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arxiv: 2606.10845 · v1 · pith:PA4U3SAKnew · submitted 2026-06-09 · 💰 econ.TH

Iterative Elimination of Borda Losers: Axiomatizations of the Baldwin and Nanson Rules

Leo Goto , Satoshi Nakada This is my paper

Pith reviewed 2026-06-27 10:54 UTC · model grok-4.3

classification 💰 econ.TH
keywords voting rulesaxiomatic characterizationBorda ruleBaldwin ruleNanson ruleCondorcet winneriterative elimination
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The pith

Baldwin and Nanson voting rules receive unified axiomatic characterizations comparable to the Borda rule.

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

The paper seeks to establish axiomatic foundations for the Baldwin and Nanson rules, which identify Condorcet winners through iterative elimination based on Borda scores. Baldwin successively drops the lowest-scoring alternative while Nanson drops all below the mean score. It shows these procedures admit unified characterizations whose axioms closely match Young's 1974 axioms for the Borda rule itself. A reader would care because the characterizations make the normative properties of these rules directly comparable to Borda scoring, clarifying when each rule is justified in group decisions.

Core claim

Both the Baldwin rule, which successively eliminates the alternative with the lowest Borda score, and the Nanson rule, which eliminates all alternatives whose Borda scores do not exceed the average, admit unified axiomatic characterizations whose axioms are closely comparable to Young's 1974 characterization of the Borda rule.

What carries the argument

Unified set of axioms capturing the iterative Borda-elimination procedures that define both the Baldwin and Nanson rules.

Load-bearing premise

The iterative Borda-elimination definitions of Baldwin and Nanson can be captured by a common set of axioms that are both necessary and sufficient.

What would settle it

A preference profile on which a rule satisfying the proposed axioms produces a winner different from the Baldwin or Nanson elimination outcome.

Figures

Figures reproduced from arXiv: 2606.10845 by Leo Goto, Satoshi Nakada.

Figure 1
Figure 1. Figure 1: Tournament 𝑆 𝑥 𝑥 𝑎1 𝑎2 · · · 𝑎𝑚−1 1 1 1 [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Majority margin matrix of 𝐶𝑥𝑦𝑧 −1 1 1 −1 −1 1 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Tournament 𝑆 𝑥 𝑦 𝑥 𝑎1 𝑎2 · · · 𝑎𝑚−2 𝑦 1 [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Tournament 𝐷 {𝑖} Then, we can see that 𝑇 [≻{𝑖} ] 𝑦𝑧 =    0 if 𝑦 = 𝑧 = 𝑥, 1 if 𝑦 = 𝑥 and 𝑧 ≠ 𝑥, 1 if 𝑦 = 𝑎𝑖 and 𝑧 = 𝑎 𝑗 s.t. 𝑖 < 𝑗, −1 otherwise. Therefore, we have 𝐷 {𝑖} = ∑︁ 1≤𝑖≤𝑚−1 𝑆ˆ 𝑥 𝑎𝑖 + ∑︁ 1≤𝑖, 𝑗≤𝑚−1: 𝑖<𝑗 𝑆ˆ 𝑎𝑖 𝑎 𝑗 . By Bottom Consistency, 𝜙(𝐷 {𝑖} ) = (𝐴), which contradicts Faithfulness. Case 2: 𝑅 𝑥 ≠ 𝑅 −𝑥 . Suppose that (iii) is applied for 𝑅 𝑥 or 𝑅 −𝑥 . Then, by Bottom Consi… view at source ↗
read the original abstract

The Baldwin and Nanson rules are two voting rules proposed to identify the Condorcet winner whenever one exists. Both rules operate as recursive Borda elimination procedures: the Baldwin rule successively eliminates the alternatives with the lowest Borda score, whereas the Nanson rule eliminates all alternatives whose Borda scores do not exceed the mean. This paper investigates the axiomatic properties of the Baldwin and Nanson rules and provides unified axiomatic characterizations. In particular, our axioms are closely comparable to Young's (1974) characterization of the Borda rule.

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

0 major / 3 minor

Summary. The paper claims to provide unified axiomatic characterizations of the Baldwin and Nanson rules—defined as recursive Borda-elimination procedures (Baldwin successively drops the lowest Borda scorer; Nanson drops all below the mean Borda score)—showing that both rules admit necessary-and-sufficient axiom sets that are closely comparable in structure to Young's (1974) characterization of the Borda rule itself.

Significance. If the derivations hold, the work supplies the first unified axiomatizations linking these two iterative elimination rules to the Borda scoring foundation, extending Young's approach beyond pure scoring rules. This strengthens the ability to compare scoring versus elimination procedures within a common axiomatic framework and may facilitate further results on Condorcet consistency and related properties.

minor comments (3)
  1. [Introduction] The abstract states the characterizations exist and are 'closely comparable' to Young (1974), but the introduction should explicitly list the new axiom sets (e.g., the common axioms plus the distinguishing ones for Baldwin versus Nanson) before the proofs begin, to make the unified structure immediately visible.
  2. [Preliminaries] Notation for the iterative elimination steps (e.g., the recursive definition of the choice set after k rounds) is introduced without a dedicated preliminary section; a short subsection collecting the Borda score function, mean threshold, and elimination operator would improve readability for readers familiar with Young but not with these rules.
  3. [Section 4] The paper cites Young (1974) but does not include a side-by-side table or explicit mapping of which axioms are shared verbatim, which are weakened, and which are new; adding such a comparison table after the main theorems would strengthen the 'closely comparable' claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive summary and recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

Axiomatic characterizations independent of target rules

full rationale

The paper states and proves axiomatic characterizations for the Baldwin and Nanson rules that are formulated independently of the rules themselves and shown to be necessary and sufficient. These axioms are explicitly compared to Young's 1974 external characterization of the Borda rule, with no self-citations, self-definitional reductions, fitted parameters renamed as predictions, or ansatzes smuggled via prior work by the same authors. The derivation relies on standard axiomatic methods that remain self-contained against external benchmarks, producing no load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are described in the abstract; the ledger is therefore empty.

pith-pipeline@v0.9.1-grok · 5610 in / 1094 out tokens · 26118 ms · 2026-06-27T10:54:26.276138+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Note on an Axiomatization of the Baldwin Rule

    econ.TH 2026-06 unverdicted novelty 3.0

    A combinatorial proof using permutations and amplified profiles is provided for the characterization of the Baldwin rule by Neutrality, Bottom Consistency, Faithfulness, Cancellation, and Bottom Independence.

Reference graph

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