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arxiv: 2605.19521 · v1 · pith:Y55N7GN5new · submitted 2026-05-19 · 💻 cs.AI · cs.GT

Efficient Elicitation of Collective Disagreements

Mohamed Ouaguenouni , Felipe Garrido-Lucero , Umberto Grandi , C\'esar Hidalgo , Magdalena Tydrichova This is my paper

Pith reviewed 2026-05-20 06:13 UTC · model grok-4.3

classification 💻 cs.AI cs.GT
keywords collective disagreementpreference elicitationplurality matrixdisagreement measuresvoter surveyssocial choice theoryranking aggregation
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The pith

Many disagreement measures in voting require information from groups of three alternatives rather than pairwise comparisons alone.

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

This paper studies the structure of disagreements among voters choosing among multiple alternatives. Surveys usually collect either simple pairwise preferences or complete rankings, but the authors demonstrate that pairwise data alone cannot separate genuine structural disagreements from mere noise. They introduce the plurality matrix, which for each possible group of alternatives records how often each one is preferred most within that group. By defining the level of disagreement measures as the smallest group size needed to compute them, they show that popular measures such as rank-variance and divisiveness operate at level three. The work then proposes practical protocols to gather the necessary data efficiently by balancing the number of participants with the complexity of questions asked to each.

Core claim

The authors establish that a plurality matrix capturing first-place probabilities within every subset of alternatives suffices to compute disagreement measures at their minimal level. In particular, they prove that measures like rank-variance and divisiveness sit at level 3, which means pairwise comparisons are insufficient to distinguish structural disagreement from noise, and they design elicitation methods to estimate this matrix with controlled participant numbers and cognitive demands.

What carries the argument

The plurality matrix, a table that for every subset S of alternatives gives the probability that each alternative in S is ranked first by a random voter within S, together with the level of a disagreement measure defined as the smallest subset size sufficient to express that measure.

If this is right

  • Existing disagreement measures can be classified by their level to determine the minimal survey complexity required.
  • Elicitation protocols can focus on subsets of size three to accurately compute level-3 measures without full rankings.
  • The trade-off between participant count and question difficulty allows tailored survey designs for different applications.
  • Theoretical analysis shows the value of higher-level information for capturing collective disagreement more precisely.

Where Pith is reading between the lines

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

  • This approach could improve preference aggregation in AI systems by incorporating deeper disagreement signals.
  • Social scientists might adopt triplet-based questions in large-scale surveys to get more reliable disagreement estimates.
  • The framework might generalize to other collective choice problems like ranking aggregation or consensus finding.

Load-bearing premise

The plurality matrix entries for subsets of size three can be estimated reliably from sampled voter responses without systematic bias from question ordering or participant fatigue.

What would settle it

A controlled experiment where full rankings are collected from all participants and then compared to estimates from the proposed protocols to check if the computed disagreement measures match within statistical error.

Figures

Figures reproduced from arXiv: 2605.19521 by C\'esar Hidalgo, Felipe Garrido-Lucero, Magdalena Tydrichova, Mohamed Ouaguenouni, Umberto Grandi.

Figure 1
Figure 1. Figure 1: Distribution of ra for each profile represented as probability mass functions [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Skewness versus Excess kurtosis produced by seven different preferences profiles from [ [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Population N (number of voters) required as a function of maximum cognitive load λ, across degrees k ∈ {2, 3, 4, 5}, on an impartial-culture profile with m = 10 alternatives and accuracy ε = 0.05. For each pair (λ, N) the optimal protocol in terms of budget is highlighted. The intervals over the points represent the 5th to the 95th percentile. 5 Conclusions To identify the disagreements of a population ove… view at source ↗
Figure 4
Figure 4. Figure 4: Pearson moment plane for three real-election datasets: Glasgow STV city council elections (208 [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
read the original abstract

We analyze the structure of the disagreement among a population of voters over a set of alternatives. Surveys typically ask either for pairwise comparisons, simple and intuitive for participants, or full rankings over alternatives, eliciting the entire voters' preferences. Building on the observation that pairwise comparisons cannot distinguish structural disagreement from noise, we propose a stratified framework to identify the minimal aggregated preference information needed to compute a number of disagreement measures from the literature. Specifically, we introduce the plurality matrix, a generalization of pairwise comparisons that records, for every subset $S$ of alternatives, the probability that each $a \in S$ ranks first in $S$. We define the level of a disagreement measure as the smallest subset size needed to express it, showing that many existing notions, including rank-variance and divisiveness, sit at level $3$, proving that pairwise comparisons are not enough. In addition, we demonstrate the interest of going beyond level $3$ both theoretically and experimentally. To make these results actionable, we design two elicitation protocols to estimate the plurality matrix, exploring the trade-off between the number of required participants and the cognitive load requested to each of them.

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 / 2 minor

Summary. The paper introduces the plurality matrix as a generalization of pairwise comparisons that records first-place probabilities for every subset S of alternatives. It defines the level of a disagreement measure as the smallest subset size needed to express the measure, shows that several existing measures (including rank-variance and divisiveness) are at level 3, and argues that pairwise data are therefore insufficient to distinguish structural disagreement from noise. The authors further demonstrate the value of information beyond level 3 both theoretically and experimentally, and design two elicitation protocols that trade off the number of participants against the cognitive load per participant.

Significance. If the central claims hold, the work provides a principled, minimal-information approach to preference elicitation for disagreement analysis in social choice and AI systems. The theoretical reduction of multiple measures to the plurality matrix for triples, together with the experimental validation on synthetic and real data, supplies a concrete tool for survey design that avoids both under-elicitation and unnecessary cognitive burden. The independence of the level definitions from fitted parameters is a notable strength.

major comments (2)
  1. [Elicitation protocol design and experimental validation sections] Elicitation protocol design and experimental validation sections: the claim that level-3 measures can be reliably recovered rests on the assumption that plurality-matrix entries for |S|=3 can be estimated from sampled rankings without systematic bias induced by question order or participant fatigue. No explicit diagnostic or robustness check for such bias is provided; any ordering or fatigue effect would render the computed level-3 values (and thus the distinction from noise) unreliable even if the theoretical reduction to the matrix is correct.
  2. [Experimental results] Experimental results: while synthetic and real-data experiments illustrate the elicitation trade-offs, the manuscript does not report full statistical tests (e.g., confidence intervals or hypothesis tests) for all protocol variants. This weakens the empirical support for the practical advantage of the proposed protocols over simpler baselines.
minor comments (2)
  1. [Definition of the plurality matrix] Notation for the plurality matrix could be clarified with an explicit small example showing how entries for |S|=2 recover ordinary pairwise probabilities and how |S|=3 entries extend them.
  2. [Introduction] A few sentences in the introduction repeat the motivation for moving beyond pairwise comparisons; tightening this paragraph would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We respond to each major comment below and describe the revisions we will incorporate to address the concerns raised.

read point-by-point responses

  1. Referee: [Elicitation protocol design and experimental validation sections] Elicitation protocol design and experimental validation sections: the claim that level-3 measures can be reliably recovered rests on the assumption that plurality-matrix entries for |S|=3 can be estimated from sampled rankings without systematic bias induced by question order or participant fatigue. No explicit diagnostic or robustness check for such bias is provided; any ordering or fatigue effect would render the computed level-3 values (and thus the distinction from noise) unreliable even if the theoretical reduction to the matrix is correct.

    Authors: We agree that the absence of explicit diagnostics for ordering or fatigue effects represents a gap in the current experimental validation. While the protocols described in the manuscript randomize question presentation and are explicitly designed to control cognitive load per participant, we did not report targeted robustness checks such as response-time correlations or cross-order comparisons. In the revised manuscript we will add these diagnostics, including an analysis of potential fatigue effects in the real-data experiments and a sensitivity check across different question orderings, to confirm that the recovered level-3 estimates remain stable. revision: yes

  2. Referee: [Experimental results] Experimental results: while synthetic and real-data experiments illustrate the elicitation trade-offs, the manuscript does not report full statistical tests (e.g., confidence intervals or hypothesis tests) for all protocol variants. This weakens the empirical support for the practical advantage of the proposed protocols over simpler baselines.

    Authors: We accept that the experimental results would be strengthened by systematic statistical reporting. The current version presents point estimates and qualitative comparisons but omits confidence intervals and formal hypothesis tests for the protocol variants. We will revise the experimental section to include bootstrap confidence intervals for all key metrics and paired statistical tests comparing each proposed protocol against the simpler baselines, thereby providing quantitative evidence for the reported trade-offs. revision: yes

Circularity Check

0 steps flagged

Theoretical levels defined via matrix expressions; independent of sampling estimates

full rationale

The paper defines the plurality matrix directly from first-place probabilities over subsets and defines the level of each disagreement measure as the minimal subset size k sufficient to express the measure using those matrix entries. It then shows algebraically that rank-variance and divisiveness require k=3. These steps are definitional and algebraic reductions from the original measure formulas; they do not depend on any fitted parameters or sampled data. The elicitation protocols are introduced afterward as a practical tool to recover the matrix entries, but the level-3 claim and the conclusion that pairwise comparisons are insufficient are established prior to and without reference to the estimation procedure. No self-citation chain, ansatz smuggling, or renaming of known results is used to support the core theoretical result. The derivation therefore remains self-contained against external benchmarks and receives only a minor score for the presence of any empirical component at all.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The framework rests on standard assumptions from social choice theory about transitive preferences and the existence of a well-defined ranking distribution; no new entities are postulated and only minor free parameters appear in the estimation protocols.

free parameters (1)
  • sample size per subset
    Chosen to achieve desired estimation accuracy in the protocols; values are not fixed by theory but selected for the experiments.
axioms (1)
  • domain assumption Voters possess complete transitive rankings over the full set of alternatives
    Invoked when defining the plurality matrix entries from underlying preferences.

pith-pipeline@v0.9.0 · 5742 in / 1327 out tokens · 37830 ms · 2026-05-20T06:13:53.631534+00:00 · methodology

discussion (0)

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