Connection-Oriented Quadratic Funding (CO-QF)

• CO-QF is a variant of quadratic funding that reallocates matching funds based on group-level interactions instead of treating contributors as isolated agents.
• It modifies the subsidy formula to reward cross-group contributions while downplaying dense, within-cluster donations.
• Simulations and Gitcoin deployments show that CO-QF enhances fairness and efficiency, mitigating strategic manipulation in funding public projects.

Connection-Oriented Quadratic Funding (CO-QF) is a modification of standard Quadratic Funding (QF) that reallocates matching funds using information about connections, groups, or communities rather than treating contributors as isolated agents. In the Gitcoin deployment studied in 2025, CO-QF was designed to favor projects that are jointly supported by different groups of contributors—described as “cooperation across difference”—instead of simply rewarding projects that mobilize large, tightly knit, or heavily coordinated crowds (Miller et al., 22 Sep 2025). It emerged against the background of the canonical QF rule $F^p = \left(\sum_i \sqrt{c_i^p}\right)^2$, which in the standard quasi-linear public-goods model attains the Samuelson condition $V^{p\prime}(F^p)=1$ and, within a symmetric analytic class $F(\vec c)=g\!\left(\sum_i h(c_i)\right)$, is uniquely selected by the welfare-maximizing and boundary conditions $h(c)=\sqrt c$, $g(x)=x^2$ (Buterin et al., 2018, Freedman, 2022).

1. Standard Quadratic Funding as the baseline

Quadratic Funding was introduced for decentralized funding of public goods in environments with quasi-linear utility, independent values across goods, smooth increasing concave valuations, and a subsidy pool or tax system that covers the deficit (Buterin et al., 2018). For a project $p$, contributors choose $c_i^p \ge 0$, and the mechanism assigns

$F^p=\left(\sum_i \sqrt{c_i^p}\right)^2.$

The resulting deficit is

$$F^p-\sum_i c_i^p = \left(\sum_i \sqrt{c_i^p}\right)^2-\sum_i c_i^p = 2\sum_{i<j}\sqrt{c_i^p c_j^p}.$$

Under the standard model, the social planner’s benchmark is the Samuelson condition. If total value for good $p$ is $V^{p\prime}(F^p)=1$0, efficient funding satisfies $V^{p\prime}(F^p)=1$1 whenever the good is funded at all (Buterin et al., 2018). The canonical QF first-order condition yields

$V^{p\prime}(F^p)=1$2

and summing across contributors recovers $V^{p\prime}(F^p)=1$3, which is the standard optimality result (Buterin et al., 2018).

A later note sharpened this baseline by showing that, within the class of symmetric mechanisms of the form

$V^{p\prime}(F^p)=1$4

with analytic $V^{p\prime}(F^p)=1$5 and natural boundary conditions such as $V^{p\prime}(F^p)=1$6, QF is not merely optimal but unique: the only solution satisfying the welfare condition for arbitrary increasing concave $V^{p\prime}(F^p)=1$7 is $V^{p\prime}(F^p)=1$8 and $V^{p\prime}(F^p)=1$9, up to the normalization fixed by self-funding in a one-agent society (Freedman, 2022). This uniqueness is restricted to that structurally symmetric, analytic class, but it defines the theoretical point of departure for any connection-oriented variant.

2. Why a connection-oriented variant was proposed

The motivation for CO-QF comes from two distinct but related problems. The first is theoretical: once agents are not purely selfish, the original QF optimality theorem no longer holds. The second is empirical: in capital-constrained settings such as Gitcoin, QF can exhaust matching funds quickly and can amplify strategic or popularity-driven patterns (Miller et al., 22 Sep 2025, Pasquini, 2020).

The 2025 CO-QF paper models prosociality through sympathy coefficients $F(\vec c)=g\!\left(\sum_i h(c_i)\right)$0, defining agent $F(\vec c)=g\!\left(\sum_i h(c_i)\right)$1’s prosocial utility as

$F(\vec c)=g\!\left(\sum_i h(c_i)\right)$2

Agents choose contributions based on $F(\vec c)=g\!\left(\sum_i h(c_i)\right)$3, but welfare is evaluated using the personal utilities $F(\vec c)=g\!\left(\sum_i h(c_i)\right)$4 to avoid double counting (Miller et al., 22 Sep 2025). Under QF, the equilibrium condition becomes

$F(\vec c)=g\!\left(\sum_i h(c_i)\right)$5

so that $F(\vec c)=g\!\left(\sum_i h(c_i)\right)$6. If $F(\vec c)=g\!\left(\sum_i h(c_i)\right)$7 for at least one off-diagonal pair, then the aggregate marginal benefit from personal utilities is strictly less than $F(\vec c)=g\!\left(\sum_i h(c_i)\right)$8, which implies that $F(\vec c)=g\!\left(\sum_i h(c_i)\right)$9 does not maximize social welfare (Miller et al., 22 Sep 2025). In the paper’s interpretation, QF becomes over-generous once dense prosocial concern is present.

The same paper presents a stylized fairness failure under a bounded subsidy pool: large, coordinated groups can drag a disproportionately large share of the matching pool toward their favorite projects, even when doing so is inefficient relative to broader welfare. This is the setting in which Gitcoin participants described QF as a “popularity contest” (Miller et al., 22 Sep 2025).

Independent evidence from Gitcoin’s capital-constrained implementation identified a complementary resource-allocation problem. Under unconstrained QF, per-project matching is

$h(c)=\sqrt c$0

so matching requirements scale with contributor pairs, approximately $h(c)=\sqrt c$1 in the number of contributors $h(c)=\sqrt c$2 (Pasquini, 2020). The paper explicitly states that “while total individual contributions scale linearly, target matching amounts scale quadratically following the number of pairs of contributors” (Pasquini, 2020). It also documents small, scattered contributions across many projects, rapid growth of the capital-constraint factor $h(c)=\sqrt c$3, and reciprocal backing consistent with strategic behavior (Pasquini, 2020). These observations do not by themselves define CO-QF, but they explain why a mechanism that is blind to connection structure can perform poorly in practice.

3. Formal definition of CO-QF

CO-QF introduces an explicit group structure. Let $h(c)=\sqrt c$4 be the set of agents and $h(c)=\sqrt c$5 a set of groups. If agent $h(c)=\sqrt c$6 belongs to group $h(c)=\sqrt c$7, the membership weight $h(c)=\sqrt c$8 measures how strongly $h(c)=\sqrt c$9 belongs to that group, with normalization

$g(x)=x^2$0

For a given project with contributions $g(x)=x^2$1, the unnormalized CO-QF subsidy is defined by iterating over ordered pairs of groups and aggregating contributions from the symmetric difference (Miller et al., 22 Sep 2025). In compact notation,

$g(x)=x^2$2

where

$g(x)=x^2$3

Total funding is then

$g(x)=x^2$4

The mechanism changes the combinatorics of the subsidy term. Standard QF rewards pairwise agreement between individuals through $g(x)=x^2$5. CO-QF rewards pairwise agreement between groups, and only contributions from the symmetric difference $g(x)=x^2$6 contribute to each group-pair term (Miller et al., 22 Sep 2025). Support from donors who are already shared across groups is discounted in that term.

In the Gitcoin deployment, the group construction was deliberately simple. The set $g(x)=x^2$7 was taken to be the set of projects in the round. A donor who contributed to project $g(x)=x^2$8 was placed into group $g(x)=x^2$9, and the membership weight was defined by donation share:

$p$0

so that a donor’s total influence is distributed across the projects to which they gave (Miller et al., 22 Sep 2025). The paper notes that if one instead sets $p$1, so that each agent is its own group and $p$2, one can recover standard QF behavior (Miller et al., 22 Sep 2025).

4. Analytical rationale: plurality, prosociality, and group proxies

The conceptual basis of CO-QF is plurality in Weyl’s sense: a project is especially valuable when it is supported by diverse groups that do not share the same social or interest cluster (Miller et al., 22 Sep 2025). On this view, the defect in standard QF is not only that it assumes selfish agents, but that it rewards dense within-cluster mobilization even when such mobilization partly reflects prosocial concern internal to a group rather than independent evidence of broad value.

CO-QF therefore uses group memberships as a proxy for the latent structure encoded by $p$3. The paper is explicit that it does not directly use $p$4 in the mechanism. It also states that designs based on explicitly eliciting social preferences or estimating them from richer social-network data were considered and rejected because $p$5 are not directly observable, and because collecting such data would harm usability and raise privacy and complexity issues (Miller et al., 22 Sep 2025). The operative choice was to use lightweight group data that were already available or easy to gather.

This perspective clarifies the mechanism’s comparative logic. QF amplifies many small donors because each new donor adds interactions with all others. CO-QF dampens the effect of many donors within the same cluster and amplifies cross-group agreement instead (Miller et al., 22 Sep 2025). A plausible implication is that CO-QF does not reject the square-root/quadratic insight of QF so much as relocate it: the key interaction is moved from individual pairs to group pairs.

The paper also records a discarded first design for CO-QF. Its desideratum was that both individuals and groups should experience only sub-linear matching returns: if one person or all members of a group increase their contribution by $p$6, the subsidy should scale as $p$7, not $p$8 (Miller et al., 22 Sep 2025). That earlier rule had stronger theoretical guarantees, but in Gitcoin-like small-world networks with overlapping memberships it attenuated contributions too heavily and produced very flat funding, which stakeholders found uninformative. The final deployed design therefore relaxes the strong group-sublinearity requirement in favor of a simpler and more interpretable group-pair formula (Miller et al., 22 Sep 2025).

5. Simulation evidence and Gitcoin deployment

The principal quantitative evaluation in the CO-QF paper is simulation-based rather than a closed-form welfare theorem. The simulated environment has $p$9 agents, partitioned into $c_i^p \ge 0$0 groups of $c_i^p \ge 0$1 agents each, with utility

$c_i^p \ge 0$2

Prosociality is parameterized by $c_i^p \ge 0$3 and total prosocial budget $c_i^p \ge 0$4 per agent satisfying $c_i^p \ge 0$5, with $c_i^p \ge 0$6 (Miller et al., 22 Sep 2025). A parameter $c_i^p \ge 0$7 controls the fraction of prosocial concern directed toward in-group members, and a variance parameter $c_i^p \ge 0$8 controls how predictive group membership is of valuations. For each $c_i^p \ge 0$9, the authors perform $F^p=\left(\sum_i \sqrt{c_i^p}\right)^2.$0 trials, solve the Nash equilibrium contribution game under direct donations, QF, and CO-QF, compute the induced funding level, and compare utilitarian welfare to the welfare-optimal benchmark through the approximation ratio

$F^p=\left(\sum_i \sqrt{c_i^p}\right)^2.$1

The reported qualitative pattern is consistent across experiments. CO-QF provides higher welfare ratios than QF and direct donations when $F^p=\left(\sum_i \sqrt{c_i^p}\right)^2.$2; as $F^p=\left(\sum_i \sqrt{c_i^p}\right)^2.$3, both QF and CO-QF converge to nearly optimal behavior; and CO-QF’s advantage persists across variations in $F^p=\left(\sum_i \sqrt{c_i^p}\right)^2.$4 and $F^p=\left(\sum_i \sqrt{c_i^p}\right)^2.$5 (Miller et al., 22 Sep 2025). The paper interprets this as evidence that CO-QF remains performant even when group definitions are noisy proxies for the underlying prosocial structure.

Gitcoin provides the real-world case study. The paper reports that CO-QF was introduced in late 2023 as an optional mechanism alongside QF and hybrid variants. From its introduction until the time of the paper, there were $F^p=\left(\sum_i \sqrt{c_i^p}\right)^2.$6 total rounds, $F^p=\left(\sum_i \sqrt{c_i^p}\right)^2.$7 of which used CO-QF or a CO-QF/QF hybrid, amounting to approximately $F^p=\left(\sum_i \sqrt{c_i^p}\right)^2.$8; the abstract separately reports an $F^p=\left(\sum_i \sqrt{c_i^p}\right)^2.$9 adoption rate for “grant-awarding ventures” (Miller et al., 22 Sep 2025). The paper also states that approximately $$F^p-\sum_i c_i^p = \left(\sum_i \sqrt{c_i^p}\right)^2-\sum_i c_i^p = 2\sum_{i<j}\sqrt{c_i^p c_j^p}.$$0 million via CO-QF or hybrid rounds (Miller et al., 22 Sep 2025).

The qualitative component uses semi-structured interviews with $$F^p-\sum_i c_i^p = \left(\sum_i \sqrt{c_i^p}\right)^2-\sum_i c_i^p = 2\sum_{i<j}\sqrt{c_i^p c_j^p}.$$1 non-employee round managers. On the distributive dimension, all interviewees considered CO-QF an improvement over QF. In the interview material, CO-QF is described as more “fair” and “equitable,” more reflective of “true sentiment of the broader group,” and better at reducing fake support, exploitation, and mechanism farming (Miller et al., 22 Sep 2025). These claims are not a formal theorem, but they are central to the paper’s empirical interpretation of deployment success.

6. Trade-offs, controversies, and broader significance

CO-QF is presented neither as a universal replacement for QF nor as a pure anti-fraud mechanism. The paper states that Gitcoin still uses separate anti-Sybil and anti-fraud tools, and that CO-QF is not primarily an anti-fraud mechanism (Miller et al., 22 Sep 2025). Its main purpose is to reorient subsidy allocation toward plural support across groups.

The principal controversy is the trade-off between fairness and engagement. Standard QF generated strong excitement because a small donation could have a very large apparent matching effect, and it strongly incentivized outreach and marketing. CO-QF reduces the payoff from pure marketing and cluster mobilization, which several interviewees worried could lower donor excitement, reduce grantee incentives to advertise, and slow ecosystem growth (Miller et al., 22 Sep 2025). Gitcoin’s operational response was to preserve pure QF, pure CO-QF, and hybrid options, with round managers able to choose a weighted average of QF and CO-QF subsidies according to local priorities (Miller et al., 22 Sep 2025).

A second controversy is mis-penalization of genuine consensus. One interviewee argued that if many donors independently converge on the same projects round after round, the project-as-group construction can treat them as overly coordinated and effectively penalize them (Miller et al., 22 Sep 2025). This is not a peripheral objection; it identifies a structural risk in any mechanism that discounts correlated behavior.

The broader significance of CO-QF lies in its generalization of quadratic mechanism design from isolated individuals to grouped or relational agents. The paper explicitly proposes extensions beyond Gitcoin: in firms, groups could be departments or geographic units; in cities, neighborhoods, income brackets, or demographic clusters; in national or civic settings, political parties, regions, or stakeholder groups (Miller et al., 22 Sep 2025). It also outlines an analogous “Connection-Oriented QV” by reinterpreting Quadratic Voting through the square root of a QF-like funding formula (Miller et al., 22 Sep 2025). This suggests a larger mechanism-design program in which the central design variable is not merely the curvature of the funding rule, but the social unit whose agreement is amplified.

In that sense, CO-QF marks a substantive shift in the interpretation of quadratic mechanisms. Standard QF assumes isolated and selfish agents and, in its classical theory, is uniquely optimal within a narrow functional class (Freedman, 2022). CO-QF begins from the claim that real contributors are social and often prosocial, and that the relevant unit of agreement may be a structured group rather than an atomized individual (Miller et al., 22 Sep 2025). The resulting mechanism is therefore best understood as a connection-oriented reformulation of quadratic matching for settings where plurality, prosociality, and bounded matching pools are empirically central.