pith. sign in

arxiv: 2605.19264 · v1 · pith:AA73L6BOnew · submitted 2026-05-19 · 💻 cs.AI · cs.MA

Swimming with Whales: Analysis of Power Imbalances in Stake-Weighted Governance

Yuzhe Zhang , Manvir Schneider , Qin Wang , Davide Grossi This is my paper

Pith reviewed 2026-05-20 06:10 UTC · model grok-4.3

classification 💻 cs.AI cs.MA
keywords stake-weighted votingpower imbalancesPenrose-Banzhaf indexProof-of-Stakeblockchain governanceProject Catalystpower index
0
0 comments X

The pith

Stake-weighted voting cannot perfectly align power with ownership but can approximate it in expectation under specific conditions.

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

The paper analyzes stake-weighted voting in Proof-of-Stake blockchains and shows that large stakeholders can dominate decisions even without owning most of the total stake. It quantifies this using the Penrose-Banzhaf power index and proves analytically that exact matching between power and relative stake is impossible in general. The work then identifies conditions under which power aligns with stake ownership in expectation and tests the approach on real governance data from Project Catalyst. Readers care because these imbalances affect who controls upgrades, funding, and rules in decentralized networks.

Core claim

While a perfect alignment between power and relative stake ownership is generally unattainable, it can be approximated in expectation under specific conditions.

What carries the argument

Penrose-Banzhaf power index applied to stake-weighted voting.

Load-bearing premise

The Penrose-Banzhaf index correctly measures power in stake-weighted blockchain votes and the conditions for expected approximation are achievable in real systems.

What would settle it

A real stake-weighted voting system that satisfies the paper's specific conditions yet shows power deviating from expected stake alignment, or where observed voting outcomes contradict Penrose-Banzhaf predictions.

Figures

Figures reproduced from arXiv: 2605.19264 by Davide Grossi, Manvir Schneider, Qin Wang, Yuzhe Zhang.

Figure 1
Figure 1. Figure 1: Means of the power-stake ratios and single-agent variances of agent [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Power-stake ratio means and within-vector variances for [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Aggregate behavior of normalized power–stake ratios under varying quotas and stake distributions. [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Mean and variance of the first agent’s normalized power–stake ratio across quotas for [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Mean and variance of the first agent’s normalized power–stake ratio across quotas for [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
read the original abstract

Voting methods weighted by stakes are the fundamental governance paradigm in Proof-of-Stake (PoS) blockchains. Such a paradigm is known to be prone to power distortions: a few users possessing large stakes may completely control decision making, even without owning the totality of the stakes. We study this phenomenon through the lens of computational social choice, focusing on the extent of power imbalances in stake-weighted voting when power is quantified using the Penrose-Banzhaf power index. Our work presents both analytical and empirical contributions. Analytically, we demonstrate that while a perfect alignment between power and relative stake ownership is generally unattainable, it can be approximated in expectation under specific conditions. Empirically, using data from a real-world on-chain governance system (Project Catalyst), we provide a more fine-grained understanding of the power imbalances that are likely to occur in current stake-weighted governance systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper analyzes power imbalances in stake-weighted voting for Proof-of-Stake blockchain governance using the Penrose-Banzhaf power index. It presents an analytical result claiming that perfect alignment between power and relative stake ownership is generally unattainable yet can be approximated in expectation under specific conditions, alongside an empirical analysis of imbalances using data from the Project Catalyst on-chain governance system.

Significance. If the analytical approximation result holds under well-specified conditions and the empirical findings are robust, the work would offer a useful bridge between computational social choice theory and practical blockchain governance design, highlighting when stake-weighted systems can mitigate whale dominance. The combination of theoretical analysis with real-world data strengthens its potential contribution to the field.

major comments (2)
  1. [Analytical contribution paragraph] Analytical contribution (abstract and corresponding section): The claim that perfect power-stake alignment is unattainable but approximable in expectation under specific conditions is load-bearing for the paper's central thesis, yet the abstract provides no equations, precise statement of those conditions, or derivation details. This makes it impossible to verify whether the result relies on the Penrose-Banzhaf index's independence and equiprobability assumptions, which the skeptic correctly notes are unlikely to hold given correlated whale behavior, low turnout, and related-proposal voting patterns in Project Catalyst data.
  2. [Empirical analysis] Empirical analysis section: The transfer of any analytical approximation guarantee to the Project Catalyst dataset is undermined if the specific conditions require the standard Banzhaf independence assumption, as real on-chain voting exhibits coordination among large stakeholders and turnout far below 50%. This directly affects the paper's claim of providing a fine-grained understanding of likely power imbalances.
minor comments (1)
  1. [Abstract] The abstract would benefit from a brief parenthetical statement of the specific conditions or a reference to the relevant theorem/equation to improve readability for readers unfamiliar with power index derivations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address each major comment below, clarifying the separation between our analytical and empirical contributions and committing to revisions that improve precision and transparency without altering the core claims.

read point-by-point responses

  1. Referee: [Analytical contribution paragraph] Analytical contribution (abstract and corresponding section): The claim that perfect power-stake alignment is unattainable but approximable in expectation under specific conditions is load-bearing for the paper's central thesis, yet the abstract provides no equations, precise statement of those conditions, or derivation details. This makes it impossible to verify whether the result relies on the Penrose-Banzhaf index's independence and equiprobability assumptions, which the skeptic correctly notes are unlikely to hold given correlated whale behavior, low turnout, and related-proposal voting patterns in Project Catalyst data.

    Authors: We agree that the abstract is too concise and should better indicate the scope of the result. The analytical claim is derived under the standard Penrose-Banzhaf assumptions of vote independence and equiprobable coalitions; we prove that exact alignment between the power index and relative stake is impossible for any finite voter set with heterogeneous stakes, yet the expected value of the index equals relative stake when the number of voters tends to infinity or under specific stake-distribution conditions that make small-stake voters dominant in expectation. The full derivation appears in Section 3. We will revise the abstract to reference the assumptions explicitly, add a short derivation sketch and statement of conditions to the introduction, and include a new paragraph discussing how correlated voting and low turnout (as observed in the data) limit the practical applicability of the approximation. revision: yes

  2. Referee: [Empirical analysis] Empirical analysis section: The transfer of any analytical approximation guarantee to the Project Catalyst dataset is undermined if the specific conditions require the standard Banzhaf independence assumption, as real on-chain voting exhibits coordination among large stakeholders and turnout far below 50%. This directly affects the paper's claim of providing a fine-grained understanding of likely power imbalances.

    Authors: The empirical section does not transfer or rely on the analytical approximation guarantee. We apply the Penrose-Banzhaf index directly to the recorded votes in the Project Catalyst dataset to compute realized power shares and compare them against stake proportions, thereby quantifying observed imbalances under actual voting behavior. The theoretical result is presented separately as a benchmark under idealized conditions. We will revise the manuscript to state this separation more explicitly, add discussion of how coordination and sub-50% turnout affect the computed index values, and include sensitivity checks that relax the independence assumption where feasible with the available data. revision: yes

Circularity Check

0 steps flagged

No circularity: analytical result derived from standard external power index

full rationale

The paper's core analytical claim—that perfect alignment between power and relative stake is generally unattainable yet approximable in expectation under specific conditions—is presented as a mathematical demonstration using the Penrose-Banzhaf index, a pre-existing tool from computational social choice theory. No equations or steps in the provided abstract or description reduce this result to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain. The empirical component draws on independent Project Catalyst data for validation rather than circularly confirming the analytical approximation. The derivation remains self-contained against external benchmarks, with the independence assumption of the index treated as a modeling choice rather than an output derived from the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on the standard definition of the Penrose-Banzhaf index and the modeling choice of stake-weighted voting as the governance mechanism; no free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption Penrose-Banzhaf power index is a valid measure of voting power in weighted voting systems
    Invoked as the lens for quantifying power imbalances throughout the abstract.
  • domain assumption Stake-weighted voting is the fundamental governance paradigm in PoS blockchains
    Stated as background for the power distortion phenomenon.

pith-pipeline@v0.9.0 · 5683 in / 1198 out tokens · 32492 ms · 2026-05-20T06:10:51.947794+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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

22 extracted references · 22 canonical work pages · 1 internal anchor

  1. [1]

    1948.Handbook of mathematical functions with formulas, graphs, and mathematical tables

    Milton Abramowitz and Irene A Stegun. 1948.Handbook of mathematical functions with formulas, graphs, and mathematical tables. Vol. 55. US Government printing office

    work page 1948

  2. [2]

    Noga Alon and Paul H Edelman. 2010. The inverse Banzhaf problem.Social Choice and Welfare34, 3 (2010), 371–377

    work page 2010

  3. [3]

    Yoram Bachrach, Evangelos Markakis, Ezra Resnick, Ariel D Procaccia, Jeffrey S Rosenschein, and Amin Saberi. 2010. Approximating power indices: theoretical and empirical analysis.Autonomous Agents and Multi-Agent Systems20, 2 (2010), 105–122

    work page 2010

  4. [4]

    2016.Handbook of computational social choice

    Felix Brandt, Vincent Conitzer, Ulle Endriss, Jérôme Lang, and Ariel D Procaccia. 2016.Handbook of computational social choice. Cambridge University Press

    work page 2016

  5. [5]

    2022.Computational aspects of cooperative game theory

    Georgios Chalkiadakis, Edith Elkind, and Michael Wooldridge. 2022.Computational aspects of cooperative game theory. Springer Nature

    work page 2022

  6. [6]

    Giulia Fanti, Leonid Kogan, Sewoong Oh, Kathleen Ruan, Pramod Viswanath, and Gerui Wang. 2019. Compounding of wealth in proof-of-stake cryptocurrencies. InInternational Conference on Financial Cryptography and Data Security (FC). Springer, 42–61

    work page 2019

  7. [7]

    Mohak Goyal, Sukolsak Sakshuwong, Sahasrajit Sarmasarkar, and Ashish Goel. 2023. Low Sample Complexity Participatory Budgeting. InInternational Colloquium on Automata, Languages, and Programming (ICALP), Vol. 261. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, 70

    work page 2023

  8. [8]

    Davide Grossi. 2022. Social Choice Around the Block: On the Computational Social Choice of Blockchain. In International Conference on Autonomous Agents and Multiagent Systems (AAMAS). 1788–1793

    work page 2022

  9. [9]

    Artyom Jelnov and Yair Tauman. 2014. Voting power and proportional representation of voters.International Journal of Game Theory43, 4 (2014), 747–766

    work page 2014

  10. [10]

    1995.Continuous univariate distributions, volume

    Norman L Johnson, Samuel Kotz, and Narayanaswamy Balakrishnan. 1995.Continuous univariate distributions, volume

    work page 1995

  11. [11]

    Vol. 2. John wiley & sons

    work page

  12. [12]

    Aggelos Kiayias and Philip Lazos. 2022. SoK: blockchain governance. InProceedings of the 4th ACM Conference on Advances in Financial Technologies. 61–73

    work page 2022

  13. [13]

    Sascha Kurz. 2012. On the inverse power index problem.Optimization61, 8 (2012), 989–1011

    work page 2012

  14. [14]

    Steven P Lalley and E Glen Weyl. 2018. Quadratic voting: How mechanism design can radicalize democracy. InAEA Papers and Proceedings, Vol. 108. American Economic Association 2014 Broadway, Suite 305, Nashville, TN 37203, 33–37

    work page 2018

  15. [15]

    Lalley and E

    Steven P. Lalley and E. Glen Weyl. 2018. Quadratic Voting: How Mechanism Design Can Radicalize Democracy.AEA Papers and Proceedings108 (May 2018), 33–37. https://doi.org/10.1257/pandp.20181002

    work page doi:10.1257/pandp.20181002 2018

  16. [16]

    Maaike Los, Zoé Christoff, and Davide Grossi. 2022. Proportional Budget Allocations: Towards a Systematization. In International Joint Conference on Artificial Intelligence (IJCAI). 398–404

    work page 2022

  17. [17]

    Lionel S Penrose. 1946. The elementary statistics of majority voting.Journal of the Royal Statistical Society109, 1 (1946), 53–57

    work page 1946

  18. [18]

    Dominik Peters, Grzegorz Pierczyński, and Piotr Skowron. 2021. Proportional participatory budgeting with additive utilities.Advances in Neural Information Processing Systems34 (2021), 12726–12737

    work page 2021

  19. [19]

    Simon Rey and Jan Maly. 2023. The (computational) social choice take on indivisible participatory budgeting.arXiv preprint arXiv:2303.00621(2023)

    work page arXiv 2023

  20. [20]

    Jayaram Sethuraman. 1994. A constructive definition of Dirichlet priors.Statistica sinica(1994), 639–650

    work page 1994

  21. [21]

    Wojciech Slomczynski and Karol Zyczkowski. 2006. Penrose voting system and optimal quota.arXiv preprint physics/0610271(2006)

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  22. [22]

    Yuzhe Zhang and Davide Grossi. 2021. Power in liquid democracy. InProceedings of the AAAI Conference on Artificial Intelligence (AAAI), Vol. 35. 5822–5830. 22 Zhang et al. 0 1 2 3 4Mean Ratio (Unnormalized Banzhaf / Normalized Weight) Agent 1 Mean Ratio vs Quota Gamma(alpha=1, theta=0.5) n=30 n=40 n=60 n=80 n=150 0.0 0.2 0.4 0.6 0.8 1.0 Quota 0.00 0.05 0....

    work page 2021