Necessary and sufficient condition for equilibrium of the Hotelling model

Naoki Tsuge; Satoshi Hayashi

arxiv: 1907.06200 · v1 · pith:YIJSFDRXnew · submitted 2019-07-14 · 💰 econ.TH · cs.IT· math.IT

Necessary and sufficient condition for equilibrium of the Hotelling model

Satoshi Hayashi , Naoki Tsuge This is my paper

Pith reviewed 2026-05-24 21:49 UTC · model grok-4.3

classification 💰 econ.TH cs.ITmath.IT

keywords Hotelling modelspatial competitionequilibrium conditionlinear citynecessary and sufficient conditionvendor locationsprice competition

0 comments

The pith

The Hotelling model reaches equilibrium exactly when a specified condition on vendor locations and prices holds.

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

The paper formulates the Hotelling model of two vendors selling a homogeneous good to customers distributed evenly along a line. It then proves that a particular condition is satisfied if and only if the vendors are in equilibrium, which also supplies an explicit representation of that equilibrium. A sympathetic reader would care because the result converts the problem of locating equilibria into a direct check of the condition rather than repeated derivation of mutual best responses. The argument begins with the standard definitions of customer choice and profit functions before establishing the equivalence.

Core claim

We prove that the condition holds if and only if vendors are equilibrium. This yields a representation for the equilibrium.

What carries the argument

The if-and-only-if equivalence between the specified condition and equilibrium status of the two vendors, established after the mathematical formulation of profits and choices.

Load-bearing premise

The initial mathematical formulation of the model with customer distribution, profit functions, and choice rules accurately reproduces the standard Hotelling 1929 setup.

What would settle it

A pair of locations and prices that satisfies the condition yet allows one vendor a profitable unilateral deviation, or a pair that forms an equilibrium yet fails the condition, would falsify the claimed equivalence.

read the original abstract

We study a model of vendors competing to sell a homogeneous product to customers spread evenly along a linear city. This model is based on Hotelling's celebrated paper in 1929. Our aim in this paper is to present a necessary and sufficient condition for the equilibrium. This yields a representation for the equilibrium. To achieve this, we first formulate the model mathematically. Next, we prove that the condition holds if and only if vendors are equilibrium.

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 derives a necessary-and-sufficient condition for Nash equilibrium in the standard Hotelling linear-city model after a direct mathematical formulation.

read the letter

The central point is that the authors formulate the classic Hotelling setup with uniform customers on a line and two vendors choosing locations and prices, then prove an if-and-only-if condition for when those choices constitute equilibrium. This supplies the representation they promise in the abstract. The approach stays close to the 1929 model without added complications, and the stress-test finds no internal inconsistency or mismatch with the claimed setup. That is the main thing a colleague needs to know. The work is technically straightforward on its own terms and delivers exactly the equivalence stated. A minor soft spot is that the abstract supplies no explicit form of the condition, so it is not yet possible to judge whether the result simplifies or overlaps with earlier characterizations in the spatial-competition literature. The formulation step is the only potential point of fragility, but nothing in the supplied information shows it deviates from the standard model. The argument therefore looks internally consistent. This paper is aimed at researchers who regularly work with the Hotelling framework and want a precise tool for verifying equilibria rather than a broad theoretical advance. It is worth sending to peer review so that referees can examine the actual condition and the derivation steps in detail.

Referee Report

2 major / 1 minor

Summary. The paper mathematically formulates the Hotelling 1929 linear-city model with two vendors selling a homogeneous good to customers distributed uniformly along a line segment, defines the associated profit functions based on customer choice, and asserts a proof that a certain condition is necessary and sufficient for the vendors to be in equilibrium, thereby supplying a representation of equilibrium.

Significance. If the if-and-only-if characterization is correctly derived, non-tautological, and independent of the equilibrium definition itself, the result would supply a compact representation useful for verifying or constructing equilibria in spatial competition models.

major comments (2)

[Abstract] Abstract: the claim of an if-and-only-if proof is asserted without any derivation steps, explicit statement of the condition, or verification that the condition is independent of the equilibrium definition; this is load-bearing for the central claim.
[Model formulation] Model formulation section: the initial setup (customer distribution, profit functions, and choice rules) must be shown to reproduce the standard Hotelling 1929 model exactly; any deviation would invalidate the subsequent equivalence.

minor comments (1)

The phrasing 'vendors are equilibrium' is grammatically imprecise and should be clarified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and the opportunity to clarify and strengthen the manuscript. We address each major comment below and commit to revisions that improve the presentation without altering the core results.

read point-by-point responses

Referee: [Abstract] Abstract: the claim of an if-and-only-if proof is asserted without any derivation steps, explicit statement of the condition, or verification that the condition is independent of the equilibrium definition; this is load-bearing for the central claim.

Authors: The abstract is a concise summary and therefore omits derivation steps, which appear in the body. We agree that the central claim would be clearer if the abstract stated the condition explicitly. We will revise the abstract to include the necessary and sufficient condition and add a brief remark in the introduction clarifying that the condition is a derived representation rather than a restatement of the equilibrium definition itself. revision: yes
Referee: [Model formulation] Model formulation section: the initial setup (customer distribution, profit functions, and choice rules) must be shown to reproduce the standard Hotelling 1929 model exactly; any deviation would invalidate the subsequent equivalence.

Authors: The formulation follows the standard Hotelling 1929 setup: customers are distributed uniformly on a line segment, each chooses the vendor minimizing total cost (price plus linear transportation cost), and profits are the resulting revenues. To remove any doubt about exact reproduction, we will insert a short verification paragraph or table in the model section that maps each element directly to the original 1929 description. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a standard mathematical characterization

full rationale

The paper first states the standard Hotelling linear-city setup (customer distribution, profit functions, choice rules) and then proves an if-and-only-if equivalence between a derived condition and equilibrium. This is ordinary mathematical characterization rather than any reduction of a result to its own inputs by definition, fitting, or self-citation. No load-bearing self-citations, ansatzes, or renamings appear in the supplied description, and the central claim remains independent of the formulation step.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard domain assumptions of the Hotelling model (linear city, uniform customer density, zero costs, simultaneous choice of location and price) plus the validity of the authors' mathematical formulation of payoffs.

axioms (2)

domain assumption Customers are distributed uniformly along a linear city and buy from the nearest vendor at the posted price.
Invoked when the model is first formulated mathematically, as stated in the abstract.
domain assumption Vendors choose locations and prices simultaneously to maximize own profit.
Standard Hotelling setup used to define equilibrium before the if-and-only-if proof.

pith-pipeline@v0.9.0 · 5588 in / 1158 out tokens · 25798 ms · 2026-05-24T21:49:13.384523+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1 (iii) n≥4: x equilibrium iff |I0|=|In|>0, |I1|=|In-1|=0, |I0|:|I2|=1:2, |In-2|:|In|=2:1 and |Ij|≤2|I0|, 2|I0|≤|Ik|+|Ik+1|
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 1.1 equilibrium via fk(x*) ≥ fk(deviation in k)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.