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An algorithm returns an EJR+ committee in the ARRV spatial model using O(d log d k) Planar queries per voter in expectation, independent of candidate count, for any distribution over rectangular preferences when the electorate is large enough.

2026-07-02 04:02 UTC pith:EWY2KXL4

arxiv 2607.01036 v1 pith:EWY2KXL4 submitted 2026-07-01 cs.GT

Multiwinner Voting with Spatial Preferences under Incomplete Information

Drew Springham , Edith Elkind , Bart de Keijzer , Maria Polukarov This is my paper
classification cs.GT
keywords votercandidatesissueonlypreferencesalgorithmaxis-alignedcandidate
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Voters in big elections cannot rate every candidate. The authors model each voter as approving a rectangle in a d-dimensional space of issues: on each issue the voter has an interval of acceptable positions. Instead of asking for the full rectangle, the algorithm asks only a few 'planar' questions that compare one candidate against the voter's tolerance on one issue. It uses a verify-or-fallback structure that first tries to confirm enough information with few queries and falls back to more queries only when needed. The total queries per voter stay logarithmic in the committee size and linear in the dimension, no matter how many candidates exist. Separate modules handle cases where the distribution of voter rectangles is known, unknown, or smooth. The output committee still meets the EJR+ fairness standard that would normally require complete approval ballots from everyone.

Core claim

We give an algorithm returning an EJR+ committee for any distribution over rectangular preferences, using only O(d log d k) Planar queries per voter in expectation given a sufficiently large electorate, independent of the number of candidates m.

Load-bearing premise

The electorate is sufficiently large (abstract, paragraph 3) so that the verify-or-fallback framework with interchangeable modules can achieve the stated query bound for any distribution over ARRV preferences.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the ARRV model definition, the EJR+ fairness axiom, and the existence of interchangeable modules that satisfy the verify-or-fallback properties for the three distribution classes. No free parameters or invented physical entities are introduced.

axioms (2)
  • domain assumption EJR+ is the target proportional fairness notion for multiwinner committees
    Invoked in the abstract as the guarantee the algorithm must return.
  • domain assumption Voter preferences are drawn from the ARRV model of axis-aligned rectangular approvals in d-dimensional space
    Stated as the setting in which the query bound holds.
invented entities (1)
  • ARRV (Axis-aligned Random Rectangle Voter) model no independent evidence
    purpose: To represent incomplete spatial preferences via axis-aligned hyper-rectangles revealed only through Planar queries
    New modeling choice introduced to capture the incomplete-information setting; no independent evidence supplied beyond the model definition itself.

pith-pipeline@v0.9.1-grok · 5755 in / 1442 out tokens · 31743 ms · 2026-07-02T04:02:43.248924+00:00 · methodology

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In multiwinner elections with many candidates, as in participatory budgeting or large-scale recommendation, voters cannot plausibly evaluate every candidate, yet standard proportional-fairness guarantees such as EJR+ are stated for fully specified approval ballots. We ask whether strong proportional representation can still be guaranteed while eliciting only a little from each voter. We study this in a spatial model, the Axis-aligned Random Rectangle Voter (ARRV) model, in which candidates occupy a $d$-dimensional issue space and each voter approves an axis-aligned hyper-rectangle: a tolerance interval on every issue. Preferences are revealed only through Planar queries, each comparing a voter's tolerance to a candidate on a single issue. We give an algorithm returning an EJR+ committee for any distribution over rectangular preferences, using only $\mathcal{O}(d\log dk)$ Planar queries per voter in expectation given a sufficiently large electorate, independent of the number of candidates $m$, where $d$ is the number of issues and $k$ the committee size. The algorithm rests on a dimension-agnostic verify-or-fallback framework whose query cost is governed by two properties supplied by interchangeable modules. We describe such modules, yielding end-to-end guarantees for known, unknown, and smooth distributions.

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    to the (PW) notion and the empirical statisticζc used here. Proof.Fix a levelℓ∈[k], callc∈Ctinyifp c,ℓ ≤ℓ/(k+ 1)andlargeifp c,ℓ ≥ℓ/k−δ 1, and write g :=ℓ/(2k(k+ 1))−δ 1/2for the gap separatingq ∗ from each of these two bounds. At this level, NGJCR admits a candidate only via the testζc/h1 ≥q ∗, so it can violate (PW) with marginδ1 only by admitting some t...

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