REVIEW 1 major objections 4 minor 3 references
How Many Votes is a Lie Worth? Measuring Strategyproofness through Resource Augmentation
T0 review · 1 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash
Pith's one-line read This paper measures how many truthful extra ballots outweigh one lie, proves Borda Count needs only m-2 (independent of voter count), and shows that in large electorates it outperforms every other positional scoring rule and every anonymous
desk verdict Genuinely new manipulation measure and mostly careful proofs, but Theorem 4.13 has a real WLOG gap and the abstract oversells. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the manipulation potential MP(f), the smallest k such that f is k'-ASP for every k' >= k. k-augmentation strategyproofness (k-ASP) requires that for every profile, every voter weakly prefers the winner after adding k truthful copies of her own ballot to the winner of any misreport. The paper's lower-bound proofs build worst-case profiles — often three equal-sized groups that form a Condorcet cycle, or a scoring gap between two candidates — in which a lie moves the winner to the voter's second choice while truthful copies cannot help until a majority or a score threshold is reached; upper-bound proofs show that at the threshold the voter's top choice becomes unbeatable.
What would settle it
For n=5, m=3, enumerate all anonymous, biranking-majority-consistent single-valued voting rules and compute MP for each. If any such rule has MP ≤ 1, the theorem's lower bound ceil((n-1)/2)=2 is false. The 'WLOG group 1' step is the only gap, so a non-neutral counterexample found this way would directly falsify the claimed theorem.
Extended reading notes
Core claim
The central discovery is a sharp separation under the new worst-case measure. Borda Count's manipulation potential is exactly m-2, constant in the electorate size; Plurality's is ceil((n-1)/2); Instant Runoff, Plurality with Runoff, and Black's rule all reach n-1; and any neutral, majoritarian biranking-majority-consistent rule has potential at least n-2. The paper further proves that every positional scoring rule other than Plurality and Borda Count has potential either strictly larger than m-2 or at least ceil((n-1)/2), and that every anonymous biranking-majority-consistent SCF has potential at least (n-1)/2 when n is odd. Thus, when voters outnumber candidates, Borda Count is the unique b
Load-bearing premise
The load-bearing premise is the 'WLOG, say this is Group 1' step in the proof of Theorem 4.13: treating the three cyclic voter groups symmetrically requires a candidate-neutrality that anonymity and biranking-majority-consistency do not imply, and without it the lower bound for all anonymous rules of that class is not established.
Editorial extensions
If this is right
- Borda Count's manipulation potential is exactly m-2, so in large electorates it is lower than Plurality's ceil((n-1)/2), Instant Runoff's n-1, Plurality-with-Runoff's n-1, Black's rule's n-1, and Maximin's roughly (m-2)n/(m-1).
- No positional scoring rule outside {Plurality, Borda Count} can do better than both: for every such rule the manipulation potential is either strictly greater than m-2 or at least ceil((n-1)/2).
- When n is odd, every anonymous biranking-majority-consistent SCF — a class containing every anonymous Condorcet extension — has manipulation potential at least (n-1)/2, so it cannot beat Plurality; the paper leaves even n open.
- Every neutral, majoritarian biranking-majority-consistent rule has manipulation potential at least n-2, which is near the worst possible value n-1.
- Any positional scoring rule with equal top-two weights s1=s2 has infinite manipulation potential: no finite number of truthful copies can neutralize a lie.
Reading between the lines
- The same resource-augmentation measure transfers to approval-based multiwinner rules, where proportionality axioms conflict with strategyproofness; a plausible outcome is that strong proportionality forces large or infinite manipulation potential, mirroring the Condorcet-extension results here.
- The odd-n restriction in Theorem 4.13 is likely removable. Because the paper proves the analogous bound for rules that pick weak Condorcet winners, a natural conjecture is that all anonymous Condorcet extensions satisfy MP >= (n-1)/2 for even n as well; a counterexample for even n would be a meaningful discovery.
- Manipulation potential is a worst-case exchange rate between lying and recruiting like-minded voters. Embedding it in a cost model — where finding voters and computing lies both have prices — could turn the ranking into an operational recommendation for a voter with a fixed budget; the paper sketches this budgeted setting as future work.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a new measure of the manipulability of social choice functions, the manipulation potential MP(f), defined via resource augmentation. A rule is k-augmentation strategyproof if no voter ever strictly prefers the outcome of a misreport to the outcome obtained by adding k truthful copies of her own vote to the original profile; MP(f) is the smallest k such that this holds for all larger k as well. The authors compute or bound MP for Plurality, IRV, PWR, Borda, Black's rule, Maximin, all positional scoring rules, and broad classes of Condorcet extensions. The headline results are that among positional scoring rules the minimum MP is attained by either Borda (m−2, independent of n) or Plurality (≈ n/2), and that under a weak majority-consistency axiom no Condorcet extension can beat Plurality (MP ≥ (n−1)/2 for odd n). The paper includes detailed appendix proofs and a discussion of interpretations and open problems.
Significance. If the main theorems hold, the paper makes a genuine contribution to the quantitative study of manipulability. The resource-augmentation angle is natural and connects voting theory to competition-complexity ideas from mechanism design. The results are broad, the proofs are mostly detailed and self-contained, and the separation between Borda and Condorcet-consistent rules is crisp and interesting. The paper is also honest about limitations (e.g., the dependence on parity in Theorem 4.13 and the open even-n case). The one load-bearing issue is a gap in the proof of Theorem 4.13, which is likely repairable but is not repaired in the manuscript as written.
major comments (1)
- [Section 4.3.3, Theorem 4.13] The proof's step 'WLOG, say this is Group 1' is not justified by the stated assumptions. The three groups in Profile 1 are related by a cyclic permutation of the candidate labels, but the theorem assumes only anonymity and biranking-majority-consistency, neither of which implies neutrality under cyclic relabeling. An anonymous biranking-majority-consistent rule may, for example, always select c1 on this profile; then the losing group is Group 2, not Group 1. For a Group 1 voter, c1 is the top choice, so the manipulation used in the proof (report the Group 2 ranking) yields c2, which is worse than c1, and the argument collapses. The gap is repairable by a three-case analysis: if the losing group is Group j, take a voter from that group as the manipulator in the reduced n-voter profile and have her report the ranking of the next group, making that group's top choice a strict-majority winne
minor comments (4)
- [Abstract / Section 4.3.3] The abstract states that 'any rule satisfying a weak form of majority consistency ... cannot outperform Plurality,' but Theorem 4.13 is proved only for anonymous rules and odd n. Remark 4.14 acknowledges the parity issue, but the abstract should also state the anonymity assumption or justify why it can be dropped.
- [Section 2 / Definition 2] The definition of MP(f) does not explicitly quantify the initial electorate size n (and candidate set size m), although all theorems state MP as a function of n and m. The text should say that n and m are fixed parameters of the instance class.
- [Appendix A.5, proof of Proposition A.2] In the definition of b_k[c_m], the text reads 'b_k[c_m] := min_{c∈C\{c_{m-1}}} M_k[c_{m-1}, c]'; the second subscript should be c_m, not c_{m-1}.
- [Throughout] Expressions such as 'n−1/4 voters' in Example 1.1 and Theorem 4.3 should be typeset as (n−1)/4 to avoid ambiguity.
Circularity Check
No circularity: the manipulation potential is a new definition with self-contained proofs; the Theorem 4.13 'WLOG' gap is a proof-completeness issue, not a circular reduction.
full rationale
MP(f) and k-ASP are introduced as new definitions, and the paper's results are derived by constructing explicit profiles and proving score or margin inequalities; no parameter is fitted to data and no quantity called a prediction is recovered from a fitted input. The paper's self-citations (e.g., Hartman et al. 2025, Berker et al. 2025, Conitzer et al. 2009) appear as background, tie-breaking conventions, or future-work pointers, and none carries the central claims. The only load-bearing concern I found is a correctness gap in Theorem 4.13, not circularity: the proof says 'at least one out of the three groups of voters will get neither their first nor their second choice as the winner. WLOG, say this is Group 1,' but the theorem assumes only anonymity and biranking-majority-consistency, not neutrality, so for a non-neutral rule the three cyclic groups need not be interchangeable. This may leave a case uncovered unless repaired by casework, but it does not make the claimed bound equivalent to an input or to a prior result by construction. Since no circular step is exhibited, the score is 0.
Assumptions & free parameters
assumptions (6)
- domain assumption Voter preferences are strict total orders over candidates and SCFs are defined on all profiles with any number of voters.
- domain assumption For set-valued SCCs, the paper uses a linear tie-breaker and requires k-ASP for all tie-breakers.
- domain assumption All analyzed SCCs are anonymous and neutral.
- standard math McGarvey's theorem (1953) is used to construct profiles realizing any sign-consistent margin matrix.
- standard math Gibbard–Satterthwaite theorem is used to handle the k=0 base case in Theorem 4.12.
- ad hoc to paper In Theorem 4.13, the proof assumes (without stating) that the rule's output is invariant under cyclic relabeling of the three candidates (neutrality).
Cite this review
Pith. "Pith review of How Many Votes is a Lie Worth? Measuring Strategyproofness through Resource Augmentation." pith.science (2026). https://pith.science/paper/AZGNE55U
@misc{pith2026260222838,
author = {Pith},
title = {Pith review of: How Many Votes is a Lie Worth? Measuring Strategyproofness through Resource Augmentation},
year = {2026},
howpublished = {\url{https://pith.science/paper/AZGNE55U}},
note = {Machine review of arXiv:2602.22838}
}
read the original abstract
It is well known, by the Gibbard-Satterthwaite Theorem, that when there are more than two candidates, any non-dictatorial voting rule can be manipulated by untruthful voters. But how much stronger is an untruthful voter over a truthful one under different voting rules? We suggest measuring the potential comparative advantage of a strategic voter by asking how many copies of their (truthful) vote must be added to the election in order to achieve an outcome as good as their best manipulation. Intuitively, this definition quantifies what a voter can gain by manipulating in comparison to what they would have gained by finding like-minded voters to join the election (equivalently, by increasing their voter weight). The higher the former is, the more incentive a voter will have to manipulate, even when it is computationally costly. Using this framework, we obtain a principled method to measure and compare the manipulation potential for different voting rules. We analyze and report this potential for well-known and broad classes of social choice functions. In particular, we show that the positional scoring rule with the smallest manipulation potential will always be either Borda Count (if the number of voters outweighs the number of candidates) or Plurality (vice versa). Further, we prove that any rule satisfying a weak form of majority consistency (and therefore any Condorcet extension) cannot outperform Plurality, and that any majoritarian Condorcet rule will perform significantly worse. Consequently, out of the voting rules we analyze, Borda Count stands out as the only one with a manipulation potential that does not grow with the number of voters. By establishing a clear separation between different rules in terms of manipulation potential, our work paves the way for the search for rules that provide voters with minimal advantage from manipulating.
Figures
Reference graph
Works this paper leans on
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[2]
As shown by McGarvey [1953], we can indeed construct a profile≻ −S that is consistent with these margins
For all other voters (N\S), we will first specify the margin matrixM(≻ −S). As shown by McGarvey [1953], we can indeed construct a profile≻ −S that is consistent with these margins. First, we specify the margins among the first five candidates: 1 =M(≻ −S)[c1, c3] =M(≻ −S)[c1, c4] =M(≻ −S)[c1, c5] =M(≻ −S)[c2, c1] =M(≻ −S)[c2, c5] =M(≻ −S)[c3, c2] =M(≻ −S)...
1953
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Hannaneh Akrami, Alon Eden, Michal Feldman, Amos Fiat, and Yoav Gal-Tzur
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Reviewed August 2, 2026 · model on record in the stance chip above.
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