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arxiv: 1907.10381 · v1 · pith:4BF2CLRBnew · submitted 2019-07-22 · 💰 econ.TH · cs.LO

Arrow's Theorem Through a Fixpoint Argument

Frank M. V. Feys (Delft University of Technology) , Helle Hvid Hansen (Delft University of Technology) This is my paper

Pith reviewed 2026-05-24 18:09 UTC · model grok-4.3

classification 💰 econ.TH cs.LO
keywords Arrow's theoremfixed point theoremBanach theoremsocial choice theorydictatorshipsocial welfare functionsmetric spacecontractive map
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The pith

Arrow's theorem can be proved by viewing dictatorships as the only fixed points of a contractive map on social welfare functions.

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

The paper offers a new proof of Arrow's theorem in social choice theory. It constructs a contractive map on the space of social welfare functions equipped with a suitable metric, such that the space is complete. By Banach's fixed point theorem, this map has a unique fixed point, which turns out to be a dictatorship. A reader would care because this recasts the impossibility of fair aggregation as the convergence to a dictatorial fixed point under iteration of the map.

Core claim

We show that dictatorships are the fixed points of a certain process defined by a contractive map on the complete metric space of social welfare functions, and thus by Banach's theorem they are the only possible outcomes satisfying the Arrow axioms.

What carries the argument

A contractive map on the complete metric space of social welfare functions whose fixed points are exactly the dictatorial social welfare functions.

If this is right

  • Only dictatorial functions remain unchanged under repeated application of the map.
  • The proof relies on the existence of a metric that makes the map contractive.
  • Social welfare functions can be seen as elements in a metric space where iteration leads to a dictatorship.
  • Arrow's conditions imply that any function satisfying them must be a fixed point of this map.

Where Pith is reading between the lines

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

  • Similar fixed-point arguments might be applicable to other impossibility results in social choice or decision theory.
  • This perspective could suggest ways to approximate social welfare functions by iterating the map until convergence.
  • Extensions might involve relaxing the contractivity to obtain approximate dictatorships or other fixed points.

Load-bearing premise

The map defined on the space of social welfare functions must be contractive with respect to some metric that makes the space complete.

What would settle it

Finding a non-dictatorial social welfare function that satisfies the standard Arrow conditions but is not a fixed point of the proposed map, or demonstrating that no metric exists making the map contractive.

read the original abstract

We present a proof of Arrow's theorem from social choice theory that uses a fixpoint argument. Specifically, we use Banach's result on the existence of a fixpoint of a contractive map defined on a complete metric space. Conceptually, our approach shows that dictatorships can be seen as fixpoints of a certain process.

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 / 0 minor

Summary. The manuscript claims to prove Arrow's theorem via a fixed-point argument: it invokes Banach's theorem to assert that dictatorships are the fixed points of a contractive map defined on a complete metric space of social welfare functions (or preference profiles).

Significance. If the construction were supplied and verified, the result would supply an alternative topological proof of Arrow's theorem and a conceptual link between social choice and contraction mappings. No such construction, metric, or Lipschitz-constant verification appears in the manuscript, so the claimed significance cannot be assessed.

major comments (1)
  1. [Abstract] Abstract (and entire manuscript): the central claim requires a contractive map T on the space of social welfare functions such that d(T(f),T(g)) ≤ k·d(f,g) with k<1 and fixed points exactly the dictatorships; no definition of T, no metric, and no proof that the Lipschitz constant is strictly less than 1 are supplied, so Banach's theorem cannot be applied and the equivalence between fixed points and dictatorships does not follow.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The referee correctly notes that the submitted manuscript does not supply an explicit contractive map, metric, or Lipschitz-constant verification. We will revise the manuscript to include these elements.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and entire manuscript): the central claim requires a contractive map T on the space of social welfare functions such that d(T(f),T(g)) ≤ k·d(f,g) with k<1 and fixed points exactly the dictatorships; no definition of T, no metric, and no proof that the Lipschitz constant is strictly less than 1 are supplied, so Banach's theorem cannot be applied and the equivalence between fixed points and dictatorships does not follow.

    Authors: We agree that the manuscript as submitted contains no definition of the map T, no metric d on the space of social welfare functions, and no verification that T is contractive with constant strictly less than 1. These omissions prevent any application of Banach's theorem or any demonstration that the fixed points coincide with dictatorships. We will revise the paper to supply a complete metric space, an explicit map T, a proof that T is a contraction, and a proof that its fixed points are exactly the dictatorships. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation applies external Banach fixed-point theorem

full rationale

The paper presents a new proof of Arrow's theorem by constructing a complete metric space of social welfare functions and a contractive map whose fixed points are dictatorships, then invoking Banach's theorem (an independent result from functional analysis) to conclude existence. No self-citation chains, no self-definitional steps where the conclusion is presupposed in the map definition, and no fitted parameters renamed as predictions appear. The approach is self-contained against external mathematical benchmarks; contractiveness and completeness are standard prerequisites for the cited theorem rather than reductions to the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on Banach's fixed-point theorem and on the unshown claim that a contractive map whose fixed points correspond to dictatorships can be defined on the relevant space.

axioms (1)
  • standard math Banach's fixed-point theorem: every contractive map on a complete metric space has a unique fixed point.
    Invoked explicitly in the abstract to guarantee existence of the fixpoint.

pith-pipeline@v0.9.0 · 5578 in / 1193 out tokens · 23788 ms · 2026-05-24T18:09:54.951440+00:00 · methodology

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Reference graph

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