Enhancing Decentralization in Blockchain Decision-Making Through Quadratic Voting and Its Generalization

Andrii Nastenko; Dmytro Kaidalov; Kenric Nelson; Lyudmila Kovalchuk; Mariia Rodinko; Roman Oliynykov

REVIEW 5 major objections 5 minor 3 references

Enhancing Decentralization in Blockchain Decision-Making Through Quadratic Voting and Its Generalization

T0 review · 5 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper claims that square-root weighting of blockchain stakes—Types 2 and 3 quadratic voting and any power-law generalization—strictly improves decentralization measured by the Gini and Nakamoto coefficients, while preserving the…

desk verdict A mix of correct elementary results and a false headline claim: the strict Nakamoto improvement (Prop. 5) fails on a two-voter example, and the paper also gets the Sybil direction and a Lagrange multiplier step wrong. read the letter →

arxiv 2504.12859 v1 pith:5TK25PWP submitted 2025-04-17 cs.GT

Lyudmila Kovalchuk , Mariia Rodinko , Roman Oliynykov , Andrii Nastenko , Dmytro Kaidalov , Kenric Nelson This is my paper · ORCID

classification cs.GT MSC 91B1291B14

keywords quadraticvotingblockchaingovernancedecentralizationGinicoefficientNakamotorelativeratiogamma-powerstakeweighting

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that replacing one-coin-one-vote with square-root weighting in blockchain governance makes decision power measurably less concentrated. It distinguishes three flavours of quadratic voting and proves that, in the split-stake Type 2 and the unsplit-stake Type 3, small stakeholders gain relative influence while the largest stakeholders lose it. The same formalism is extended to any power-law stake weighting with exponent between 0 and 1, for which the paper proves lower Gini coefficients, higher Nakamoto coefficients, and the existence of a threshold stakeholder below whom everyone gains influence. Governance designers can tune the exponent to hit a desired cap on the largest participant's impact, such as 'the richest 40% hold no more than 60% of voting credit.'

What carries the argument

The central object is the relative voting ratio $r_g^{(i)}=\frac{g(s^{(i)})}{\sum_j g(s^{(j)})}$, and the key mechanism is the power-law stake transformation $T_\gamma(s)=s^\gamma$, called gamma-power voting (GPV), with QV as the $\gamma=1/2$ case. The proof machinery is Lemma 2, a summation-by-parts dominance lemma: if the partial sums of normalized linear shares are at most the partial sums of gamma-normalized shares, then for any nondecreasing sequence of weights the linear weighted sum is at least the gamma weighted sum. Applied to weights $y_i=i$, this yields the Gini decrease; applied to a step function selecting the top $k$ shares, it yields the top-$k$ dominance inequality behind the Nakamoto claim. The threshold-stakeholder behaviour is carried by $\eta^{(i)}=r_{QV}^{(i)}/r_{LV}^{(i)}$, which is shown to be strictly decreasing in stake.

What would settle it

For stakes $\{1,100\}$ and threshold $a=0.51$, the largest linear share is $100/101\approx0.9901$ and the largest square-root share is $10/11\approx0.9091$; both exceed $0.51$, so the Nakamoto coefficient is $1$ under both linear and square-root weighting, contradicting the strict inequality $N_{\gamma V}(a)>N_{LV}(a)$ of Proposition 5.

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Extended reading notes

Core claim

The paper's central claim is that Types 2 and 3 quadratic voting, and more generally any gamma-power voting with $0<\gamma<1$, enhance decentralization as measured by the Gini and Nakamoto coefficients. In the paper's terminology, Type 2 QV splits the stake first and then takes square roots of the parts; Type 3 QV gives the square root of the total stake to each supported proposal. For both, the relative voting ratio of the smallest stakeholder rises and that of the largest falls, while the ordering of influence by stake size is preserved. The paper identifies a threshold stakeholder: those below gain relative voting ratio, those above lose it, and this threshold can be moved by changing the exponent. It further proves a strict decrease of the Gini coefficient, claims a strict increase of the Nakamoto coefficient at every control threshold, and shows that without privacy-preserving cryptographic ballots a late voter can maximize utility against observed results, while Type 1 allows collusion by large stakeholders splitting stakes across proposals.

Load-bearing premise

The proof that gamma-power weighting strictly increases the Nakamoto coefficient for every threshold assumes that every normalized top-$k$ share under gamma weighting is strictly smaller than the corresponding linear top-$k$ share; if that pointwise dominance fails near the threshold boundary, the strict increase can fail.

Editorial extensions

If this is right

The largest stakeholder's relative voting ratio strictly decreases and the smallest's strictly increases under Type 2 and Type 3 QV, while the order of influence by stake is preserved.
The Gini coefficient of voting-credit distribution is strictly smaller under any gamma-power weighting with $0<\gamma<1$ than under linear one-coin-one-vote weighting.
The Nakamoto coefficient is strictly larger under gamma-power weighting at every control threshold, so reaching any controlling share requires more stakeholders.
For any desired cap $\alpha$ between $1/n$ and the largest linear share, there is a unique exponent $\gamma$ that makes the largest transformed stake share exactly $\alpha$, enabling algorithmic decentralization tuning.
Deploying QV without privacy-preserving ballots lets voters who vote last optimize their utility against observed results, and Type 1 QV is vulnerable to collusion attacks where large stakeholders split stakes evenly across proposals.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A concrete before/after test in a real DAO switching from linear to quadratic voting should show voters below the threshold gaining relative voting share and those above losing it, with the ordering of influence unchanged.
The same gamma-tuning algorithm could be used as a constitutional guardrail: specify the number $k$ of largest stakeholders and the maximum total influence $\alpha$ they may control, then solve for $\gamma$ to enforce that cap.
The dominance proof is not tied to power laws, so the decentralization conclusion should also hold for other sublinear weighting functions such as a capped logarithm; testing one would show whether the power-law form is essential.
Type 3 QV makes voting power depend on how many proposals a voter supports, so participation breadth itself changes influence; that creates both an engagement incentive and a bribery surface the paper does not fully explore.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things. First, the paper is not empty: the RVR propositions, the Gini coefficient proposition for gamma-power weighting, and the algorithm for tuning maximal stake impact to a desired bound are correct and moderately useful. Second, the central claim in the abstract—that QV types 2 and 3 strictly improve the Nakamoto coefficient—is false. The stress-test note is right, and the counterexample is embarrassingly simple: stakes {1,100}, threshold a = 0.51. Linear voting needs 1 voter (100/101 > 0.51); QV also needs 1 voter (10/11 > 0.51). So N_QV = N_LV = 1, not N_QV > N_LV. Proposition 5's proof uses inequality (11), but Lemma 2 only gives non-strict dominance at the threshold index. The paper quietly turns '≥' into '>'. That sinks the abstract's advertised result. This is a load-bearing flaw, not a typo. The utility-maximization section also divides by s(i) instead of sqrt(s(i)) in equation (14) for QV-1, and the claim that QV-3 is not Sybil-vulnerable is backwards—with unsplit stakes, splitting one large stake into many small ones actually increases total voting power because sqrt is concave. On top of that, the abstract contradicts Definition 1 on which QV type applies sqrt before versus after splitting. These are not subtle issues. What the paper does well: Proposition 1 and 2 give clean monotonicity properties of relative voting ratios, and the Gini result (Proposition 3/4) is correctly proven for arbitrary gamma in (0,1). The algorithm in Section 6, choosing gamma to cap the largest stakeholder's transformed impact, is a genuine and practical idea that survives the other mistakes. The related work section is adequate, though it leans heavily on blog posts and informal sources. Who is this for? Governance designers working on DAO or Cardano Catalyst might read Section 6 with profit, but they should ignore the Nakamoto claims until they are reworked. As is, the paper needs major revision. I would not take it, but I also would not desk-reject without a referee: the gamma-power parameterization is worth a serious look, and a competent referee can separate the salvageable core from the false headline. My honest recommendation: reject, but invite a resubmission that drops or replaces Proposition 5, fixes the Lagrange error, and retracts the Sybil claim for QV-3.

Referee Report

5 major / 5 minor

Summary. The paper studies Quadratic Voting (QV) in blockchain governance, distinguishing three QV types depending on whether the stake is split and whether the square root is applied before or after splitting, and introduces a generalized 'gamma-power voting' (GPV) scheme. The authors prove properties of the relative voting ratio (RVR) under QV-2 and QV-3, show that the Gini coefficient decreases for QV-2 and GPV compared to linear voting, and claim that the Nakamoto coefficient strictly increases for every threshold. They also analyze utility maximization for voters who vote after observing others, concluding that privacy-preserving voting protocols are necessary, and propose an algorithm to calibrate the gamma parameter to limit the influence of the largest stakeholders.

Significance. If the main claims were correct, the paper would provide a useful formal toolkit for comparing voting schemes by decentralization metrics and for tuning a family of concave weighting schemes. The RVR results (Propositions 1-2) and the Gini inequality (Propositions 3-4) are elementary but correct in substance, with the gamma-power parametrization being a natural extension. However, the headline result that QV and GPV strictly increase the Nakamoto coefficient (Proposition 5) is false, and the utility-maximization section contains algebraic and logical errors. Because these issues affect the central contribution, the paper in its current form is not publishable, even though parts of the formal framework could be salvaged after substantial revision.

major comments (5)

[Section 3.2, Proposition 5 and inequality (11)] The claimed strict inequality N_{γV}(a) > N_{LV}(a) for every threshold a is false. The proof through Lemma 2 with the indicator weights y_i yields only the non-strict dominance ∑_{i=n-k+1}^n (s(i))^γ / ∑_{i=1}^n (s(i))^γ ≤ ∑_{i=n-k+1}^n s(i) / ∑_{i=1}^n s(i), with equality at k=n. Even if this dominance were strict, it would not imply strict increase of the Nakamoto coefficient: when the linear top-k share first reaches a, the gamma top-k share may already be at or above a. For stakes {1,100} and a=0.51, both the linear top-1 share (100/101) and the quadratic top-1 share (10/11) exceed 0.51, so both Nakamoto coefficients equal 1. The counterexample directly disproves Proposition 5 as stated, and with it the abstract claim that Types 2 and 3 QV and their generalizations strictly enhance decentralization as measured by the Nakamoto coefficient.
[Section 4.2, Proposition 6 and equations (16)-(17)] The Lagrange multiplier derivation for QV-2 is incorrect. For the constraint √s(i) = ∑_r s_r^{(i)}, the derivative of the Lagrange function (17) with respect to s_r^{(i)} is -λ, not -2λ, so the stationarity condition should be π_r^{(i)}(b_r-a_r)/(s_r^{(i)}+b_r)^2 = λ. Consequently the expression for λ and equation (16) do not follow from the stated Lagrangian. Similarly, the second derivative ∂²L/∂s_r^{(i)}∂λ equals -2 in the paper, but for this linear constraint it should be -1. These errors undermine the utility-maximization and last-voter manipulation arguments in Section 4.2.
[Section 4.1, Lemma 3 and equation (14)] The utility-maximization proof for QV-1 is also incomplete: equation (14) is not the stationarity condition derived from the Lagrangian (15) except under a restrictive proportionality assumption on the A_r = π_r^{(i)}(b_r-a_r)/(s_r^{(i)}+b_r)^2 terms, and the paper does not justify that assumption. Moreover, Lemma 3 asserts that a symmetric matrix with negative diagonal entries, negative off-diagonal entries in the last row/column, and a zero bottom-right entry is negatively definite; this is false in general, as such a matrix need not be negative definite even for n=2. The negative-definiteness of the Hessian is therefore not established by the given argument.
[Section 2, paragraph before Proposition 1] The statement 'both types of QV, type 1 and type 2, are vulnerable to Sybil attacks, but QV-3 is not' is false. In QV-3, an attacker with total stake S split into k accounts can contribute k·√(S/k) = √(kS) votes to a proposal, which is strictly larger than √S for k>1. Thus QV-3 is also Sybil-vulnerable; splitting improves an attacker's influence. This claim should be corrected or removed. The decentralization theorems in Section 3 do not rely on it, but it appears as a stated advantage of QV-3 and is currently misleading.
[Section 3.1, proof of Proposition 3] The proof of the Gini inequality contains a normalization error. The text defines b_k = (s(k))^γ / (∑ s(i))^γ and then claims A_n = B_n = 1, but ∑ b_i = ∑ (s(i))^γ / (∑ s(i))^γ, which is not 1 in general. Lemma 2 cannot be applied as written without normalizing b_i by ∑ (s(i))^γ. The Gini statement itself is true and can be proved by the same majorization argument once this normalization is fixed, but the formal proof in the manuscript is not correct as it stands.

minor comments (5)

[Throughout] There are two different results numbered Proposition 5: the Nakamoto coefficient statement in Section 3.2 and the utility-maximization statement in Section 4.1. The numbering should be corrected throughout.
[Section 2.2] The text refers to 'QV-5' in the informal discussion following Proposition 2; this should be 'QV-3'.
[Section 3.2, Definition 4] The threshold is written as 'a ∈ 0,1', which should be 'a ∈ (0,1)'.
[Section 4.1, equation (13)] The sum in the utility function is indexed by i over the number of proposals, but the voter index is also i; this creates a notational clash. The proposal index should be r throughout.
[Section 4.2] The phrase 'the maximum of a continuous function is a closed area' should say that the maximum is attained on a compact feasible set; the feasible set is compact when the s_r are nonnegative or when the sphere constraint is used.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: QV decentralization theorems are derived from explicit definitions and standard inequalities; the Proposition 5 issue is a proof gap, not circularity.

full rationale

The paper's derivation chain is self-contained. QV and GPV are explicitly defined by the functions g(x)=x^gamma with gamma in (0,1) (Definition 1 and Definition 2), and the decentralization measures are explicitly defined as the Gini coefficient and Nakamoto coefficient (Definitions 3 and 4). Propositions 1 and 2 derive RVR properties from monotonicity and elementary inequalities. Proposition 3 and its generalization Proposition 4 establish the Gini comparison via partial-sum dominance and Lemma 2, with no fitted parameters or externally assumed targets. Proposition 5 attempts to prove the Nakamoto-coefficient comparison from the same partial-sum dominance; even if the proof is flawed because non-strict dominance of top-k shares does not by itself imply a strict increase in the Nakamoto coefficient, this is a mathematical correctness issue, not circularity. The conclusion is not identical to the definition of either QV or the Nakamoto coefficient, and it is not obtained by renaming a known result. The paper invokes no load-bearing self-citation: citations to previous QV and blockchain governance literature are background and do not supply the central equivalence. The Section 6 algorithm that chooses gamma to achieve a desired maximum stake impact is explicitly a construction/calibration from the monotone function t_S(gamma), not a hidden fit presented as an independent prediction. Therefore no circular step is identifiable under the stated criteria, and the honest finding is no significant circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The results depend on the standard QV model and the new gamma-power parameterization; the only free design parameter is the exponent gamma. No empirical data are used, and no new physical or conceptual entities are introduced.

free parameters (2)

gamma exponent = chosen by user per use case
The generalized model (Definition 2, Section 2) uses g(x)=x^gamma; Proposition 7 and the algorithm in Section 6 tune gamma to make the largest stake impact equal a desired bound alpha. It is a design parameter, not estimated from data.
alpha = user-specified target
Input to the algorithm in Section 6; the goal is to ensure s_rel,T(i) <= alpha for all i. It is a specification parameter rather than a fitted value.

assumptions (4)

domain assumption Stakes are positive and ordered, and the credit and score functions f,g are increasing.
Used throughout Section 1 and in Propositions 1 and 2 to define RVR and preserve stake order.
domain assumption Voter utility has the form U_i = sum_r pi_r (s_r+a_r)/(s_r+b_r), with other voters' totals a_r and b_r known.
Equations (12)-(13) in Section 4.1; this behavioral model is inherited from Dimitri (2022) and Posner-Weyl, not derived in the paper.
standard math Standard calculus and real analysis facts, including Lagrange multipliers, Hessian negative definiteness, the intermediate value theorem, and monotonic functions.
Used in Sections 4 and 5 for utility maximization and for existence and uniqueness of the tuning exponent.
domain assumption Decentralization is measured by the Gini and Nakamoto coefficients as defined in Section 3.
The proofs establish properties of these specific indices; the paper does not argue these indices capture all aspects of decentralization.

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Pith. "Pith review of Enhancing Decentralization in Blockchain Decision-Making Through Quadratic Voting and Its Generalization." pith.science (2026). https://pith.science/paper/5TK25PWP

@misc{pith2026250412859,
  author       = {Pith},
  title        = {Pith review of: Enhancing Decentralization in Blockchain Decision-Making Through Quadratic Voting and Its Generalization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5TK25PWP}},
  note         = {Machine review of arXiv:2504.12859}
}

read the original abstract

This study explores the application of Quadratic Voting (QV) and its generalization to improve decentralization and effectiveness in blockchain governance systems. The conducted research identified three main types of quadratic (square root) voting. Two of them pertain to voting with a split stake, and one involves voting without splitting. In split stakes, Type 1 QV applies the square root to the total stake before distributing it among preferences, while Type 2 QV distributes the stake first and then applies the square root. In unsplit stakes (Type 3 QV), the square root of the total stake is allocated entirely to each preference. The presented formal proofs confirm that Types 2 and 3 QV, along with generalized models, enhance decentralization as measured by the Gini and Nakamoto coefficients. A pivotal discovery is the existence of a threshold stakeholder whose relative voting ratio increases under QV compared to linear voting, while smaller stakeholders also gain influence. The generalized QV model allows flexible adjustment of this threshold, enabling tailored decentralization levels. Maintaining fairness, QV ensures that stakeholders with higher stakes retain a proportionally greater voting ratio while redistributing influence to prevent excessive concentration. It is shown that to preserve fairness and robustness, QV must be implemented alongside privacy-preserving cryptographic voting protocols, as voters casting their ballots last could otherwise manipulate outcomes. The generalized QV model, proposed in this paper, enables algorithmic parametrization to achieve desired levels of decentralization for specific use cases. This flexibility makes it applicable across diverse domains, including user interaction with cryptocurrency platforms, facilitating community events and educational initiatives, and supporting charitable activities through decentralized decision-making.

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Reference graph

Works this paper leans on

3 extracted references · 2 canonical work pages

[1]

Akinbohun S, Apeh S, Olaye EO et al (2023) Literature Review of Blockchain -Based Voting System: Framework and Concept. Journal of Civil and Environmental Systems Engineering 20(1):72-83 Benjamin D, Heffetz O, Kimball M et al (2017) The relationship between the normalized gradient addition mechanism and quadratic voting. Public choice 172 (2017):233-263 B...

arXiv 2023
[12]

SoK: Measuring Blockchain Decentralization

Springer Berlin Heidelberg Lei T, Zhang Q, Qiu W et al (2025) An enhanced DPoS consensus mechanism using quadratic voting in Web 3.0 ecosystem. Blockchain, 3(1), 1-2 Miller J, Weyl EG, Erichsen L (2022) Beyond Collusion Resistance: Leveraging Social Information for Plural Funding and Voting. Available at SSRN 4311507 Ovezik C, Karakostas D, Kiayias A et a...

work page Pith review arXiv 2025
[108]

In: Information and Communications Security: 12th International Conference, ICICS 2010, Barcelona, Spain, 15 -17 December 2010, Proceedings

Broadway, Suite 305, Nashville, TN 37203: American Economic Association Langer L, Jonker H, Pieters W (2010) Anonymity and verifiability in voting: understanding (un) linkability. In: Information and Communications Security: 12th International Conference, ICICS 2010, Barcelona, Spain, 15 -17 December 2010, Proceedings

work page 2010

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Akinbohun S, Apeh S, Olaye EO et al (2023) Literature Review of Blockchain -Based Voting System: Framework and Concept. Journal of Civil and Environmental Systems Engineering 20(1):72-83 Benjamin D, Heffetz O, Kimball M et al (2017) The relationship between the normalized gradient addition mechanism and quadratic voting. Public choice 172 (2017):233-263 B...

arXiv 2023

[2] [12]

SoK: Measuring Blockchain Decentralization

Springer Berlin Heidelberg Lei T, Zhang Q, Qiu W et al (2025) An enhanced DPoS consensus mechanism using quadratic voting in Web 3.0 ecosystem. Blockchain, 3(1), 1-2 Miller J, Weyl EG, Erichsen L (2022) Beyond Collusion Resistance: Leveraging Social Information for Plural Funding and Voting. Available at SSRN 4311507 Ovezik C, Karakostas D, Kiayias A et a...

work page Pith review arXiv 2025

[3] [108]

In: Information and Communications Security: 12th International Conference, ICICS 2010, Barcelona, Spain, 15 -17 December 2010, Proceedings

Broadway, Suite 305, Nashville, TN 37203: American Economic Association Langer L, Jonker H, Pieters W (2010) Anonymity and verifiability in voting: understanding (un) linkability. In: Information and Communications Security: 12th International Conference, ICICS 2010, Barcelona, Spain, 15 -17 December 2010, Proceedings

work page 2010