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arxiv: 2602.14816 · v1 · pith:4Q77X3QRnew · submitted 2026-02-16 · 💰 econ.TH · cs.GT· cs.MA

Majoritarian Assignment Rules

Felix Brandt , Haoyuan Chen , Chris Dong , Patrick Lederer , Alexander Schlenga This is my paper

Pith reviewed 2026-05-21 13:06 UTC · model grok-4.3

classification 💰 econ.TH cs.GTcs.MA
keywords assignment problemmajoritarian rulestop cyclemajority graphPareto optimalitysemi-popularitysocial choice
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The pith

In assignment problems the top cycle of majority preferences is restricted to one, two, all-but-two, all-but-one, or all allocations.

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

The paper studies majoritarian rules for assigning objects to agents and shows that the assignment domain has a special structure that produces a near one-to-one link between preference profiles and majority graphs. Because of this link, whether an assignment is Pareto optimal, least unpopular, or mixed popular can be read directly from its majority graph. All Pareto-optimal assignments turn out to be semi-popular and to lie inside the top cycle, so serial dictatorships already produce members of the top cycle. The main result gives a complete characterization of the top cycle itself: its size can only be one, two, all but two, all but one, or the full set of assignments.

Core claim

We establish a near one-to-one correspondence between preference profiles and majority graphs in the assignment domain. This correspondence implies that Pareto-optimality, least unpopularity, and mixed popularity are properties of the majority graph alone. All Pareto-optimal assignments are semi-popular and belong to the top cycle; members of the top cycle can therefore be obtained by serial dictatorships. The top cycle admits a complete characterization: it consists of exactly one, two, all-but-two, all-but-one, or all assignments. By contrast, the uncovered set is always very small.

What carries the argument

the near one-to-one correspondence between preference profiles and majority graphs that arises from the structure of the assignment domain

If this is right

  • Every Pareto-optimal assignment is semi-popular.
  • Every Pareto-optimal assignment lies in the top cycle.
  • Serial dictatorships produce members of the top cycle.
  • The uncovered set contains only a very small number of assignments.

Where Pith is reading between the lines

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

  • The same structural correspondence may make it possible to compute popular or stable assignments by operating only on the majority graph rather than on the full preference profile.
  • Other restricted domains in social choice may exhibit similar restrictions on the size of the top cycle once their majority-graph structure is examined.
  • The characterization suggests that checking membership in the top cycle for a given assignment can be reduced to checking a small number of graph-theoretic conditions.

Load-bearing premise

The assignment domain creates a near one-to-one correspondence between preference profiles and majority graphs.

What would settle it

A concrete preference profile over assignments whose top cycle contains three assignments, or any number other than one, two, all-but-two, all-but-one, or all of them.

Figures

Figures reproduced from arXiv: 2602.14816 by Alexander Schlenga, Chris Dong, Felix Brandt, Haoyuan Chen, Patrick Lederer.

Figure 1
Figure 1. Figure 1: Size distributions of UCs for 𝑛 = 5. The high peak is at size 2 for all of them. In total, there are 9, 078, 630 profiles (up to symmetries) and there are 5! = 120 different assign￾ments. 100 200 300 400 0 5 10 15 20 25 Cardinality Frequency (as percentage) McKelvey Bordes Gillies Pareto 20 40 60 80 100 0 1 2 3 4 5 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: It turns out that the Bordes-UC is almost indistinguishable from the McKelvey-UC, while the Gillies-UC is, on average, the most discriminating one. This can be explained as follows: Under the Gillies-UC, for an assignment 𝜆 to not be covered despite 𝜇 ≻ 𝜆, there needs to exist another assignment 𝜂 such that 𝜆 ¥ 𝜂 ≻ 𝜇. How￾ever, for small numbers of agents, there are many profiles admitting popular assignme… view at source ↗
Figure 3
Figure 3. Figure 3: Cumulative distributions of the ratio of UC and PO [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
read the original abstract

A central problem in multiagent systems is the fair assignment of objects to agents. In this paper, we initiate the analysis of classic majoritarian social choice functions in assignment. Exploiting the special structure of the assignment domain, we show a number of surprising results with no counterparts in general social choice. In particular, we establish a near one-to-one correspondence between preference profiles and majority graphs. This correspondence implies that key properties of assignments -- such as Pareto-optimality, least unpopularity, and mixed popularity -- can be determined solely by the associated majority graph. We further show that all Pareto-optimal assignments are semi-popular and belong to the top cycle. Elements of the top cycle can thus easily be found via serial dictatorships. Our main result is a complete characterization of the top cycle, which implies the top cycle can only consist of one, two, all but two, all but one, or all assignments. By contrast, we find that the uncovered set contains only very few assignments.

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 initiates the study of majoritarian social choice functions applied to the assignment of indivisible objects to agents. Exploiting the special structure of the assignment domain, it claims a near one-to-one correspondence between strict preference profiles and majority graphs. This correspondence is used to show that Pareto optimality, least unpopularity, and mixed popularity are determined solely by the majority graph. All Pareto-optimal assignments are shown to be semi-popular and to lie in the top cycle, so that top-cycle elements can be found via serial dictatorships. The central result is a complete characterization of the top cycle implying that its size must be 1, 2, n-2, n-1, or n; by contrast the uncovered set is shown to contain only very few assignments.

Significance. If the structural correspondence and the top-cycle characterization hold, the paper delivers results with no direct counterparts in general social choice theory. The ability to read Pareto optimality and top-cycle membership directly from the majority graph, together with the sharp size restrictions on the top cycle, constitutes a substantive contribution to the theory of assignment mechanisms. The explicit link to serial dictatorships for locating top-cycle elements is a practical strength.

major comments (2)
  1. [Section 3 (correspondence result)] The central claim that key properties (Pareto optimality, semi-popularity, top-cycle membership) can be read off the majority graph alone rests on the asserted near one-to-one correspondence between preference profiles and majority graphs. The manuscript must supply an explicit construction of this mapping or a proof that rules out counterexamples in which two distinct profiles induce identical majority graphs yet differ in their Pareto sets or top-cycle membership; without such a demonstration the implications for Pareto optimality and the size characterization do not follow.
  2. [Main characterization theorem] Theorem establishing the top-cycle size restriction (sizes 1, 2, n-2, n-1, or n): the proof relies on the domain-specific structure of assignment problems. The argument should be checked for any unstated restrictions on the preference domain; if the structural correspondence fails for even one profile, the exhaustive list of admissible sizes would be incomplete.
minor comments (2)
  1. [Abstract and Section 2] Define 'mixed popularity' and 'least unpopularity' explicitly on first use; the abstract employs these terms without prior definition.
  2. [Section 4] Clarify the precise notion of 'semi-popular' used in the statement that all Pareto-optimal assignments are semi-popular; confirm it coincides with the standard definition in the majoritarian literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments focus on strengthening the presentation of the correspondence result and confirming the domain of the top-cycle characterization. We address each point below and will incorporate clarifications and an explicit construction in the revised manuscript.

read point-by-point responses

  1. Referee: [Section 3 (correspondence result)] The central claim that key properties (Pareto optimality, semi-popularity, top-cycle membership) can be read off the majority graph alone rests on the asserted near one-to-one correspondence between preference profiles and majority graphs. The manuscript must supply an explicit construction of this mapping or a proof that rules out counterexamples in which two distinct profiles induce identical majority graphs yet differ in their Pareto sets or top-cycle membership; without such a demonstration the implications for Pareto optimality and the size characterization do not follow.

    Authors: We agree that greater explicitness will improve the exposition. The manuscript already derives the near one-to-one correspondence from the assignment domain's structure, but we will add to Section 3 an explicit construction of the mapping together with a lemma showing that any two strict preference profiles inducing the same majority graph necessarily share identical Pareto sets and the same top cycle. This addition will make the implications for Pareto optimality and the size characterization fully rigorous. revision: yes

  2. Referee: [Main characterization theorem] Theorem establishing the top-cycle size restriction (sizes 1, 2, n-2, n-1, or n): the proof relies on the domain-specific structure of assignment problems. The argument should be checked for any unstated restrictions on the preference domain; if the structural correspondence fails for even one profile, the exhaustive list of admissible sizes would be incomplete.

    Authors: The proof of the top-cycle characterization applies to the standard domain of all strict linear orders over assignments. We have verified that the structural correspondence holds without exception for every profile in this domain; no unstated restrictions exist. The exhaustive enumeration of admissible sizes (1, 2, n-2, n-1, or n) follows directly from this complete coverage. We will insert a brief clarifying remark stating the domain explicitly and confirming the absence of counterexamples. revision: partial

Circularity Check

0 steps flagged

No circularity: structural correspondence and top-cycle characterization derived independently from assignment domain properties.

full rationale

The paper establishes a near one-to-one correspondence between strict preference profiles and majority graphs by exploiting the special structure of the assignment domain. This correspondence is then used to show that Pareto-optimality, semi-popularity, and top-cycle membership can be read off the majority graph alone, followed by a direct characterization of possible top-cycle sizes (1, 2, n-2, n-1, or n). No step reduces a claimed result to a quantity defined in terms of that result, nor does any load-bearing premise collapse to a self-citation or fitted input. The derivation remains self-contained against the domain axioms.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard domain assumptions of social choice theory (strict linear orders over bundles, finite sets of agents and objects) that are inherited from prior literature rather than introduced ad hoc; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption Agents possess strict linear preferences over all possible assignments.
    Standard assumption invoked to define majority comparisons and Pareto optimality in the assignment setting.

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Reference graph

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    Case (2): Finally, assume that there are four distinct agents 𝑥, 𝑦, 𝑧, 𝑤 and two houses𝑝, 𝑞 such that𝑥 and 𝑦 top-rank 𝑝, and𝑤 and 𝑧 top-rank 𝑞

    Hence, by the same argument as before, we get that 𝜆¥ ∗ 𝜆′ and thus𝜆∈TC(𝑃). Case (2): Finally, assume that there are four distinct agents 𝑥, 𝑦, 𝑧, 𝑤 and two houses𝑝, 𝑞 such that𝑥 and 𝑦 top-rank 𝑝, and𝑤 and 𝑧 top-rank 𝑞. Once again, we let 𝜆 denote an arbitrary assignment with 𝜆(𝑥)=𝑝 and show that 𝜆∈TC(𝑃) . To this end, we define 𝜆′ as the assignment obtai...

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    Just as before, this means that BC(𝑃) ∩TC(𝑃)=∅ , so TC(𝑃)=𝑀\PP(𝑃) andBC(𝑃)=PP(𝑃)

    Further, it is straightforward to see that |PP(𝑃)|= 1. Just as before, this means that BC(𝑃) ∩TC(𝑃)=∅ , so TC(𝑃)=𝑀\PP(𝑃) andBC(𝑃)=PP(𝑃). Finally, we turn to case (v). To this end, we recall that by Lem- mas 5 and 6, both |TC(𝑃)|> 2and |BC(𝑃)|> 2if none of the first four cases apply. If 𝑛= 5, our claim follows directly from Claim (3) of Fact 1. We hence su...

    work page