A simple characterization of single-peaked domains

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Pith reviewed 2026-05-07 12:40 UTC · model grok-4.3

classification 💰 econ.TH

keywords single-peaked domainsstrategy-proofnesstreesextreme rulesdomain characterizationsocial choicepreference domainsvoting rules

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The pith

Extreme rules on a tree are strategy-proof only when preferences are single-peaked on that tree.

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

This paper shows that single-peaked domains on trees can be identified simply by checking when certain extreme rules become strategy-proof. For any tree, the rules stay unanimous and anonymous no matter which preferences are allowed. They satisfy strategy-proofness exactly on the single-peaked domain tied to that tree. Readers care because the result ties a common domain restriction in voting directly to the incentive properties of concrete rules, without needing extra conditions.

Core claim

For any tree, the extreme rules defined on that tree are unanimous and anonymous on any preference domain. In particular, these rules are strategy-proof only on the single-peaked domain associated with the tree. This yields a characterization of single-peaked domains via the strategy-proofness of the extreme rules.

What carries the argument

Extreme rules defined on a tree, which serve as the test objects whose strategy-proofness isolates the single-peaked domain.

Load-bearing premise

The extreme rules are unanimous and anonymous on any preference domain for any tree.

What would settle it

A preference domain that is not single-peaked on some tree but on which at least one extreme rule remains strategy-proof.

read the original abstract

This paper characterizes the single-peaked domain on a tree via the strategy-proofness of extreme rules defined on that tree. For any tree, these rules are unanimous and anonymous on any preference domain. In particular, we show that they are strategy-proof only on the single-peaked domain associated with that tree.

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.

Desk Editor's Note private letter to a colleague

The paper gives a clean if-and-only-if link between single-peaked domains on trees and strategy-proofness of the associated extreme rules.

read the letter

The main point is that extreme rules defined on a tree are strategy-proof exactly on the single-peaked domain for that tree, while staying unanimous and anonymous on any preference domain. This flips the usual approach by using the incentive property of specific rules to pin down the domain restriction. It keeps the statement general across arbitrary trees and avoids extra assumptions in the claim itself. The stress-test found no obvious internal contradictions or hidden circularity in the abstract structure. What the paper does well is deliver a direct characterization that could simplify checks for single-peakedness in mechanism design settings. The result feels proportionate to the setup and ties back to standard social choice concerns about non-manipulable voting. Soft spots are limited but real. The abstract supplies no proof outline or explicit definition of the extreme rules, so it is hard to judge how much this overlaps with earlier characterizations using medians or other tree-based restrictions. If the proof turns out to lean on the tree structure in a definitional way, the novelty could be modest rather than substantial. The full text would also clarify whether the literature engagement is thorough. This is for readers already working on domain restrictions and strategy-proof rules in economic theory. A specialist in social choice or mechanism design would get the most out of it for seeing whether the framing helps with applications. It deserves peer review because the central claim is precise and the approach is straightforward enough for referees to evaluate the details and contribution directly.

Referee Report

0 major / 2 minor

Summary. The paper claims to characterize single-peaked domains on a tree via the strategy-proofness of extreme rules defined on that tree. It asserts that for any tree these rules are unanimous and anonymous on any preference domain, and in particular that they are strategy-proof if and only if the domain is the single-peaked domain associated with the tree.

Significance. If the central claim holds, the result supplies a clean if-and-only-if characterization that isolates strategy-proofness as the property distinguishing the single-peaked domain. The universal unanimity and anonymity of the extreme rules is a notable strength, as it shows that the domain restriction alone governs incentive compatibility for these rules. This could streamline analysis in social choice settings with tree-structured alternatives.

minor comments (2)

The abstract is extremely concise and supplies no proof outline or definition of extreme rules; the introduction should include a short illustrative example for a small tree to make the construction immediately accessible.
Ensure that the definitions of 'extreme rules' and 'single-peaked domain on a tree' are stated with full precision before the main theorem, including any notational conventions for the tree structure.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our paper and for highlighting the clean if-and-only-if nature of the characterization. We are pleased that the universal unanimity and anonymity of the extreme rules is noted as a strength. As the major comments section contains no specific points requiring clarification or correction, we see no need for revisions at this stage.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper offers a characterization result: extreme rules defined on a tree are unanimous and anonymous on arbitrary domains and strategy-proof precisely when the domain is the single-peaked domain associated with that tree. No quoted equations or definitions in the abstract or structure reduce the target domain or the strategy-proofness property to a self-referential fit, a renamed input, or a load-bearing self-citation. Single-peakedness on trees is a standard, independently defined concept in the literature; the proof therefore proceeds from those definitions to an equivalence without the derivation collapsing into its own premises by construction. This is the normal, non-circular outcome for an if-and-only-if characterization theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard social choice concepts of strategy-proofness, unanimity, and anonymity together with the definition of single-peakedness on trees; no free parameters or invented entities are apparent from the abstract.

axioms (1)

domain assumption Extreme rules are unanimous and anonymous on any preference domain.
Stated explicitly in the abstract as holding for any tree.

pith-pipeline@v0.9.0 · 5327 in / 1076 out tokens · 60035 ms · 2026-05-07T12:40:08.600996+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references

[1]

A characterization of the single-peaked domain,

Arrow, K. J.(1951):Social Choice and Individual Values, New York: Wiley. Ballester, M. A. and G. Haeringer(2011): “A characterization of the single-peaked domain,”Social Choice and Welfare, 36, 305–322. Bandhu, S., B. Mondal, and A. Pramanik(2022): “Strategy-proofness of the unanim- ity with status-quo rule over restricted domains,”Economics Letters, 210,...

1951
[2]

The single-peaked domain revisited: A simple global characterization,

Puppe, C.(2018): “The single-peaked domain revisited: A simple global characterization,” Journal of Economic Theory, 176, 55–80. Sanver, M. R.(2009): “Strategy-proofness of the plurality rule over restricted domains,” Economic Theory, 39, 461–471. Satterthwaite, M. A.(1975): “Strategy-proofness and Arrow’s conditions: Existence and correspondence theorems...

2018