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arxiv: 2605.12717 · v1 · submitted 2026-05-12 · 💻 cs.GT · cs.AI

Recognition: 1 theorem link

· Lean Theorem

The End Justifies the Mean: A Linear Ranking Rule for Proportional Sequential Decisions

Carmel Baharav , Niclas Boehmer , Bailey Flanigan , Maximilian T. Wittmann
Authors on Pith no claims yet

Pith reviewed 2026-05-14 19:52 UTC · model grok-4.3

classification 💻 cs.GT cs.AI
keywords proportionalitylinear rankingangular meansequential decisionsindividual proportionalityvoting rulesparticipatory designfairness
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The pith

The angular mean of voter vectors satisfies long-run individual proportionality for repeated linear rankings.

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

The paper shows that when groups repeatedly rank items using a fixed linear scoring rule, the angular mean of individual preference vectors ensures each voter type agrees with the rankings in proportion to their share of the population, averaged over time. This addresses a core issue in collective decision making where the arithmetic mean often sidelines minority views. The result holds for long-run averages even though exact fairness in every single batch is impossible with any fixed rule. The gap to per-batch fairness narrows quickly as batches contain more items, and real data experiments confirm gains when preferences conflict.

Core claim

The main result is that the angular mean, obtained by normalizing the sum of the unit vectors in the directions of each voter type's preferred scoring vector, satisfies long-run individual proportionality: for each voter type i, the long-run fraction of rankings they agree with is at least their population fraction alpha_i. This is achieved because the angular mean balances the directional contributions on the unit sphere rather than allowing vector lengths to dominate.

What carries the argument

The angular mean of the preferred scoring vectors, which normalizes the sum of their unit directions to produce the collective scoring vector theta star.

If this is right

  • The arithmetic mean rule is majoritarian and fails proportionality when preferences diverge.
  • No fixed linear scoring vector can achieve exact per-batch IP for all voter types.
  • The difference between per-batch agreement and long-run agreement decreases rapidly with increasing batch size.
  • Performance of all rules is similar under homogeneous preferences, but angular mean improves outcomes substantially under high disagreement.

Where Pith is reading between the lines

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

  • Normalizing vectors before averaging may generalize to other repeated decision settings where direction matters more than magnitude.
  • The approach could extend to cases with evolving voter preferences by periodically recomputing the angular mean.
  • Testing on larger-scale participatory systems might reveal whether the long-run guarantee translates to perceived fairness.

Load-bearing premise

Long-run average proportionality suffices even if individual batches show large deviations from ideal shares, assuming voter preferences stay constant over batches.

What would settle it

Simulate repeated batches drawn from the item space using the angular mean rule and measure whether the empirical long-run agreement rate for each voter type falls below their population fraction alpha_i.

read the original abstract

AI alignment and participatory design motivate a new democratic design problem: how to collectively choose a decision rule to use repeatedly. We study this problem for linear ranking rules, which repeatedly rank items $x_j$ within batches $X=(x_1,\dots,x_m)\in(\mathbb{R}^d)^m$, where each item's ranking is dictated by its score $\langle \theta^*,x_j\rangle$ according to a fixed scoring vector $\theta^*$. Given voters' preferred scoring vectors $\theta^{(1)},\dots,\theta^{(n)}$ and their population fractions $\alpha^{(1)},\dots,\alpha^{(n)}$, we ask how to choose a collective vector $\theta^*$ satisfying individual proportionality (IP): every voter type $i$ should agree with the resulting rankings to an $\alpha^{(i)}$-proportional degree, either on average over time (long-run IP) or even within each batch (per-batch IP). The default rule, the arithmetic mean of the $\theta^{(i)}$, has been shown to be severely majoritarian; more generally, it is not clear that any fixed linear rule can balance many voters' disparate opinions. Our main result is that, surprisingly, there is a simple rule that does satisfy long-run IP: the angular mean, the spherical analog of the arithmetic mean. We then show that exact per-batch IP is impossible for fixed linear rules, but that the gap between per-batch and long-run IP shrinks quickly with batch size. Experiments on three real-world preference datasets show that all rules perform similarly when voters' preferences are homogeneous, while the angular mean substantially improves proportionality in high-disagreement regimes.

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 studies aggregation of voter preferences θ^(i) with weights α^(i) into a fixed linear scoring vector θ* for repeated ranking of item batches X. It claims that the angular mean θ* = normalize(∑ α^i θ^i) satisfies long-run individual proportionality (IP), meaning each voter type i agrees with the induced rankings (measured by a ranking metric such as fraction of agreeing pairs) to an expected degree exactly proportional to α^i over random batches. It proves that exact per-batch IP is impossible for any fixed linear rule and shows that the per-batch/long-run gap shrinks with batch size m. Experiments on three real-world datasets indicate that the angular mean improves proportionality relative to the arithmetic mean in high-disagreement regimes while performing similarly when preferences are homogeneous.

Significance. If the long-run IP claim holds, the angular mean supplies a simple, parameter-free aggregation rule grounded in spherical geometry that mitigates majoritarianism for sequential linear ranking decisions, with direct relevance to AI alignment and participatory design. The impossibility result for per-batch IP and the quantitative shrinkage with batch size clarify fundamental limits of fixed linear rules. The experimental validation on real preference data strengthens the practical case.

major comments (2)
  1. [§3 (main theorem)] §3 (main theorem on long-run IP): the claim that θ* = normalize(∑ α^i θ^i) yields exact α^i-proportional expected agreement is not obviously guaranteed. Ranking agreement (e.g., fraction of pairs where sign(⟨θ*, x_a - x_b⟩) matches sign(⟨θ^i, x_a - x_b⟩)) is a nonlinear function of the angle between θ* and θ^i. The vector-sum construction produces linear averaging of directions, but the paper must explicitly verify that the expectation of this nonlinear metric remains proportional to α^i for general item distributions; no such verification or counter-example check is indicated in the provided derivation outline.
  2. [§4] §4 (per-batch impossibility and gap analysis): while the impossibility of exact per-batch IP for fixed linear rules is correctly established, the rate at which the gap to long-run IP shrinks with batch size m requires an explicit quantitative bound (e.g., via concentration or Lipschitz arguments on the agreement metric). The current statement that the gap “shrinks quickly” is too qualitative to support the practical claim that long-run IP is a sufficient substitute for moderate m.
minor comments (2)
  1. [Throughout] Notation for voter vectors alternates between θ^{(i)} and θ^i; standardize throughout.
  2. [Experiments section] The three real-world datasets are referenced only by name; add a brief description of dimensionality d, number of items per batch, and how voter vectors θ^i were estimated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper accordingly.

read point-by-point responses

  1. Referee: [§3 (main theorem)] §3 (main theorem on long-run IP): the claim that θ* = normalize(∑ α^i θ^i) yields exact α^i-proportional expected agreement is not obviously guaranteed. Ranking agreement (e.g., fraction of pairs where sign(⟨θ*, x_a - x_b⟩) matches sign(⟨θ^i, x_a - x_b⟩)) is a nonlinear function of the angle between θ* and θ^i. The vector-sum construction produces linear averaging of directions, but the paper must explicitly verify that the expectation of this nonlinear metric remains proportional to α^i for general item distributions; no such verification or counter-example check is indicated in the provided derivation outline.

    Authors: We agree that the current presentation of the main theorem does not make the verification of the nonlinear expectation fully explicit. The proof sketch in §3 relies on the reduction of expected agreement to a function of the angle under isotropic item distributions, together with the variational property of the angular mean, but this step is not expanded. In the revision we will insert a dedicated lemma deriving the closed-form expected agreement (1 - φ_i/π) and confirming that the resulting values are proportional to α^i as required for long-run IP. revision: yes

  2. Referee: [§4] §4 (per-batch impossibility and gap analysis): while the impossibility of exact per-batch IP for fixed linear rules is correctly established, the rate at which the gap to long-run IP shrinks with batch size m requires an explicit quantitative bound (e.g., via concentration or Lipschitz arguments on the agreement metric). The current statement that the gap “shrinks quickly” is too qualitative to support the practical claim that long-run IP is a sufficient substitute for moderate m.

    Authors: We concur that an explicit rate is needed. We will add to §4 a formal theorem stating that the gap between per-batch and long-run agreement is O(1/sqrt(m)) with high probability. The proof uses the Lipschitz continuity of the pairwise agreement metric (constant independent of m) together with McDiarmid’s inequality applied to the m independent items in the batch. revision: yes

Circularity Check

0 steps flagged

No significant circularity in angular mean derivation

full rationale

The angular mean is introduced as an independent definition (normalized weighted sum of voter vectors) and the long-run IP property is claimed as a derived result from spherical geometry rather than by construction or tautology. No quoted steps reduce the claimed proportionality to a fitted input, self-citation chain, or renamed known result. The derivation is presented as self-contained against the stated assumptions on fixed preferences and batch sampling; any concerns about nonlinearity of agreement metrics pertain to correctness, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the mathematical definition of long-run IP and the geometric properties of the unit sphere; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Voter preferences are fixed vectors that remain constant across batches
    Setup assumes theta^(i) are given and stable.
  • domain assumption Long-run average agreement is a sufficient fairness criterion
    Paper distinguishes long-run IP from per-batch IP and focuses on the former.

pith-pipeline@v0.9.0 · 5611 in / 1184 out tokens · 67360 ms · 2026-05-14T19:52:24.096293+00:00 · methodology

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