Doctor-Optimal Stability in Unitary Many-to-Many Markets

Yi-You Yang

In bilaterally unitary many-to-many matching with contracts, unilateral substitutability restores a unique doctor-optimal stable allocation that every doctor-proposing cumulative-offer path reaches.

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2026-07-14 09:29 UTC pith:3WG4P6V2

load-bearing objection Clean domain-boundary result: bilateral unitarity restores doctor-optimal COP and a new quasi-stable lattice under unilateral substitutability.

arxiv 2607.10756 v1 pith:3WG4P6V2 submitted 2026-07-12 econ.TH

Doctor-Optimal Stability in Unitary Many-to-Many Markets

Yi-You Yang This is my paper

classification econ.TH

keywords matching with contractsunilateral substitutabilitybilateral unitaritydoctor-optimal stabilitycumulative offer processweak hospital-quasi-stabilityrural hospitals theorem

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper asks when a doctor-optimal stable matching still exists once both doctors and hospitals can hold several partners but each pair may sign at most one contract. Under doctor substitutability and hospital unilateral substitutability (both with unitarity and irrelevance of rejected contracts), every trajectory of the doctor-proposing cumulative-offer process ends at the same stable allocation: the greatest one under the doctor Blair order. The same allocation is also the top of a new lattice of weakly hospital-quasi-stable allocations, which relax full stability to relationship retention rather than contract retention. That common point is hospital-pessimal in the revealed-choice sense, and under the law of aggregate demand every agent keeps the same number of relationships at every stable outcome. The result shows that the many-to-one doctor-optimal guarantee survives portfolio choice across partners once contracts inside each bilateral relationship are forced to be mutually exclusive.

Core claim

Under bilateral unitarity, doctor substitutability and hospital unilateral substitutability (with IRC) guarantee that the doctor-proposing cumulative-offer process always terminates at the unique greatest stable allocation under the doctor Blair order; that allocation coincides with the greatest weakly hospital-quasi-stable allocation and is hospital-pessimal.

What carries the argument

Weakly hospital-quasi-stable allocations (doctor-individually rational allocations in which each hospital, when offered all doctor-acceptable contracts, retains some contract with every currently employed doctor) form a finite lattice under the doctor Blair order; an improvement operator T on that lattice has fixed points exactly at the stable allocations, so its greatest element is stable.

Load-bearing premise

Hospitals must treat alternative contracts with the same doctor as mutually exclusive; once several contracts can coexist inside one bilateral relationship, the doctor-optimal conclusion can fail.

What would settle it

Exhibit a bilaterally unitary market satisfying the stated choice conditions whose stable set has two or more elements that are incomparable under the doctor Blair order, or a cumulative-offer trajectory that ends at a stable allocation that is not greatest.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Clean domain-boundary result: bilateral unitarity restores doctor-optimal COP and a new quasi-stable lattice under unilateral substitutability.

read the letter

This paper closes a clean gap. Under bilateral unitarity (doctors substitutable+unitary+IRC; hospitals unilaterally substitutable+unitary+IRC), every doctor-proposing cumulative-offer trajectory hits the unique greatest stable allocation in the doctor Blair order. That same allocation is the greatest element of a new lattice of weakly hospital-quasi-stable allocations (relationship retention, not contract retention). The three objects coincide, and the common one is hospital-pessimal in the revealed-choice sense. Rural-hospitals follows under LAD.

What is new is the unitarity boundary itself and the order-theoretic route. Hatfield-Kojima give doctor-optimality for unit-demand doctors; Bando-Hirai show the conclusion fails once multiple contracts can coexist inside one bilateral pair; Kasuya shows US is maximal on the unitary side. Yang keeps portfolio choice across partners on both sides, forces mutual exclusivity inside each pair, and recovers the doctor-optimal selection. The weakly hospital-quasi-stable lattice is a genuine adaptation of firm-quasi-stability: hospitals may rewrite terms with a doctor but must keep the identity. The improvement operator T needs no isotonicity; fixed points inside Qw are exactly the stable allocations, so the greatest element is stable. That is independent of the procedural argument.

The math is self-contained. Lemma 1 (irrevocable rejections) is the technical core: identity preservation + doctor unitarity blocks simultaneous same-pair contracts + infinite descent. Proposition 2 (stability = CH(Γ(Y))) and the lattice join are standard once unitarity is in place. Appendices fill the steps. No free parameters, no circular redefinition of the target.

Soft spots are minor and explicit. Hospital unitarity is load-bearing; the paper itself cites Bando-Hirai Example 3 as the counter-example once it is dropped. That is a domain restriction, not a hidden gap. Significance is solid inside matching theory, not earth-shaking outside it. Citation pattern is tight and honest.

This is for people who work on matching with contracts or market design with multi-contract relationships. It deserves a serious referee. I would engage with it and expect to cite the lattice construction and the unitarity boundary.

Referee Report

0 major / 4 minor

Summary. The paper studies bilaterally unitary many-to-many doctor–hospital matching with contracts under choice-function primitives. Doctors have substitutable, unitary choice functions satisfying IRC; hospitals have unilaterally substitutable, unitary choice functions satisfying IRC. The main results are that every trajectory of the doctor-proposing cumulative offer process terminates at the unique greatest stable allocation under the doctor Blair order (Theorem 1), that weakly hospital-quasi-stable allocations form a finite lattice whose greatest element is stable (Theorems 2–3), and that these two constructions coincide with the doctor-optimal stable allocation (Corollary 3). The common allocation is hospital-pessimal in the revealed-choice sense (Proposition 5), and under the law of aggregate demand the rural hospitals theorem holds for relationship counts (Corollary 4). The domain restriction of bilateral unitarity is shown to be essential by reference to the nonunitary counterexample of Bando and Hirai (2026) and is maximal for a universal guarantee of doctor-optimal stability by Kasuya’s necessity result (Corollary 2).

Significance. The paper cleanly isolates bilateral unitarity as the boundary that restores doctor-optimal stability under unilateral substitutability once doctors are allowed portfolio choice. It connects three strands—Hatfield–Kojima (2010) sufficiency in many-to-one, Kasuya (2021) maximal-domain necessity, and the Bando–Hirai (2026) nonunitary failure—into a sharp domain statement. The dual characterizations (procedural via cumulative offers and order-theoretic via the weakly hospital-quasi-stable lattice) are independent and mutually reinforcing; the lattice construction is a genuine conceptual contribution that replaces contract retention with identity retention. Full proofs of the key lemmas (especially irrevocability of rejections via infinite descent) are supplied, and the maximality corollary is carefully scoped as a domain-level rather than market-level claim. If the results hold, the paper supplies a usable sufficient condition for doctor-optimal selection in many-to-many markets with contracts that is both necessary (within the unitary hospital domain) and tight.

minor comments (4)

In the proof of Lemma 3 (Appendix C), the reference “By Corollary 2” appears to be a typographical slip for Lemma 2 (cumulative rejections); the same slip recurs later in the proof of Lemma 4. Correcting the cross-references would remove a small source of confusion.
Section 2.1, after Proposition 1: a one-sentence reminder that hospital choice need not be path-independent (already stated later) would help readers who expect the usual Hatfield–Milgrom toolkit to apply symmetrically.
Definition 1 of weak hospital-quasi-stability could be accompanied by a short parenthetical example of a relationship-preserving but contract-changing renegotiation; this would make the distinction from classical firm-quasi-stability more immediate for readers coming from Sotomayor or Yang.
The arXiv date stamp (July 2026) and the Bando–Hirai (2026) citation are consistent internally but may need updating if the paper is revised for journal submission after those works appear in print.

Circularity Check

0 steps flagged

No significant circularity: doctor-optimal stable allocation is derived from independent choice axioms, not defined as the cumulative-offer or lattice outcome.

full rationale

The paper defines stability (individual rationality + no blocking set), the doctor Blair order, the cumulative-offer process, and weakly hospital-quasi-stability independently of one another. Theorem 1 proves that every COP trajectory ends at a stable allocation that Blair-dominates every other stable allocation, using irrevocability of rejections (Lemma 1), terminal identities (Lemma 3), and the stability characterization (Proposition 2). Theorem 2–3 and Proposition 4 construct a lattice of weakly hospital-quasi-stable allocations and an improvement operator T whose fixed points are exactly the stable allocations; the greatest element is therefore stable and coincides with the COP outcome (Corollary 3). These are genuine derivations under Assumption 1 (substitutability/unitarity/IRC), not tautologies. Self-citations to the author’s earlier quasi-stability papers supply background motivation for the relaxation but are not load-bearing for the coincidence result; the proofs are self-contained in the appendices. Hospital unitarity is an explicit domain restriction whose necessity is documented via Bando–Hirai Example 3, not a circular smuggling of the conclusion. No fitted parameters, self-definitional identities, or uniqueness theorems imported from the same author appear. Score 0 is therefore appropriate.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 1 invented entities

The paper is pure theory. Its load-bearing content consists of standard choice-function axioms plus the domain restriction of bilateral unitarity. No free parameters or fitted constants appear. The only invented entity is the weakly hospital-quasi-stable domain, introduced as a technical intermediate whose greatest element is later shown to be stable.

axioms (5)

domain assumption Doctor choice functions are substitutable, unitary, and satisfy IRC (Assumption 1).
Standard conditions from Hatfield-Milgrom / Aygün-Sönmez; used throughout for path independence and Blair order.
domain assumption Hospital choice functions are unilaterally substitutable, unitary, and satisfy IRC (Assumption 1).
Unilateral substitutability from Hatfield-Kojima; unitarity is the paper's key domain restriction that restores doctor-optimality.
standard math Under unitarity, unilateral substitutability is equivalent to substitutability across doctors (Proposition 1, citing Bando-Hirai).
Used to convert identity-level substitution into the rejection-irreversibility argument of Lemma 1.
standard math Stability characterization: Y stable iff Y = C_H(Γ(Y)) under doctor substitutability, hospital unitarity, and IRC (Proposition 2).
Technical lemma proved in Appendix B; bridges cumulative-offer terminal states to stability.
domain assumption Law of aggregate demand for the rural-hospitals corollary only.
Imposed solely for Corollary 4; not needed for the main doctor-optimality theorems.

invented entities (1)

Weakly hospital-quasi-stable allocations no independent evidence
purpose: Relax contract retention to doctor-identity retention so that the set of relationship-preserving states forms a lattice whose greatest element is stable.
Defined in Definition 1; the lattice and improvement operator T are the order-theoretic half of the main characterization. No independent empirical handle is claimed or needed.

pith-pipeline@v1.1.0-grok45 · 18276 in / 2654 out tokens · 32097 ms · 2026-07-14T09:29:20.546832+00:00 · methodology

0 comments

read the original abstract

We study bilaterally unitary many-to-many doctor--hospital matching with contracts, taking choice functions as primitives. Doctor choices are substitutable and satisfy irrelevance of rejected contracts, while hospital choices are unilaterally substitutable and satisfy the same condition. Every trajectory of the doctor-proposing cumulative offer process terminates at the greatest stable allocation under the doctor Blair order. We also introduce weakly hospital-quasi-stable allocations and show that they form a finite lattice whose greatest element is stable. Hence, the cumulative-offer outcome, the greatest weakly hospital-quasi-stable allocation, and the doctor-optimal stable allocation coincide. The common allocation is hospital-pessimal in the revealed-choice sense. Under the law of aggregate demand, every agent signs the same number of contracts at all stable allocations.

discussion (0)

Reference graph

Works this paper leans on

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Yi-You Yang , title =

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Yusuke Kasuya , doi =. Unilateral substitutability is necessary for doctor-optimal stability , volume =. Economics Letters , month =

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Substitutes and stability for many-to-many matching with contracts , volume =

Keisuke Bando and Toshiyuki Hirai and Jun Zhang , doi =. Substitutes and stability for many-to-many matching with contracts , volume =. Games and Economic Behavior , month =

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Cumulative offer process is order-independent , volume =

Daisuke Hirata and Yusuke Kasuya , doi =. Cumulative offer process is order-independent , volume =. Economics Letters , month =

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Flanagan , doi =

Francis X. Flanagan , doi =. The substitutes condition and the lattice structure of the set of stable allocations , volume =. Journal of Mathematical Economics , month =

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Matching with contracts: Comment , volume =

Orhan Aygün and Tayfun Sönmez , doi =. Matching with contracts: Comment , volume =. American Economic Review , pages =

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Substitutes and stability for matching with contracts , volume =

John William Hatfield and Fuhito Kojima , doi =. Substitutes and stability for matching with contracts , volume =. Journal of Economic Theory , keywords =

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John William Hatfield and Paul R. Milgrom , doi =. Matching with contracts , volume =. American Economic Review , month =

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A Non-constructive Elementary Proof of the Existence of Stable Marriages , volume =

Marilda Sotomayor , doi =. A Non-constructive Elementary Proof of the Existence of Stable Marriages , volume =. Games and Economic Behavior , month =

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The Lattice Structure of the Set of Stable Matchings with Multiple Partners , volume =

Charles Blair , doi =. The Lattice Structure of the Set of Stable Matchings with Multiple Partners , volume =. Mathematics of Operations Research , month =

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Kelso and Vincent P

Alexander S. Kelso and Vincent P. Crawford , doi =. Job Matching, Coalition Formation, and Gross Substitutes , volume =. Econometrica , month =

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Aizerman and Andrew V

Mark A. Aizerman and Andrew V. Malishevski , doi =. IEEE Transactions on Automatic Control , title =

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Plott , doi =

Charles R. Plott , doi =. Path Independence, Rationality, and Social Choice , volume =. Econometrica , month =