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arxiv: 2606.08098 · v2 · pith:JEMQLYZEnew · submitted 2026-06-06 · 💻 cs.AI · cs.LG

When Does Delegation Beat Majority? A Delegation-Based Aggregator for Multi-Sample LLM Inference

Yasushi Sakai , Allen Song , Kent Larson This is my paper

Pith reviewed 2026-06-27 19:55 UTC · model grok-4.3

classification 💻 cs.AI cs.LG
keywords LLM aggregationmajority votingdelegationMMLU-Prounsupervised consensussemantic entropyembedding similarity
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The pith

A delegation aggregator using entropy and embedding geometry beats majority voting on MMLU-Pro by 1.5 points.

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

The paper establishes that majority voting over LLM samples discards two usable signals: within-group letter entropy and between-group reasoning geometry captured by embeddings. Propagational Proxy Voting (PPV) turns these signals into per-sample delegation rules that decide how much weight each sample keeps for itself and how it splits the rest across peers. The method partitions 128 samples into 16 groups per question, builds a stochastic delegation matrix from the two signals, and takes the stationary distribution as the consensus answer. On MMLU-Pro this yields a 1.5 pp gain overall and 2.24 pp on the non-trivial subset, with statistical significance. The approach requires no gold labels or auxiliary training.

Core claim

Propagational Proxy Voting partitions 128 sampled generations into 16 groups, computes each group's letter-level semantic entropy and reasoning embedding centroid, and feeds both into a stochastic delegation matrix whose stationary distribution selects the consensus answer. On MMLU-Pro it improves over majority by 1.5 percentage points overall and 2.24 points on the non-trivial subset (paired McNemar p ~ 1.0e-14, n=8,099). The method overturns a 10-6 majority for the wrong answer when the majority cluster shows low internal cosine similarity while the minority cluster is tight.

What carries the argument

Stochastic delegation matrix whose 'When' weights are set by letter entropy and whose 'Whom' targets are set by per-question-centered embedding cosine similarity.

If this is right

  • Majority voting is suboptimal because it ignores entropy and geometry signals that delegation can consume.
  • Negative results on alternative delegation strategies narrow the design space for future unsupervised aggregators.
  • No within-question ensemble of confidence modes closes the oracle gap to perfect aggregation.

Where Pith is reading between the lines

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

  • The same geometry-driven delegation could be tested on open-ended generation tasks where cluster tightness might predict answer quality.
  • If embedding centroids prove stable across model families, PPV might transfer without retraining the embedding step.
  • Combining the delegation matrix with cross-question information could further reduce variance on hard subsets.

Load-bearing premise

That per-question-centered embedding cosine similarity reliably captures between-group reasoning geometry that should control delegation weights.

What would settle it

Run PPV on a fresh multi-sample benchmark and check whether the stationary distribution still outperforms majority when the geometric-coherence signal is removed or reversed.

Figures

Figures reproduced from arXiv: 2606.08098 by Allen Song, Kent Larson, Yasushi Sakai.

Figure 1
Figure 1. Figure 1: Simplified network of direct voting and delegation [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Accuracy on the 8,099 non-trivial MMLU-Pro questions. Bars zoomed to highlight the ∼6 pp gap between majority and oracle. PPV (confidence) closes 38% of that gap unsupervised [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Where the disagreements land on the non-trivial subset. Left (pink) = majority correct, PPV wrong; right (green) = PPV correct, majority wrong. div fires rarely but is right 82.7% of the time; confidence fires more and rescues the most absolute questions (+181 net) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
read the original abstract

Majority voting over sampled answers is the dominant unsupervised aggregator for multi-sample LLM inference. In this paper, we show a delegation-based aggregator (Propagational Proxy Voting, PPV; Sakai et al., 2025) yields an unsupervised consensus rule that beats majority on MMLU-Pro by +1.5 pp overall and +2.24 pp on the non-trivial subset (paired McNemar p ~ 1.0e-14, n = 8,099). Majority discards two signals that every sample carries: within-group letter entropy and between-group reasoning geometry. PPV exposes per-voter levers that consume exactly these two signals: When (how much weight a voter keeps on its own pick) and Whom (how it splits the remainder across peers). We drive When with letter entropy and Whom with per-question-centered embedding cosine. Our method needs no gold labels and no auxiliary training: per-question, we partition 128 sampled generations into 16 groups, compute each group's letter-level semantic entropy and reasoning embedding centroid, and feed both into a stochastic delegation matrix whose stationary distribution selects the consensus answer. We walk through an example in which PPV overturns a clear 10-6 majority for the wrong letter: the 10-voter majority cluster is geometrically incoherent (mean within-cluster cosine -0.02) while the 6-voter minority is tight (+0.26), so propagated delegation mass concentrates on the minority's answer even though entropy alone would keep the majority ahead. We further report delegation strategies with negative results that constrain the design space for unsupervised LLM aggregation. No within-question ensemble of confidence modes closes the oracle gap.

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

3 major / 2 minor

Summary. The paper proposes Propagational Proxy Voting (PPV), a delegation-based aggregator for multi-sample LLM inference. It partitions 128 samples per question into 16 groups, computes letter-level semantic entropy within groups and reasoning embedding centroids, then constructs a stochastic delegation matrix driven by entropy for 'When' (self-weight) and per-question-centered cosine similarities for 'Whom' (peer delegation). The stationary distribution yields the consensus answer. PPV is shown to outperform majority voting on MMLU-Pro by +1.5 pp overall and +2.24 pp on the non-trivial subset (paired McNemar p ≈ 1.0e-14, n=8099), with an illustrative example where geometry overrides a 10-6 majority. Negative results on other delegation strategies are also reported.

Significance. If the result holds, the work is significant for demonstrating that delegation can improve unsupervised consensus by consuming signals (entropy and geometry) discarded by majority voting. The large-scale evaluation with a paired statistical test on 8099 questions, the fully unsupervised per-question design, and the explicit reporting of negative results that constrain the design space are strengths. These elements provide concrete guidance for future multi-sample LLM aggregation methods.

major comments (3)
  1. [Abstract / illustrative example] Abstract and the example section: The central claim that geometry-driven delegation produces the reported gains rests on the assumption that per-question-centered embedding cosine similarity reliably encodes substantive reasoning differences rather than artifacts. The single 10-voter vs. 6-voter example (mean within-cluster cosine -0.02 vs. +0.26) illustrates the mechanism but provides no systematic validation (e.g., ablation removing the cosine component, correlation with human reasoning judgments, or checks across the 8099 questions) that the signal is load-bearing for the +2.24 pp non-trivial-subset improvement.
  2. [Methods (partition size and weight mapping)] Methods description of the stochastic matrix: The partition into exactly 16 groups from 128 samples and the specific functional mapping from entropy and cosine values to the entries of the delegation matrix are free parameters. The manuscript does not report sensitivity analysis or confirm that these choices were fixed without reference to the MMLU-Pro labels; this directly affects whether the +1.5 pp gain can be attributed to the delegation principle itself versus parameter selection.
  3. [Results / experimental setup] Results section on the McNemar test: While the paired test on n=8099 is reported, the exact construction of the delegation matrix (including any normalization, damping factor, or convergence criterion for the stationary distribution) is not specified in sufficient detail to reproduce the geometry contribution or to isolate it from other components of the pipeline.
minor comments (2)
  1. [Methods] Clarify the precise definition of 'per-question-centered' embedding cosine (e.g., whether centering subtracts the global mean or the question-specific mean) and provide the equation for the stochastic matrix entries.
  2. [Abstract] The abstract states 'no auxiliary training' but the design choices (group count, mapping functions) function as hyperparameters; a brief discussion of how these were selected without labels would improve clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below, committing to revisions where the manuscript can be strengthened while defending the current evidence on its merits.

read point-by-point responses

  1. Referee: [Abstract / illustrative example] The central claim that geometry-driven delegation produces the reported gains rests on the assumption that per-question-centered embedding cosine similarity reliably encodes substantive reasoning differences rather than artifacts. The single 10-voter vs. 6-voter example illustrates the mechanism but provides no systematic validation (e.g., ablation removing the cosine component, correlation with human reasoning judgments, or checks across the 8099 questions) that the signal is load-bearing for the +2.24 pp non-trivial-subset improvement.

    Authors: The example is intended only to illustrate the mechanism. The primary evidence that the full PPV method (including geometry) improves over majority is the paired McNemar test on all 8099 questions. We will revise the paper to add an ablation isolating the cosine component and to report aggregate statistics on within-cluster cosines across the full dataset. Correlation with human reasoning judgments would require new annotations outside the current unsupervised study and is left for future work. revision: yes

  2. Referee: [Methods (partition size and weight mapping)] The partition into exactly 16 groups from 128 samples and the specific functional mapping from entropy and cosine values to the entries of the delegation matrix are free parameters. The manuscript does not report sensitivity analysis or confirm that these choices were fixed without reference to the MMLU-Pro labels; this directly affects whether the +1.5 pp gain can be attributed to the delegation principle itself versus parameter selection.

    Authors: The group count and functional forms were fixed a priori on computational grounds and on separate unlabeled data with no access to MMLU-Pro labels. We will add a sensitivity analysis in the revision varying both the partition size and the mapping hyperparameters to demonstrate robustness of the reported gains. revision: yes

  3. Referee: [Results / experimental setup] Results section on the McNemar test: While the paired test on n=8099 is reported, the exact construction of the delegation matrix (including any normalization, damping factor, or convergence criterion for the stationary distribution) is not specified in sufficient detail to reproduce the geometry contribution or to isolate it from other components of the pipeline.

    Authors: We agree that the matrix construction requires more explicit detail for reproducibility. The revised methods section will fully specify the stochastic delegation matrix, including normalization, any damping, and the convergence criterion for the stationary distribution. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents an empirical benchmark result (PPV outperforming majority by +1.5 pp on MMLU-Pro) obtained via direct evaluation on held-out data with no gold labels used in the aggregator. The method description (partitioning into groups, computing letter entropy for When and per-question-centered embedding cosine for Whom, then building a stochastic delegation matrix) is a fixed unsupervised procedure whose outputs are measured against external ground truth rather than being forced by fitted parameters or self-referential equations. The citation to Sakai et al. 2025 defines the PPV framework but does not bear the load of the performance claim, which rests on independent McNemar-tested accuracy numbers. No load-bearing step reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

Ledger extracted from abstract only; full paper may introduce additional parameters or assumptions.

free parameters (2)
  • Partition size (16 groups from 128 samples)
    Arbitrary choice of how to group samples for entropy and centroid computation.
  • Mapping from entropy/cosine to delegation weights
    Exact functional form converting signals into When/Whom values not specified.
axioms (1)
  • standard math Stationary distribution of the stochastic delegation matrix yields the consensus answer.
    Invoked to select the final answer from the delegation process.

pith-pipeline@v0.9.1-grok · 5833 in / 1121 out tokens · 29712 ms · 2026-06-27T19:55:58.225687+00:00 · methodology

discussion (0)

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Reference graph

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