence that works at ten thousand donors is not obviously the defence that works at ten million.

Established The mechanism depends first on a scientific result nobody is reliably producing: a proof of unique personhood that works across a whole population. This is upstream of everything — the theorems, the software, the legitimacy all assume it — and it is not an engineering deliverable so much as an open problem in identity, biometrics and social-graph analysis whose progress this field cannot itself force.

Frontier It depends on a mechanism-design result that does not yet exist: a quadratic variant that is collusion-resistant under common knowledge of its own rule. Cluster matching and pairwise bounding are engineering patches with no proved guarantee; a theorem that bounds manipulation when the adversary knows the algorithm would move the field from patching to design.

Established It depends on causal-inference infrastructure the field has never built — the ability to compare allocation quality against a counterfactual. The observational round data is rich, but a discipline that wants to claim its mechanism allocates better needs a comparison group, blinded scoring, and a pre-registered outcome. That dependency is methodological, cheap, and unmet, which is the same shape section 7 finds in Civic Technology.

Frontier It depends on a legal reconciliation between anonymous intensity-voting and open-government law. Where a public body adopts the rule, the record of who valued what is either public — defeating coercion-resistance — or secret — defeating open-meetings and open-records law. No jurisdiction in this record has a settled answer; Colorado has a court ruling that the secret version is unlawful.

Speculative And it depends on a durable, uncaptured source of matching capital. Every field deployment of QF runs on donated pools; a version embedded in public budgeting would need a standing appropriation and a rule insulating the pool from the interests it funds, neither of which exists.

Frontier It depends on a settled answer to who sets the slate. The pricing rule is defined over a fixed set of questions or eligible projects, and no theorem in the family governs how that set is chosen; a discipline of agenda-formation and eligibility-curation is a genuine prerequisite that sits outside the mechanism, and no brief in this corpus, this one included, can supply it from the pricing side.

Speculative It depends on a participatory culture that treats the result as legitimate. A pricing rule produces a number; whether a community defers to that number rather than to seniority, volume or the loudest advocate is a matter of adopted norm, not of the formula. Every deployment in this record leaned on an unusually committed community to supply that legitimacy, and it is unclear the property survives transplant to an ordinary one.

Frontier The decisive experiment is a randomised comparison inside a single funding body: the same pool split three ways, once by quadratic matching, once by an expert panel and once by simple proportional matching, with the funded projects scored blind by independent assessors at 24 months, and the same round seeded with sybil clusters at known budgets to measure how much matching they capture and how much the defence claws back. This is the test the whole subject is missing, and it is decisive because it attacks the two claims the field actually makes — that quadratic allocation is better, and that it can be defended — at once, with an outcome measure and a control instead of a theorem and an assertion.

Frontier The seeded-sybil arm is the part no existing study has run, and it is the one that tests the load-bearing dependency directly. Retrospective audits like BlockScience’s estimate sybil incidence after the fact and under uncertainty about ground truth; injecting clusters at known budgets converts that estimate into a measurement, yielding a capture rate and a clawback rate for the defence against an attacker whose size the experimenter knows exactly. It requires no new hardware and no new theory, only a funder willing to spend part of a real pool on being attacked on purpose.

Established A cheaper experiment, available now, is to randomise the voice-credit budget across comparable QV decisions. The credit budget is a configuration value; varying it across matched committees and measuring how the outcome and the spread of credits respond would answer whether the free parameter that currently does the quiet work can be set on evidence rather than by feel. Nothing technical or ethical blocks it and no one has run it.

Frontier The natural experiment already running is Colorado, and it is a control pair the field is not exploiting. One caucus adopted QV in 2019 and expanded it; another chamber and the minority party did not, or did so later. A difference-in-differences on bill survival, budget composition and member satisfaction across adopting and non-adopting groups is observational, imperfect, and far more than the anecdotes the subject currently runs on.

Established The sybil audits are the one experiment already done repeatedly, and their result should reframe the field’s self-description. Across Gitcoin rounds the measured fake-account share ran from roughly 5% to nearly 28% of contributors while the matching distortion was held to low single digits only by continuous detection. A mechanism whose optimality proof assumes distinct participants, and which in practice is one-quarter fake accounts absent intervention, is a mechanism whose field performance is a property of its fraud team as much as its formula.

Speculative A confirmatory experiment worth pre-registering is the collusion sting: offer a documented bribe for coordinated contributions and measure detection. Buterin reports open bribery has already appeared unbidden; a controlled version at a known price would measure the bribe elasticity of the outcome and test whether cluster matching detects real-person coordination as opposed to fake-person duplication, which is the harder and less-studied half of the threat.

Handwave The experiment that would settle the deepest question — whether stated intensity is a real cardinal quantity — is the one nobody knows how to run cleanly. Recovering a person’s true willingness-to-shift-the-outcome without the elicitation creating the number it measures is closer to a problem in the foundations of preference than to a field trial, and this brief does not pretend a protocol for it exists.

Frontier A within-platform A/B test is available to any operator and has not been published. Randomly assign otherwise-comparable projects to base quadratic funding or to cluster matching inside the same round, and compare the resulting allocations and the downstream project outcomes; the operator already runs both formulas and already has the pipeline, so the missing ingredient is the will to publish a comparison that might not flatter the newer mechanism.

Frontier A cheap census would replace anecdote with a denominator. Nobody has published how many quadratic-voting and quadratic-funding deployments were launched, how many survived a second cycle, and how many bound a real decision; the field quotes its successes without a base rate. Compiling that register needs no funding and no new instrumentation, and it would let the subject state its own hit rate instead of its highlights.

Established The core software is undemanding and that is part of the trap. A voice-credit ledger, a square-root-and-square matching computation, and a web front end are ordinary engineering; the same low barrier that let a Colorado caucus and a crypto platform stand the mechanism up in weeks is why the difficult parts — identity, auditing, dispute resolution — get deferred. The lesson is the same one drawn in Civic Technology: the platform is easy and the institution around it is the whole problem.

Frontier The identity stack is the real engineering deliverable and it is a pipeline, not a component. A deployment needs enrolment, a uniqueness check, a revocation path for compromised identities, and a privacy layer that lets the system count distinct people without publishing who they are — the sort of credential infrastructure that Digital Citizenship treats as its subject. Gitcoin’s passport of stacked verification stamps is the most mature attempt and is still, on the operator’s own numbers, permeable by a quarter of a round.

Frontier The matching computation must be auditable and coercion-resistant at the same time, which is a cryptographic requirement, not a UI one. Proving to a sceptic that no cluster captured the pool wants the contribution graph; protecting a voter from a buyer of votes wants that graph hidden. Cluster matching, pairwise bounding and any future collusion-resistant variant are engineering that lives entirely inside this tension, and none of the deployed systems resolves it — they pick a side per round.

Established The fraud-detection function is a permanent operational cost, not a launch task. BlockScience’s evaluations describe a per-round pipeline of machine-learning scores, heuristics and human review tuned to a chosen aggressiveness; running a quadratic mechanism means staffing that pipeline indefinitely, because the attackers adapt between rounds. A budget for the mechanism that omits a standing anti-fraud budget has not costed the mechanism.

Frontier The detector has a dial, and the dial is a policy. BlockScience’s evaluations describe raising the machine-learning aggressiveness from 20% in one round to 30% in the next; a higher setting removes more sybils and more real users, so the false-reject rate the ethical section worries about is set by an operational parameter chosen by whoever runs the round. No value of the dial is simply correct.

Frontier Verifiable computation of the match is within reach and mostly skipped. Publishing a proof that the announced allocation is the correct output of the formula on the recorded contributions is achievable with existing cryptographic techniques, yet most rounds ask participants to trust the operator’s tally. Until the match is verifiable, the mechanism’s claim to neutrality rests on the honesty of whoever runs the script.

Established Proof-of-personhood and reputation systems are the adjacent technology the whole family waits on. Everything a quadratic mechanism promises is contingent on the identity layer that Reputation Economies and Digital Citizenship study directly; a breakthrough there is a breakthrough here, and a failure there caps this field regardless of how good the pricing rule is.

Frontier Deliberative and collective-intelligence methods are the substitute good, and the honest comparison is unflattering. The strongest causal evidence in the participation literature involves no pricing rule at all, a point Civic Technology and Future Democracies both land; deliberative minipublics and participatory budgeting have outcome studies quadratic mechanisms lack. Whether pricing intensity beats deliberating about it is exactly the untested comparison.

Frontier AI-assisted preference elicitation and aggregation is a fast-moving adjacent capability. The tooling explored in AI-Assisted Governance and Collective Intelligence could either replace quadratic elicitation with something richer or supply the sybil-detection models the mechanism depends on — and could equally industrialise the manufacture of plausible fake participants, cutting the other way.

Speculative Standards and cryptographic voting infrastructure are the enabling substrate for any serious public deployment. Coercion-resistant tallying, verifiable computation of the match, and interoperable credential formats are the kind of governance-by-specification that Standards as Governance owns; without them a public quadratic process is auditable or secret but not both.

Speculative Other intensity-revealing instruments are the competitive frontier, not just a substitute. Prediction markets and conviction-voting variants price intensity through stake or time rather than through a squared budget, and whether any of them dominates quadratic pricing on the honesty-versus-manipulability trade is unsettled. The family is one entry in a crowded design space, not the only rule that reads intensity.

Established The first institutional requirement is a body willing to bind a real budget or decision to the mechanism’s output. A quadratic vote that only advises is a survey; the Colorado caucus came closest by using it to prioritise bills, and even there it informed leaders rather than binding them. Without a duty to act on the result, the mechanism inherits the fate diagnosed in Civic Technology, where input is collected and enactment is optional.

Frontier The second is a legal reconciliation with open-government law, and there is now a ruling against the naive version. On 6 January 2024 a Denver District Court judge, David Goldberg, held that the Colorado legislature’s anonymous quadratic voting — used each April over four days to rank bills against the roughly $40 million budget, with leaders seeing only anonymous bar charts — violated the state’s Open Meetings Law, constituted a “serial meeting” that “thwarts the entire purpose” of the law, and shielded from the public what it was entitled to know. Senate leadership signalled the caucus would stop using it. Any public deployment must solve what Colorado did not.

Established The third is a standing anti-fraud institution, funded and staffed between rounds. The Gitcoin experience is unambiguous that the mechanism runs only with a continuous detection function; an agency adopting quadratic funding is adopting a permanent forensic unit, with all the discretion, contestability and capture risk that implies. Who audits the auditors of the pool is a governance question the pricing rule cannot answer.

Frontier The fourth is custody of the matching pool insulated from the interests it funds. Whoever holds and curates the pool sets eligibility and therefore the outcome upstream of any contributor; an institution serious about the mechanism needs the pool governed at arm’s length from both sponsors and recipients, a separation no deployment in this record has built.

Frontier The Colorado chronology shows how quickly an advisory signal becomes a de facto gate. Introduced as an experiment in 2019, the anonymous vote had by 2022 spread across the Democratic caucus, and reporting documented a bill — a $3 million-a-year wildfire-investigation measure that had passed the Senate unanimously — dying without a House hearing after ranking low in the secret survey. A tool meant to inform leaders had become the thing deciding which bills lived, which is precisely why the transparency challenge followed.

Speculative A public quadratic-funding programme would need a line in a budget and a firewall around it. Embedding QF in government means appropriating a standing matching pool and then insulating its eligibility rules from the legislators and agencies whose constituents compete for it — an institutional design closer to an independent grant council than to a voting app, and one no jurisdiction has attempted.

Established Paid in money rather than equal credits, quadratic voting is a rule for letting the rich buy outcomes, and the design knows it. The square-root softens the advantage of wealth relative to a straight auction but does not remove it; willingness to pay is not proportional to welfare, so a money-denominated QV weights the intense-and-rich over the intense-and-poor. The democratic form depends entirely on an equal budget of artificial credits, and any drift toward real payment is a drift toward plutocracy wearing an egalitarian formula.

Frontier The anonymity that protects the voter is the same property that removes public accountability, and reasonable people weigh it oppositely. A secret intensity ballot shields a legislator from a lobbyist and a citizen from an employer; it also hides from the public which representative sank which bill, the harm the Colorado court named. There is no setting of the dial that satisfies both coercion-resistance and open government, only a choice about which failure to accept.

Frontier Sybil defence is surveillance, and the incidence falls unequally. Distinguishing a real new participant from a manufactured one means scrutinising behaviour, devices and social graphs; the false rejects — people wrongly judged fake — concentrate among those without stable devices, documents or dense online networks, which is to say the mechanism’s inclusiveness in theory is bought with exclusion in operation.

Established Control of the pool and the agenda is where the real power sits, and it is easy to launder behind the mathematics. A curated eligibility list or a sponsor-chosen slate determines the outcome before any square root is taken; presenting the result as the neutral output of an optimal rule obscures a discretionary choice made upstream. The ethical exposure of a quadratic system is less the formula than the quiet decisions the formula is used to legitimate.

Frontier Requiring proof of personhood hands power to whoever issues the proof. A universal uniqueness credential is a single point of exclusion and surveillance, and building political and philanthropic voice on top of it makes the issuer a gatekeeper over both. The mechanism that markets itself as distributing influence more fairly cannot run without first concentrating one decision — who counts as a person — in whoever controls the register.

Speculative Handing everyone an equal budget of credits is an egalitarian gesture that embeds a paternalist judgement. Someone decides how many credits, over what horizon, on which questions, and those choices shape outcomes as much as any vote. Equal credits distribute voice equally only within a frame an administrator has already drawn, and calling the result the will of the group flatters a designed artefact.

Frontier If intensity-weighting scaled to public choice, its civilizational effect would be to shift the balance between majorities and intense minorities — and it is genuinely ambiguous whether that is protection or capture. A rule that lets a group that cares a great deal outbid a lukewarm majority protects a persecuted minority in one case and empowers a determined, well-organised faction in the next; the mechanism cannot tell the two apart, because both are, formally, intense minorities.

Speculative The deepest structural implication is that a quadratic civilization would be a civilization organised around a universal identity register. Because the rule fails without one-person-one-account, adopting it at scale means building and trusting a population-wide personhood system — a piece of infrastructure whose controller holds a chokepoint over political and philanthropic voice alike. The mechanism advertised as decentralising influence has centralisation of identity as its precondition.

Speculative As a funding rule, QF at civilizational scale would tilt the public-goods landscape toward the broadly-liked and away from the narrowly-vital. Rewarding breadth of support systematically favours goods with many shallow beneficiaries over goods with few deep ones — basic research with a small expert constituency, unglamorous maintenance, the interests of small populations — and a society that funded its commons this way would over-provide the popular and under-provide the essential-but-obscure.

Handwave The most expansive claim, that quadratic mechanisms could form the pricing layer of a new economic and political order, is a coherent aspiration resting on an unbuilt foundation. The Radical Markets programme sketches a society that prices many things quadratically; every step of it inherits the identity and collusion dependencies this brief documents, so the vision is best read as a research agenda whose first milestone is the proof-of-personhood it presupposes, not as a near-term settlement.

Speculative A civilization that priced intensity would make intensity legible, and legibility invites management. Once how much each group cares about each question is a recorded number, that number becomes a target for persuasion, suppression and trade; the same measurement that lets a planner respect intensity lets an operator manipulate it. Making preference intensity a first-class quantity is not a neutral act.

These horizons track when the quadratic family’s claims could actually be tested and, separately, adopted — two different clocks.

• 10 yr: Frontier The decisive comparison — quadratic matching against an expert panel against simple matching, blind-scored at 24 months, with seeded sybil clusters — is cheap, unblocked and runnable now; whether any funder chooses to be attacked on purpose to run it is the open question, and the natural experiment in Colorado can be mined regardless, alongside the deliberative baselines in Civic Technology and Future Democracies.
• 25 yr: Speculative A population-scale, privacy-preserving proof-of-personhood either exists by now, in which case quadratic funding becomes a serious instrument for public and philanthropic budgets, or it does not, in which case the family stays a niche tool run by dedicated fraud teams. The whole trajectory hinges on that single external result.
• 50 yr: Speculative Conditional on identity being solved, a mature collusion-resistant variant with a proved guarantee under public knowledge of the rule would let a public body bind real decisions to the mechanism; absent that theorem, adoption plateaus at advisory use.
• 100 / 250+ yr: Handwave Any claim that quadratic pricing becomes a standing layer of political economy is a claim about institutions and identity infrastructure a quarter-millennium out, and this brief attaches no probability to it beyond noting the dependencies would all have to have been met.

• Depends on the identity and reputation results owned by Digital Citizenship and Reputation Economies, and it shares the evaluation deficit that Civic Technology documents; as pure software nothing on this map blocks building it, but as a mechanism with guarantees it waits on a personhood result no brief here produces.
• Requires (not on this map) a proof that one account is one person, working across a whole population without excluding the poor and undocumented, which is the assumption every theorem rests on. A body willing to bind a real budget or decision to the output rather than treat it as a survey. A legal reconciliation of anonymous intensity-voting with open-meetings and open-records law, of the kind a Colorado court found the naive version failed. A causal comparison of allocation quality against expert panels and simple matching, which no one has run. And a durable matching pool insulated from the interests it funds.
• Enables a way to register preference intensity and breadth that one-person-one-vote and dollar-weighted giving both discard — but it enables nothing typed on this map until the personhood requirement is met, because every downstream use inherits that dependency.
• Adjacent Collective Intelligence and Future Democracies, which reach the intensity problem through deliberation; AI-Assisted Governance, which could supply or defeat the sybil defences; Standards as Governance, which owns the cryptographic voting substrate; and Institutional Design, which owns the question of what a body does with a signal once it has one.

Established “Quadratic voting lets the rich buy elections.” Only if it is run in money. The democratic form gives everyone an equal budget of artificial voice credits, under which QV does the opposite — it lets an intense minority outweigh a lukewarm majority without anyone spending a dollar. The claim is true of a money-denominated variant and false of the one its advocates actually propose; the confusion is worth correcting in both directions.

Frontier “Quadratic funding optimally funds public goods — it is proved.” The theorem is real and its scope is narrow: first-best provision holds under a standard model of independent, honest, rational contributors. Collusion and identity duplication are outside the model, and both are endemic in every field deployment, so the proof establishes what the rule does among honest distinct people, not what it does in the wild.

Frontier “Gitcoin proves quadratic funding works at scale.” It proves the rule is operable and that a well-staffed operator can hold fraud to low single-digit distortion — a real and non-trivial result. It does not prove the allocations are better than an expert panel or simple matching would produce, because Gitcoin runs no such comparison; “works” here means “runs and resists fraud,” not “allocates optimally.”

Frontier “Colorado shows quadratic voting works in real government.” It shows QV can be stood up and can produce a signal legislators find useful; it also produced a lawsuit and a ruling that the anonymous version is unlawful under open-meetings law, and it produced no measured improvement in outcomes because none was measured. The case is evidence of feasibility and of a legal obstacle, not of efficacy.

Established “Sybil resistance is an implementation detail that will be tidied up.” It is the assumption the entire mathematics rests on. A quadratic mechanism without robust personhood is not a slightly worse quadratic mechanism; it is an auction over who can generate identities cheapest, which is why every real system substitutes a permanent fraud team for the missing proof.

Speculative “The optimality theorems tell us how these mechanisms behave.” They tell us how idealised models behave in the large-population, price-taking, collusion-free limit. The distance between that limit and a real committee of realistic size with correlated values and public knowledge of the algorithm is exactly the quantity no field study has measured, and treating the theorem as a description of the world is the central overclaim this subject invites.

Speculative “Connection-oriented cluster matching solved the sybil problem.” Its author and operator report roughly a 50% improvement over base quadratic funding and several million dollars run through it (programme’s own figures, no independent replication); it moves detection inside the formula rather than removing the need for it, and it penalises genuine tight communities along with attackers. It is a better patch, not a proof.

Frontier “Quadratic voting and quadratic funding are basically the same mechanism.” They share a square-root heart and a sybil dependency, but they solve different problems — QV allocates a fixed decision among voters with equal credit budgets, QF allocates a variable subsidy among projects from a sponsor’s pool — and their failure modes differ: QV’s central risk is collusion among real voters, QF’s is identity duplication by real donors. Treating a result about one as a result about the other is a common and load-bearing error.

Frontier “Quadratic voting takes the politics out of budgeting.” It relocates the politics rather than removing it — to who sets the slate, who sizes the credit budget, and who curates eligibility. The Colorado organisers spoke of capturing preferences more cleanly, which is a real gain, but the decisions that most shape a budget are made before and around the vote, where the formula has no purchase.