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arxiv: 2510.09076 · v2 · submitted 2025-10-10 · 💰 econ.TH

Arrow's Impossibility Theorem as a Generalisation of Condorcet's Paradox

Ori Livson , Mikhail Prokopenko This is my paper

Pith reviewed 2026-05-18 08:20 UTC · model grok-4.3

classification 💰 econ.TH
keywords Arrow's Impossibility TheoremCondorcet's Paradoxsocial welfare functionspreference cyclesweak preferencessocial choice theoryvoting paradoxesranked choice
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The pith

Arrow's impossibility theorem can be restated as the unavoidable production of contradictory preference cycles even when voters express indifference.

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

The paper establishes that Arrow's Impossibility Theorem is equivalent to the claim that any social welfare function violating at least one of Arrow's conditions must generate contradictory preference cycles in some profile of weak preferences. It does so by extending an earlier construction method from strict rankings to the general case that permits ties between alternatives. A sympathetic reader would care because this equivalence supplies a single, cycle-based lens for understanding why no ranked-choice system can meet the usual fairness requirements simultaneously. The work also derives several further properties of social welfare functions from the same profile-construction technique.

Core claim

Arrow's Impossibility Theorem can be equivalently stated in terms of contradictory preference cycles, accounting for weak preferences. The proof generalizes D'Antoni's strict-preference construction by explicitly building, for any social welfare function that violates one or more of Arrow's conditions, a profile of individual weak orders whose aggregated social ordering contains a cycle.

What carries the argument

Explicit construction of preference profiles that induce contradictory cycles for any social welfare function violating Arrow's axioms, extended to weak orders.

If this is right

  • Any social welfare function failing independence of irrelevant alternatives produces cycles in some profiles that include indifferences.
  • The cycle equivalence continues to hold when voters may rank alternatives as tied.
  • The profile-construction technique yields additional facts about the structure of social welfare functions beyond the main equivalence.
  • The same method can be applied to study cycles in related inconsistency phenomena such as money pumps or intransitive games.

Where Pith is reading between the lines

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

  • The cycle-construction approach may supply a uniform explanation for preference inconsistencies across voting, decision theory, and betting systems.
  • Testing the construction on concrete rules such as plurality or Borda count with ties would reveal characteristic cycle patterns.
  • The framework could be extended to settings with more than three alternatives to check whether cycle length or structure changes.

Load-bearing premise

For every social welfare function that violates at least one Arrow condition, an explicit profile of weak preferences can always be constructed to produce a contradictory cycle.

What would settle it

Discovery of even one social welfare function that violates an Arrow condition yet admits no weak-preference profile generating a cycle would refute the claimed equivalence.

read the original abstract

Arrow's Impossibility Theorem is a seminal result of Social Choice Theory that demonstrates the impossibility of ranked-choice decision-making processes to jointly satisfy a number of intuitive and seemingly desirable constraints. The theorem is often described as a generalisation of Condorcet's Paradox, wherein pairwise majority voting may fail to jointly satisfy the same constraints due to the occurrence of elections that result in contradictory preference cycles. However, a formal proof of this relationship has been limited to D'Antoni's work, which applies only to the strict preference case, i.e., where indifference between alternatives is not allowed. In this paper, we generalise D'Antoni's methodology to prove in full (i.e., accounting for weak preferences) that Arrow's Impossibility Theorem can be equivalently stated in terms of contradictory preference cycles. This methodology involves explicitly constructing profiles that lead to preference cycles. Using this framework, we also prove a number of additional facts regarding social welfare functions. As a result, this methodology may yield further insights into the nature of preference cycles in other domains e.g., Money Pumps, Dutch Books, Intransitive Games, etc.

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

1 major / 1 minor

Summary. The paper generalizes D'Antoni's strict-preference construction to show that Arrow's Impossibility Theorem is equivalent to the existence of contradictory preference cycles for social welfare functions on weak orders. It does so by explicitly building profiles that force cycles whenever at least one of Arrow's conditions (IIA, Pareto, non-dictatorship, or unrestricted domain) is violated, and derives several additional facts about SWFs from the same framework.

Significance. If the explicit constructions are complete for weak preferences, the result supplies a direct, profile-based reformulation of Arrow's theorem that makes the link to Condorcet's paradox fully rigorous rather than merely intuitive. The constructive method is a strength and could be reused for related intransitivity questions in money pumps or Dutch books.

major comments (1)
  1. The central equivalence claim requires that every SWF violating an Arrow condition on weak orders admits an explicit profile whose social output contains a contradictory cycle. The manuscript's generalization of D'Antoni's decisive-set and profile-swap technique must be checked against cases in which the SWF returns indifferences on some pairs while still violating IIA or Pareto on strict pairs; if the construction only perturbs strict profiles or assumes tie-breaking that preserves the violation, the equivalence may fail for constant-tie SWFs on certain triples.
minor comments (1)
  1. Notation for weak orders and the precise definition of 'contradictory cycle' should be stated once at the beginning rather than reintroduced in each construction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying a potential gap in the coverage of weak-order cases. We address the major comment below and will revise the manuscript to strengthen the explicit constructions.

read point-by-point responses

  1. Referee: The central equivalence claim requires that every SWF violating an Arrow condition on weak orders admits an explicit profile whose social output contains a contradictory cycle. The manuscript's generalization of D'Antoni's decisive-set and profile-swap technique must be checked against cases in which the SWF returns indifferences on some pairs while still violating IIA or Pareto on strict pairs; if the construction only perturbs strict profiles or assumes tie-breaking that preserves the violation, the equivalence may fail for constant-tie SWFs on certain triples.

    Authors: We appreciate this observation on the handling of indifferences. The construction in the manuscript begins from decisive sets defined on strict preferences and extends the profile-swap argument by allowing individual indifferences on non-critical pairs while preserving the violation on the target triple. For SWFs that output indifferences on some pairs, the swaps are performed only on the strict components of the violating condition (IIA or Pareto), forcing the social preference to produce a strict cycle on the triple without requiring tie-breaking. Constant-tie SWFs violate Pareto on any unanimous strict profile; however, because they produce no strict social preferences, they do not generate cycles. We therefore acknowledge that the current statement of the equivalence requires qualification for such pathological functions. We will revise the manuscript to add an explicit lemma showing that the result holds for all SWFs that are not constant on every pair, together with a separate remark that constant-tie functions are already excluded by the standard interpretation of Pareto in the weak-order setting. revision: yes

Circularity Check

0 steps flagged

No circularity: equivalence proven via explicit profile constructions

full rationale

The paper establishes its central claim—that Arrow's theorem is equivalently stated via contradictory preference cycles under weak preferences—through direct generalization of D'Antoni's methodology by explicitly constructing violating profiles for any social welfare function breaching the axioms. This construction-based proof technique does not reduce any step to self-definition, fitted inputs renamed as predictions, or load-bearing self-citations; the cited prior work is external and the derivations remain independent mathematical arguments rather than tautological restatements of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof relies on standard axioms of preference relations and social welfare functions from social choice theory; no free parameters or invented entities are introduced.

axioms (2)
  • standard math Preference relations are complete and transitive (weak orders).
    Invoked when defining profiles and social welfare functions that must handle weak preferences.
  • domain assumption Social welfare functions map preference profiles to a social preference relation.
    Core setup for Arrow's theorem and the cycle equivalence.

pith-pipeline@v0.9.0 · 5724 in / 1207 out tokens · 26696 ms · 2026-05-18T08:20:45.027111+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. Condorcet's Paradox as Non-Orientability

    math.AT 2026-01 unverdicted novelty 7.0

    Condorcet's paradox corresponds to non-orientability of a surface homeomorphic to the Klein bottle or real projective plane in a generalized topological model of strict ordinal preferences.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · cited by 1 Pith paper

  1. [1]

    From Condorcet’s Paradox to Arrow: Yet Another Simple Proof of the Impossibility Theorem

    Massimo D’Antoni. “From Condorcet’s Paradox to Arrow: Yet Another Simple Proof of the Impossibility Theorem”. In: Social Choice and Welfare (Nov. 2024). issn: 1432-217X. doi: 10.1007/s00355-024-01557-8

    work page doi:10.1007/s00355-024-01557-8 2024

  2. [2]

    Arrow’s Theorem

    Michael Morreau. “Arrow’s Theorem”. In: The Stanford Encyclopedia of Philosophy . Ed. by Edward N. Zalta. Winter

    work page

  3. [3]

    Metaphysics Research Lab, Stanford University, 2019

    work page 2019

  4. [4]

    The Philosophy of Intransitive Preference

    Paul Anand. “The Philosophy of Intransitive Preference”. In: The Economic Journal 103.417 (1993), pp. 337–346. issn: 00130133, 14680297. url: http://www.jstor.org/stable/2234772 (visited on 09/05/2025)

    work page arXiv 1993

  5. [5]

    Intransitivity, Utility, and the Aggregation of Preference Patterns

    Kenneth O. May. “Intransitivity, Utility, and the Aggregation of Preference Patterns”. In: Econometrica 22.1 (1954), pp. 1–13. issn: 00129682, 14680262. url: http://www.jstor.org/stable/1909827 (visited on 09/05/2025)

    work page arXiv 1954

  6. [6]

    Money-pump arguments

    Johan E Gustafsson. Money-pump arguments. Cambridge University Press, 2022

    work page 2022

  7. [7]

    2013 , isbn =

    Alan H´ ajek. “Dutch Book Arguments”. In: The Handbook of Rational and Social Choice . Oxford University Press, Jan. 2009. isbn: 9780199290420. doi: 10 . 1093 / acprof : oso / 9780199290420 . 003 . 0008. eprint: https : / / academic . oup . com / book / 0 / chapter / 142646166 / chapter - ag - pdf / 45537939 / book _ 2418 _ section _ 142646166 . ag . pdf....

    work page doi:10.1093/acprof:oso/9780199290420.003.0008 2009

  8. [8]

    Acyclic Domains of Linear Orders: A Survey

    Bernard Monjardet. Acyclic Domains of Linear Orders: A Survey . Ed. by Steven J. Brams, William V. Gehrlein, and Fred S. Roberts. Berlin, Heidelberg, 2009. doi: 10.1007/978-3-540-79128-7_8 . url: https://doi.org/10.1007/978- 3-540-79128-7_8

    work page doi:10.1007/978-3-540-79128-7_8 2009

  9. [9]

    Maximal Condorcet domains. A further progress report

    Clemens Puppe and Arkadii Slinko. “Maximal Condorcet domains. A further progress report”. In: Games and Economic Behavior 145 (2024), pp. 426–450. issn: 0899-8256. doi: https : / / doi . org / 10 . 1016 / j . geb . 2024 . 04 . 001. url: https://www.sciencedirect.com/science/article/pii/S0899825624000502

    work page 2024

  10. [10]

    Social Choice Theory

    Christian List. “Social Choice Theory”. In: The Stanford Encyclopedia of Philosophy . Ed. by Edward N. Zalta and Uri Nodelman. Winter 2022. Metaphysics Research Lab, Stanford University, 2022

    work page 2022

  11. [11]

    Three Brief Proofs of Arrow’s Impossibility Theorem

    John Geanakoplos. “Three Brief Proofs of Arrow’s Impossibility Theorem”. In: Economic Theory 26.1 (2005), pp. 211–

    work page 2005

  12. [12]

    issn: 09382259, 14320479

    work page

  13. [13]

    A One-shot Proof of Arrow’s Impossibility Theorem

    Ning Neil Yu. “A One-shot Proof of Arrow’s Impossibility Theorem”. In: Economic Theory 50.2 (2012), pp. 523–525. issn: 09382259, 14320479

    work page 2012

  14. [14]

    Comparing and Contrasting Arrow’s Impossibility Theorem and G¨ odel’s Incom- pleteness Theorem

    Ori Livson and Mikhail Prokopenko. Comparing and Contrasting Arrow’s Impossibility Theorem and G¨ odel’s Incom- pleteness Theorem. 2025. arXiv: 2504.06589 [math.LO] . url: https://arxiv.org/abs/2504.06589

    work page arXiv 2025

  15. [15]

    Social Choice Theory: A Re-Examination

    Amartya Sen. “Social Choice Theory: A Re-Examination”. In: Econometrica 45.1 (1977), pp. 53–89. issn: 00129682, 14680262. url: http://www.jstor.org/stable/1913287 (visited on 10/01/2025)

    work page arXiv 1977

  16. [16]

    A complete characterization of hierarchy

    Kui Ou-Yang. “A complete characterization of hierarchy”. In: Economics Letters 136 (2015), pp. 162–164. issn: 0165-

    work page 2015

  17. [17]

    org / 10

    doi: https : / / doi . org / 10 . 1016 / j . econlet . 2015 . 09 . 033. url: https : / / www . sciencedirect . com / science / article/pii/S0165176515003912. 15

    work page 2015