Concave is the New Linear: The Impossibility of Anti-Plutocratic DAO Governance
Pith reviewed 2026-05-20 01:10 UTC · model grok-4.3
The pith
No wallet-balance voting rule prevents Sybil attacks from making power linear in token holdings on permissionless blockchains.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We prove that no voting rule that derives power solely from wallet balance can succeed on a permissionless blockchain. Through a costed model of on-chain voting that captures realistic blockchain frictions including per-wallet splitting and voting costs, fixed setup costs, and minimum-balance requirements, we show that whenever a wallet of any size yields nonzero voting power, a Sybil attacker who splits tokens across many wallets achieves total voting power that grows at least linearly in their token holdings. For concave rules actually proposed to dampen governance power those that are positive, increasing, and finite we show that the optimal strategy yields power that is asymptotically线性.
What carries the argument
The costed model of on-chain voting that incorporates per-wallet splitting costs, voting costs, fixed setup costs, and minimum-balance requirements, forcing any nonzero-power rule to admit linear total power under optimal splitting.
If this is right
- Attack costs are orders of magnitude below the value at stake when the model is instantiated on real DAOs.
- Sybil amplification factors range from 1,172 times to 4,039 times under quadratic voting and exceed 229,000 times under steeper power rules.
- Replaying the ten most recent proposals of five major DAOs shows the same linearization pattern under linear, quadratic, logarithmic, and power voting.
Where Pith is reading between the lines
- Protocols relying on these rules face the practical risk that governance can be captured at low cost by coordinated token splitting.
- This result points toward the need for governance designs that incorporate elements beyond pure on-chain wallet balance to limit Sybil strategies.
Load-bearing premise
The costed model of on-chain voting accurately represents attacker capabilities and protocol constraints in permissionless settings.
What would settle it
An experiment on a live DAO using quadratic voting that measures total voting power obtained by splitting a fixed token amount across increasing numbers of wallets and finds the resulting power remains substantially sublinear after all modeled costs.
read the original abstract
Decentralized Autonomous Organizations (DAOs) run protocol governance by letting token holders vote on proposals. The dominant rule, voting power proportional to wallet balance, concentrates control among a small number of large holders, fueling the token-control governance attacks that have already compromised real protocols. To counter this concentration, the community has turned to anti-plutocratic voting mechanisms such as Quadratic Voting (QV), which assign sublinear voting power per token with the goal of dampening the influence of large holders. We prove that no voting rule that derives power solely from wallet balance can succeed on a permissionless blockchain. Through a costed model of on-chain voting that captures realistic blockchain frictions -- including per-wallet splitting and voting costs, fixed setup costs, and minimum-balance requirements -- we show that whenever a wallet of any size yields nonzero voting power, a Sybil attacker who splits tokens across many wallets achieves total voting power that grows at least linearly in their token holdings. For concave rules actually proposed to dampen governance power -- those that are positive, increasing, and finite -- we show that the optimal strategy yields power that is asymptotically linear in token holdings, regardless of the cost scheme. Instantiating the model on real DAOs reveals attack costs orders of magnitude below the value at stake. Replaying the ten most recent finalized proposals of five major DAOs (ENS, Compound, Uniswap, Arbitrum, and ZKsync) under linear, quadratic, logarithmic, and power-($\beta = 0.25$) voting, we measure Sybil amplification factors between $1,172\times$ and $4,039\times$ under Quadratic Voting, and exceeding $229,000\times$ under steeper power rules.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that no voting rule deriving power solely from wallet balance can achieve anti-plutocratic governance on permissionless blockchains. Through a costed model incorporating per-wallet splitting/voting costs, fixed setup costs, and minimum-balance requirements, it shows that any nonzero-power wallet allows a Sybil attacker to achieve total power growing at least linearly in holdings; for positive, increasing, finite concave rules (e.g., quadratic, logarithmic, power-β), the optimal split yields asymptotically linear power independent of the specific cost scheme. Empirical replay of the ten most recent proposals from ENS, Compound, Uniswap, Arbitrum, and ZKsync quantifies Sybil amplification factors of 1,172×–4,039× under quadratic voting and >229,000× under steeper power rules.
Significance. If the central result holds, it substantially weakens the case for balance-based concave mechanisms in DAO governance and shifts attention toward identity, reputation, or off-chain solutions. The explicit cost model grounded in blockchain frictions, the parameter-free asymptotic linearity claim for the stated class of concave functions, and the reproducible empirical replay on real proposals are notable strengths that increase the work's practical impact in algorithmic game theory and decentralized systems.
major comments (2)
- [§4] §4 (proof of asymptotic linearity): the argument that n·f(t/n) approaches a linear function of t for any positive increasing finite concave f requires that costs remain sublinear in n (or are offset by the minimum-balance cap) so the attacker can reach the large-n regime; the model does not explicitly derive or bound the regime in which this holds when fixed setup costs or per-wallet minimums are non-negligible relative to t.
- [Model section] Model section (cost function definition, likely Eq. (3)–(5)): the claim that the linearity result is independent of the cost scheme assumes additive, wallet-independent costs; if realistic on-chain effects (gas-price impact from sequential transactions, nonce limits, or per-address proposal thresholds) introduce superlinear scaling, the optimal strategy may be forced to fewer wallets, breaking the guarantee for some concave f.
minor comments (2)
- [Abstract and §5] Abstract and §5: the reported amplification factors would benefit from a brief sensitivity table showing how results change under modest variations in the per-wallet cost parameter.
- [Empirical section] Figure captions (empirical section): legends distinguishing linear vs. quadratic vs. power-β curves are difficult to read at print size; increasing font size or adding a table of exact factors would improve clarity.
Simulated Author's Rebuttal
We thank the referee for the thoughtful and detailed report. The comments correctly identify points where the presentation of the cost model and asymptotic regime can be clarified. We address each major comment below and indicate the revisions we will make.
read point-by-point responses
-
Referee: [§4] §4 (proof of asymptotic linearity): the argument that n·f(t/n) approaches a linear function of t for any positive increasing finite concave f requires that costs remain sublinear in n (or are offset by the minimum-balance cap) so the attacker can reach the large-n regime; the model does not explicitly derive or bound the regime in which this holds when fixed setup costs or per-wallet minimums are non-negligible relative to t.
Authors: We agree that an explicit bound on the regime is useful. In the model, both fixed setup costs and per-wallet minimum-balance requirements are independent of n. Consequently, total cost grows at most linearly with n. For any fixed cost parameters, there therefore exists a finite threshold T such that for total holdings t > T the attacker’s optimal n is large enough for n f(t/n) to lie within any prescribed ε of its linear limit. We will insert a short lemma in the revised §4 that derives this threshold explicitly in terms of the cost parameters and the concavity modulus of f. This makes the large-n regime rigorous without changing the statement of the main theorem. revision: yes
-
Referee: [Model section] Model section (cost function definition, likely Eq. (3)–(5)): the claim that the linearity result is independent of the cost scheme assumes additive, wallet-independent costs; if realistic on-chain effects (gas-price impact from sequential transactions, nonce limits, or per-address proposal thresholds) introduce superlinear scaling, the optimal strategy may be forced to fewer wallets, breaking the guarantee for some concave f.
Authors: The model treats costs as additive across independent wallets, which matches the permissionless setting in which each address can be funded and used separately. The independence claim is therefore with respect to the functional form of the per-wallet cost, not with respect to arbitrary n-dependent cost functions. If superlinear effects (e.g., gas-price impact) are present, the attacker simply optimizes over a smaller but still positive n; for every concave f in the class we consider, even a bounded n yields total power that is a positive fraction of linear in t. We will add a clarifying paragraph in the model section that states this assumption explicitly and notes the robustness under moderate superlinearity. A short sensitivity discussion will also be appended to the empirical section. revision: partial
Circularity Check
No significant circularity; derivation is self-contained within stated cost model
full rationale
The paper defines a costed model of on-chain voting (per-wallet splitting/voting costs, fixed setup costs, minimum-balance requirements) as an explicit external assumption representing blockchain frictions. From this model it mathematically derives that any positive, increasing, finite concave voting rule yields asymptotically linear total power under optimal Sybil splitting, and that no wallet-balance-only rule can resist such attacks on permissionless chains. These steps follow directly from analyzing the function properties and cost sublinearity without reducing to fitted parameters, self-definitions, or load-bearing self-citations. The result is independent of the specific concave form chosen and does not rename known empirical patterns or smuggle ansatzes via prior work. The derivation therefore remains self-contained against the model's assumptions.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Voting power is derived solely from wallet balance with no external identity or reputation signals.
- domain assumption Attackers can create and operate arbitrarily many wallets at costs that include per-wallet splitting, voting, fixed setup, and minimum-balance requirements.
Lean theorems connected to this paper
-
IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
For concave rules ... positive, increasing, and finite ... optimal strategy yields power that is asymptotically linear in token holdings, regardless of the cost scheme (Theorem 10)
-
IndisputableMonolith/Foundation/AlphaCoordinateFixation.leanJ_uniquely_calibrated_via_higher_derivative unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
V*(W) = n f( (a-p)/n - v-s ) ... κ = sup g(x) with g(x)=f(x)/(x+c)
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Reference graph
Works this paper leans on
- [1]
- [2]
- [3]
-
[4]
Bennett, A. , title =. 2025 , type =. doi:10.25740/hj860vc2584 , url =
- [5]
-
[6]
Transaction Details , howpublished =
-
[7]
Proceedings of the 2023 Workshop on Decentralized Finance and Security (DeFi '23) , year =
Dotan, Maya and Yaish, Aviv and Yin, Hsin-Chu and Tsytkin, Eytan and Zohar, Aviv , title =. Proceedings of the 2023 Workshop on Decentralized Finance and Security (DeFi '23) , year =
work page 2023
-
[8]
Behnke, Rob , title =
-
[9]
Abrams, Zack , title =
-
[10]
6th Conference on Advances in Financial Technologies (AFT 2024) , year =
Feichtinger, Rainer and Fritsch, Robin and Heimbach, Lioba and Vonlanthen, Yann and Wattenhofer, Roger , title =. 6th Conference on Advances in Financial Technologies (AFT 2024) , year =
work page 2024
-
[11]
Zhou, Zibo and Zhang, Zongyang and Hao, Feng and Zheng, Bowen and Masyhur, Zulkarnaim , title =. 2025 , isbn =. doi:10.1145/3719027.3744810 , booktitle =
- [12]
- [13]
-
[14]
arXiv preprint arXiv:2407.10945 , year =
Blockchain Governance: An Empirical Analysis of User Engagement on DAOs , author =. arXiv preprint arXiv:2407.10945 , year =. doi:10.48550/arXiv.2407.10945 , url =
-
[15]
Louis Kaplow and Scott Duke Kominers , title =. Public Choice , year =. doi:10.1007/s11127-017-0412-5 , url =
- [16]
-
[17]
Mohan, Vijay and Khezr, Peyman and Berg, Chris , title =. Management Science , volume =. 2024 , doi =
work page 2024
- [18]
- [19]
- [20]
- [21]
- [22]
- [23]
- [24]
- [25]
- [26]
- [27]
- [28]
-
[29]
Baby Doge , author =
- [30]
-
[31]
veBAL Tokenomics and Governance , author =. 2026 , howpublished =
work page 2026
-
[32]
Minaei, Mohsen and Moreno-Sanchez, Pedro and Fang, Zhiyong and Raghuraman, Srinivasan and Alamati, Navid and Chatzigiannis, Panagiotis and Kumaresan, Ranjit and Le, Duc V. , title =. 2025 , isbn =. doi:10.1145/3708821.3736221 , booktitle =
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.