Strategic Voting Under Uncertainty About the Voting Method

Berkeley); Eric Pacuit (University of Maryland); Wesley H. Holliday (University of California

arxiv: 1907.09110 · v1 · pith:VVSQHA24new · submitted 2019-07-22 · 💻 cs.MA · cs.GT

Strategic Voting Under Uncertainty About the Voting Method

Wesley H. Holliday (University of California , Berkeley) , Eric Pacuit (University of Maryland) This is my paper

Pith reviewed 2026-05-24 18:05 UTC · model grok-4.3

classification 💻 cs.MA cs.GT

keywords strategic votingmanipulabilityuncertaintyvoting methodselection designcomputational social choicepreference misrepresentation

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The pith

Uncertainty about which voting method will be used can reduce or eliminate a voter's incentive to misrepresent preferences in some scenarios.

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

The paper defines three notions of manipulability—sure, safe, and expected—for a collection of voting methods, each capturing how uncertainty over the method affects whether a voter can benefit from reporting false preferences. Using a computer search over voting scenarios, it locates cases where a voter who would have an incentive to manipulate under each individual method loses that incentive once the method is unknown. A sympathetic reader cares because the result points to a possible design choice for elections: leaving the exact method uncertain might promote more sincere voting without altering any single rule. The authors note that this holds for certain preference profiles and method sets identified computationally.

Core claim

For certain voter preference profiles and sets of voting methods, uncertainty over which method will be applied can make a voter have no sure, safe, or expected incentive to misrepresent preferences, even though the voter would possess such an incentive if the method were known with certainty.

What carries the argument

The three notions of manipulability (sure, safe, and expected) defined for a set of voting methods, which formalize different strengths of incentive to misrepresent preferences under uncertainty.

If this is right

Election designers could sometimes reduce strategic voting by keeping the precise method uncertain among several options rather than committing to one.
Some preference profiles that are manipulable under every individual method become non-manipulable once the method is drawn from a set.
Computational enumeration can locate the preference profiles and method sets that exhibit this reduction in manipulability.
The result applies only to the specific uncertainty models introduced; other models of voter beliefs might yield different outcomes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same computational approach could be used to test whether uncertainty about turnout or about other voters' reports produces similar reductions in manipulation incentives.
Real-world election organizers might experiment with announcing a short list of possible methods in advance to observe effects on reported preferences.
Hybrid rules that randomly select among methods at counting time could be evaluated for their effect on the identified non-manipulable scenarios.

Load-bearing premise

The three defined notions of manipulability correctly model the incentives that matter to voters facing uncertainty about the voting method.

What would settle it

A concrete preference profile and set of methods where the computational enumeration reports no sure, safe, or expected manipulability, yet a direct check shows that a voter still gains by misreporting under at least one of the three uncertainty models.

Figures

Figures reproduced from arXiv: 1907.09110 by Berkeley), Eric Pacuit (University of Maryland), Wesley H. Holliday (University of California.

**Figure 1.** Figure 1: Percentage of profiles witnessing sure weak dominance manipulation for (3,4) (left) and (3,7) (right). To illustrate what happens with larger number of voters, we present in [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗

**Figure 2.** Figure 2: Percentage of (4,m)-profiles witnessing sure weak dominance manipulation for Borda alone and Borda paired with several other methods. By exhaustive search from (3,4)–(3,8), we find that {Borda, Baldwin}, {Borda, StrictNanson}, {WeakNanson, Baldwin}, and {WeakNanson, StrictNanson} are not susceptible to sure weak dominance manipulation, while each method individually is. This raises the question of whether… view at source ↗

**Figure 3.** Figure 3: Percentage of profiles witnessing sure optimistic dominance manipulation for (3,7) (left) and sure pessimistic dominance manipulation for (3,6) (right). manipulate, relative to what would happen if voters knew the method was f , even if there is no reduction relative to what would happen if the planner intended to use f 0 and voters knew this. For example, in [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

**Figure 4.** Figure 4: Percentage of profiles witnessing safe weak dominance manipulation for (3,6) (left) and (3,7) (right) [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

**Figure 5.** Figure 5: Percentage of (4,m)-profiles witnessing safe weak dominance manipulation for Coombs alone and Coombs paired with several other methods. (3) A voter who is uncertain between methods f and f 0 may have an incentive to manipulate on fewer profiles than a voter who knows the method is f but more profiles than a voter who knows the method is f 0 . E.g., this happens with (3,6) when f = Borda and f 0 = Hare. I… view at source ↗

read the original abstract

Much of the theoretical work on strategic voting makes strong assumptions about what voters know about the voting situation. A strategizing voter is typically assumed to know how other voters will vote and to know the rules of the voting method. A growing body of literature explores strategic voting when there is uncertainty about how others will vote. In this paper, we study strategic voting when there is uncertainty about the voting method. We introduce three notions of manipulability for a set of voting methods: sure, safe, and expected manipulability. With the help of a computer program, we identify voting scenarios in which uncertainty about the voting method may reduce or even eliminate a voter's incentive to misrepresent her preferences. Thus, it may be in the interest of an election designer who wishes to reduce strategic voting to leave voters uncertain about which of several reasonable voting methods will be used to determine the winners of an election.

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.

Desk Editor's Note private letter to a colleague

The paper defines three new manipulability notions under voting-rule uncertainty and computationally locates profiles where that uncertainty reduces strategic incentives.

read the letter

The core contribution is the shift from uncertainty about other voters' actions to uncertainty about which rule from a set will be used. They define sure manipulability (manipulation works no matter which rule applies), safe manipulability (manipulation works for at least one rule without risk under others), and expected manipulability (manipulation improves expected outcome under a probability distribution over rules). A computer search then turns up concrete preference profiles where one or more of these notions show lower or zero incentive to misreport compared with known-rule cases. That is the actual new material and the part that extends the existing literature on strategic voting under incomplete information. The computational examples are the evidence offered for the claim that rule uncertainty can sometimes deter strategy. The modeling choices are explicit and the search is presented as exhaustive within its scope, so there is no hidden circularity or parameter fitting. The main limitation is that the results are existence claims tied to the chosen definitions and the searched space; nothing shows how frequently these profiles arise or whether the three notions track real voter reasoning. The abstract gives no details on the search algorithm or verification, which leaves the computational step harder to assess than a general theorem would be. This is solid enough for social choice readers who already work on manipulation and mechanism design; it supplies a new formal handle and some illustrative cases without overclaiming generality. It deserves peer review so referees can check the search procedure and the robustness of the definitions.

Referee Report

0 major / 3 minor

Summary. The paper introduces three notions of manipulability—sure, safe, and expected—for a voter facing uncertainty over a set of voting methods. Using a computer program to perform a search, it identifies concrete preference profiles and uncertainty models in which this uncertainty reduces or eliminates the voter's incentive to misreport preferences, and concludes that election designers may therefore benefit from leaving voters uncertain about which of several reasonable methods will be used.

Significance. If the computational examples hold under the stated definitions, the result supplies an existence proof that uncertainty over the voting rule itself can deter strategic voting, extending prior work that focused on uncertainty about others' votes. The explicit, non-circular definitions and the use of exhaustive search to locate supporting scenarios constitute a clear strength, providing falsifiable, reproducible instances rather than a general theorem.

minor comments (3)

Abstract and introduction: the description of the computer search does not state the exact set of voting methods enumerated, the size of the preference-profile space searched, or the termination criteria; adding these details would strengthen reproducibility without altering the existence claim.
Section introducing the three manipulability notions: a small, fully worked numerical example illustrating the difference between sure, safe, and expected manipulability would improve accessibility before the computational results are presented.
The manuscript would benefit from depositing the search program (or pseudocode) as supplementary material so that the reported scenarios can be independently regenerated.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the paper, including the accurate summary of our contributions and the recommendation for minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; new definitions and computational search

full rationale

The paper defines three new notions (sure, safe, and expected manipulability) over a set of voting methods and reports the results of an exhaustive computer search for preference profiles and uncertainty models in which the incentive to misreport is reduced or eliminated. No load-bearing step reduces by construction to a fitted parameter, a self-citation chain, or an ansatz imported from prior work by the same authors. The central claim is an existence result under explicitly stated modeling choices; disagreement with those choices is not a circularity issue. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 3 invented entities

Based on abstract only; the paper relies on standard domain assumptions of voting theory and introduces three new defined concepts rather than new physical entities or fitted parameters.

axioms (1)

domain assumption Voters have complete, transitive preferences over candidates and may consider misrepresenting them.
Standard background assumption in social choice theory invoked when discussing manipulation.

invented entities (3)

sure manipulability no independent evidence
purpose: Captures when a voter benefits from misrepresenting preferences regardless of which method is used.
New definition introduced to handle method uncertainty.
safe manipulability no independent evidence
purpose: Captures manipulation that remains beneficial across possible methods.
New definition introduced to handle method uncertainty.
expected manipulability no independent evidence
purpose: Captures manipulation evaluated by expected utility over possible methods.
New definition introduced to handle method uncertainty.

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discussion (0)

Reference graph

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It follows by (4) that B(ϕ) < B(ψ) and B(ϕ) < B(χ). Thus, ϕ has a below average Borda score in P. Finally, we claim that ϕ is the only candidate with a below average Borda score in P. Since the average Borda score is m, this means B(ψ)≥ m and B(χ)≥ m. Suppose for contradiction that B(ψ) < m or B(χ) < m. Without loss of generality, suppose B(ψ) < m. By Cla...

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B(β ),B(γ)≤ m < B(α); or B(β )≤ m < B(α),B(γ) and α >M P γ; or B(γ)≤ m < B(α),B(β ) and α >M P β. Then{α} is the WN-winning set in P, so i has no incentive to transition from P to P′

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B(α)≤ m < B(β ),B(γ) and γ >M P β, so{γ} is the WN-winning set in P. In this case B′(α)≤ m≤ B′(β ), m < B′(γ), and γ >M P′ β for any P′ differing only in i’s ranking. So the winning set in P′ is again {γ}, providing no incentive under WN to transition to P′

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In this case B′(α)≤ m≤ B′(β ), m < B′(γ), and γ≥M P′ β for any P′ differ only in i’s ranking

B(α)≤ m < B(β ),B(γ) and γ =M P β, so{β , γ} is the WN-winning set in P. In this case B′(α)≤ m≤ B′(β ), m < B′(γ), and γ≥M P′ β for any P′ differ only in i’s ranking. So the WN-winning set in P′ is {γ} or{β , γ}, neither of which weakly dominates{β , γ} for i

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[42]

It follows that B(β )≤ m− 2 and hence B′(β ) < m, and γ≥M P′ α for any P′ differ only in i’s ranking

B(β )≤ m < B(α),B(γ) and γ≥M P α, so the WN-winning set inP is either{γ} or{α, γ}. It follows that B(β )≤ m− 2 and hence B′(β ) < m, and γ≥M P′ α for any P′ differ only in i’s ranking. Thus, if{γ} is the WN-winning set in P, it is the WN-winning set in P′; and if{α, γ} is the WN-winning set in P, then the WN-winning set in P′ contains γ, but no set contai...

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[43]

It follows that B(γ) < m and indeed B(γ)≤ m− 2

B(γ)≤ m < B(α),B(β ), and β≥M P α, so the WN-winning set is either{β} or{α, β}. It follows that B(γ) < m and indeed B(γ)≤ m− 2. Hence B′(γ)≤ m. Thus, γ is eliminated for WN in the ﬁrst round for P′. Since β≥M P α with i having the ranking αβ γ in P, it follows that β≥M P′ α for any P′ differing from P only in i’s ranking. Thus, if{β} is the WN-wining set ...

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[44]

B(α),B(γ)≤ m≤ B(β ), so the WN-winning set inP is{β} or{α, β , γ}. In this case B′(α)≤ m, so the WN-winning set in P′ is either{α, β , γ} or a set not containing α, and in both cases the WN-winning set in P′ does not weakly dominate the WN-winning set in P

work page
[45]

Hence B′(α)≤ m < B′(γ)

B(α),B(β )≤ m < B(γ), so {γ} is the WN-winning set in P. Hence B′(α)≤ m < B′(γ). If B′(β )≤ m, then{γ} is still the WN-winning set in P′. So suppose m < B′(β ), which implies B(β ) = m, B(α) < m, and B′(α) < m. There are only two post-manipulation rankings consistent with m < B′(β ): β αγ and β γα. Note that the majority ordering of β and γ has not change...

work page

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Thus, ϕ has a below average Borda score in P

It follows by (4) that B(ϕ) < B(ψ) and B(ϕ) < B(χ). Thus, ϕ has a below average Borda score in P. Finally, we claim that ϕ is the only candidate with a below average Borda score in P. Since the average Borda score is m, this means B(ψ)≥ m and B(χ)≥ m. Suppose for contradiction that B(ψ) < m or B(χ) < m. Without loss of generality, suppose B(ψ) < m. By Cla...

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Then{α} is the WN-winning set in P, so i has no incentive to transition from P to P′

B(β ),B(γ)≤ m < B(α); or B(β )≤ m < B(α),B(γ) and α >M P γ; or B(γ)≤ m < B(α),B(β ) and α >M P β. Then{α} is the WN-winning set in P, so i has no incentive to transition from P to P′

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Then B′(α)≤ m < B′(γ), so α is not in the WN-winning set in P′

B(α)≤ m < B(β ),B(γ) and β >M P γ, so{β} is the WN-winning set inP. Then B′(α)≤ m < B′(γ), so α is not in the WN-winning set in P′. But only sets that contain α weakly dominate{β} for i

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In this case B′(α)≤ m≤ B′(β ), m < B′(γ), and γ >M P′ β for any P′ differing only in i’s ranking

B(α)≤ m < B(β ),B(γ) and γ >M P β, so{γ} is the WN-winning set in P. In this case B′(α)≤ m≤ B′(β ), m < B′(γ), and γ >M P′ β for any P′ differing only in i’s ranking. So the winning set in P′ is again {γ}, providing no incentive under WN to transition to P′

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In this case B′(α)≤ m≤ B′(β ), m < B′(γ), and γ≥M P′ β for any P′ differ only in i’s ranking

B(α)≤ m < B(β ),B(γ) and γ =M P β, so{β , γ} is the WN-winning set in P. In this case B′(α)≤ m≤ B′(β ), m < B′(γ), and γ≥M P′ β for any P′ differ only in i’s ranking. So the WN-winning set in P′ is {γ} or{β , γ}, neither of which weakly dominates{β , γ} for i

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It follows that B(β )≤ m− 2 and hence B′(β ) < m, and γ≥M P′ α for any P′ differ only in i’s ranking

B(β )≤ m < B(α),B(γ) and γ≥M P α, so the WN-winning set inP is either{γ} or{α, γ}. It follows that B(β )≤ m− 2 and hence B′(β ) < m, and γ≥M P′ α for any P′ differ only in i’s ranking. Thus, if{γ} is the WN-winning set in P, it is the WN-winning set in P′; and if{α, γ} is the WN-winning set in P, then the WN-winning set in P′ contains γ, but no set contai...

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It follows that B(γ) < m and indeed B(γ)≤ m− 2

B(γ)≤ m < B(α),B(β ), and β≥M P α, so the WN-winning set is either{β} or{α, β}. It follows that B(γ) < m and indeed B(γ)≤ m− 2. Hence B′(γ)≤ m. Thus, γ is eliminated for WN in the ﬁrst round for P′. Since β≥M P α with i having the ranking αβ γ in P, it follows that β≥M P′ α for any P′ differing from P only in i’s ranking. Thus, if{β} is the WN-wining set ...

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B(α),B(γ)≤ m≤ B(β ), so the WN-winning set inP is{β} or{α, β , γ}. In this case B′(α)≤ m, so the WN-winning set in P′ is either{α, β , γ} or a set not containing α, and in both cases the WN-winning set in P′ does not weakly dominate the WN-winning set in P

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Hence B′(α)≤ m < B′(γ)

B(α),B(β )≤ m < B(γ), so {γ} is the WN-winning set in P. Hence B′(α)≤ m < B′(γ). If B′(β )≤ m, then{γ} is still the WN-winning set in P′. So suppose m < B′(β ), which implies B(β ) = m, B(α) < m, and B′(α) < m. There are only two post-manipulation rankings consistent with m < B′(β ): β αγ and β γα. Note that the majority ordering of β and γ has not change...

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