Mechanism, theorem, attack

The Square Root Has Accomplices

Quadratic funding makes one hundred small donors outweigh one rich donor, then lets coordinated identities turn that same square-root bonus against the matching pool. Split $100, run the two-account attack, and apply the 2019 pairwise bound. Every monetary readout is recomputed in the page, and every assumption is stated.

A project supported by many people creates benefits that no single donor can capture. Quadratic funding tries to price that breadth. For contributions cᵢ, its uncapped funding weight is the square of a sum of square roots:

funding weight = (Σ√cᵢ)²
matching subsidy = funding weight − Σcᵢ

The square roots make small contributions unusually consequential when they come from distinct people. They do not, by themselves, establish that the people are distinct or uncoordinated.

At one donor, a $100 contribution has $100 of funding weight and asks for no subsidy. At one hundred donors, each giving $1, the same donor principal has $10,000 of weight and asks the pool for $9,900. Breadth is the mechanism's information. The next run asks what happens when breadth is counterfeit.

The theorem's missing person

Buterin, Hitzig, and Weyl show that the rule reaches first-best public-goods provision in their standard model. That result is scoped to quasilinear utility, a complete-information Nash equilibrium, and participants who do not coordinate or multiply identities. Quasilinearity also abstracts from wealth effects, so one dollar is not a clean unit of care.

The coordination assumption is not an implementation footnote. It is the hinge. Let two controlled accounts contribute W each to a fake project. Their principal is 2W. Their funding weight is (√W + √W)² = 4W. The difference is a matching subsidy of 2W.

A cap made from suspicion

In June 2019, Vitalik Buterin proposed weighting each donor pair by how much their giving overlaps across projects. For accounts i and j, with pairwise bound M:

overlapᵢⱼ = Σₚ √cᵢ→ₚ √cⱼ→ₚ
κᵢⱼ = M / (M + overlapᵢⱼ)
pair subsidy = 2κᵢⱼ√cᵢ→ₚ√cⱼ→ₚ

For the same two accounts giving W to one project, overlap is W. Their subsidy becomes 2MW/(M+W). It rises with W, but never reaches 2M. The patch does not identify colluders. It discounts pairs whose contribution histories look coordinated, which can include legitimate communities.

The standard rule rewards every cross-term 2√cᵢ√cⱼ. The optimality theorem reads those terms as benefits one donor creates for another. The attack reads the same terms as transfers coordinated accounts can manufacture. Nothing in the algebra changes. Only the account relationship does.