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arxiv: 2606.28028 · v1 · pith:4CUA5CPUnew · submitted 2026-06-26 · 💰 econ.TH

Rationalizable Behavior in Matching with Externalities

Antonio Nicol\`o , Pietro Salmaso , Riccardo D. Saulle This is my paper

Pith reviewed 2026-06-29 02:00 UTC · model grok-4.3

classification 💰 econ.TH
keywords matching with externalitiesrationalizabilitystabilityGale-Shapleymatching with couplesconjecture-rationalizable stabilityepistemic foundationpairwise rationality
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The pith

Rationalizable conjectures define a stability concept for matching markets with externalities that always exists and extends Gale-Shapley stability.

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

In matching markets, agents often care about others' matches as well as their own, so stability hinges on what each agent believes would occur after a unilateral deviation. The paper defines rationalizable conjectures as those beliefs that survive iterated elimination of implausible responses, in the manner of rationalizability in games. These conjectures support conjecture-rationalizable stability, a solution concept that is always non-empty, coincides exactly with Gale-Shapley stability when externalities are absent, and supplies predictions in couple-matching problems where ordinary stability is empty. The paper further shows that every conjecture-rationalizable stable matching is rationalizable and supplies an epistemic justification linking the concepts to pairwise rationality plus common belief in pairwise rationality, with correct beliefs required for the stability component.

Core claim

Conjecture-rationalizable stability consists of matchings that remain stable when agents hold only rationalizable conjectures about how others would respond to a deviation. This notion is guaranteed to exist in every market, reduces to the classical Gale-Shapley concept in the absence of externalities, and every such matching is also rationalizable. In markets with couples the concept yields non-empty sets of outcomes even when standard stability is vacuous. Rationalizability itself follows from pairwise rationality and common belief in pairwise rationality, while conjecture-rationalizable stability additionally requires that the conjectures are correct.

What carries the argument

Rationalizable conjectures, formed by iterated elimination of non-rationalizable beliefs about others' responses to deviations, which determine whether a matching is stable against those conjectures.

If this is right

  • Conjecture-rationalizable stability is non-empty in every finite matching market with externalities.
  • The concept coincides with Gale-Shapley stability whenever agents care only about their own partners.
  • Every conjecture-rationalizable stable matching is also a rationalizable matching.
  • In matching markets with couples the concept produces stable outcomes even when the ordinary stability set is empty.
  • Rationalizability is implied by pairwise rationality together with common belief in pairwise rationality; conjecture-rationalizable stability further requires correct beliefs.

Where Pith is reading between the lines

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

  • The framework could be used to predict which matchings survive when agents reason about one another's reasoning in labor or housing markets that exhibit network effects.
  • Empirical work could elicit agents' conjectures after hypothetical deviations and test whether the surviving rationalizable set matches observed behavior.
  • Similar iterated-elimination reasoning might be applied to other cooperative solution concepts that currently lack existence guarantees under interdependent preferences.

Load-bearing premise

The iterated elimination process that defines rationalizable conjectures accurately captures the beliefs agents would actually hold after a deviation.

What would settle it

An experiment or field observation in which agents in a market with externalities form post-deviation beliefs that lie outside the set surviving iterated elimination, or in which observed stable matchings systematically differ from those selected by conjecture-rationalizable stability.

Figures

Figures reproduced from arXiv: 2606.28028 by Antonio Nicol\`o, Pietro Salmaso, Riccardo D. Saulle.

Figure 1
Figure 1. Figure 1: Matchings among a1, a2, a3, and b1. An edge between ai and b1 indicates that the two agents are matched. This market has no P-stable matching: µ 1 is blocked by a1, µ 0 is blocked by pa2, b1q, µ 2 is blocked by pa1, b1q, and µ 3 is blocked by b1. These blocking arguments, however, all rely on specific assumptions about the conjectures held by the deviating agents. Before proceeding further, it is useful to… view at source ↗
Figure 2
Figure 2. Figure 2: Left: agents’ preferences, listed from most to least preferred. Right: iterated elimination of inconceivable conjectures. Each entry is a set of residual matchings; for instance, µ 2 ´a1 denotes the restriction of µ 2 to the agents other than a1. Consider for instance the matching µ 1 . This matching is the preferred one by the employer who has no incentive to deviate. Agent a1 has incentive to deviate bec… view at source ↗
Figure 3
Figure 3. Figure 3: Agents’ preferences and possible matchings. We first observe that µ 0 ´a1 is C0 -undominated as a conjecture for a1 when he is single. Under µ 0 ´a1 both a2 and b1 are single, therefore by Definition 3.1, this con￾jecture is C0 -dominated if µ 2 ´a2,b1 ` pa2, b1q ąa2 µ 1 ´a2,a2 ` pa2q for every µ 1 ´a2 P C0 pa2, Áq. This fails because µ 1 ´a2 P C0 pa2, Áq and µ 1 ´a2 ` pa2q “ µ 1 ąa2 µ 2 “ µ 2 ´a2,b1 ` pa2… view at source ↗
Figure 4
Figure 4. Figure 4: Mutual support of conjectures. The conjectures leading to no marriage survive because each is sustained by another rationalizable conjecture. Sasaki & Toda (1996) argue that a natural requirement for stability is that the matching under consideration be compatible with agents’ conjectures. In our terminology, a matching µ is C-consistent if each agent regards as possible the residual matching generated by … view at source ↗
Figure 5
Figure 5. Figure 5: Preferences of hospitals and medical residents. Residents ps1, s2q form a couple and have joint preferences over the pairs of jobs they may obtain. In this example there are two hospitals offering one job each, and three medical 22Informally, a couple’s preferences are responsive if the unilateral improvement of one part￾ner’s job is considered beneficial for the couple as well. Klaus & Klijn (See 2005, p.… view at source ↗
Figure 6
Figure 6. Figure 6: Preferences and matchings of Example 3. First notice that µ 0 ´a1 is C0 -dominated. Given any possible conjectures that b1 and a2 may hold when they are single, they prefer matching together than remaining single. Hence µ 1 is C1 -dominated. By (1) in Definition 5.1 agent a1 prefers remaining alone than matching with b1 for all conjectures in C1 pa1, Áq and all conjectures in C1 pa1, b1, Áq. Therefore, µ 1… view at source ↗
read the original abstract

In many matching markets, agents care not only about their own partners but also about the matches formed by others. With externalities, stability depends on what agents believe would happen after a deviation. We introduce rationalizable conjectures: beliefs that survive iterated elimination, in the spirit of rationalizability in non-cooperative games. These beliefs define conjecture-rationalizable stability, a solution concept that always exists, extends Gale--Shapley stability, and coincides with it when externalities are absent. We also introduce rationalizable matchings, a non-equilibrium counterpart, and show that every conjecture-rationalizable stable matching is rationalizable. In matching with couples, our concept yields non-empty predictions even when standard stability is vacuous. Finally, we provide an epistemic foundation: rationalizability is behaviorally implied by pairwise rationality and common belief in pairwise rationality, while conjecture-rationalizable stability additionally requires belief correctness.

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

0 major / 3 minor

Summary. The paper introduces rationalizable conjectures in matching markets with externalities, defined via iterated elimination of never-best-response beliefs in the spirit of rationalizability. These beliefs underpin conjecture-rationalizable stability, which the authors prove always exists, extends Gale-Shapley stability, and coincides with it when externalities are absent. The manuscript also defines rationalizable matchings as a non-equilibrium counterpart and shows every conjecture-rationalizable stable matching is rationalizable. It provides an epistemic foundation linking rationalizability to pairwise rationality plus common belief in pairwise rationality (with stability additionally requiring correct beliefs) and demonstrates non-empty predictions in the matching-with-couples setting where standard stability is empty.

Significance. If the central results hold, the work supplies a well-grounded, non-vacuous stability concept for externalities that inherits existence from rationalizability while recovering the classical Gale-Shapley benchmark. The explicit epistemic foundation (pairwise rationality + common belief) and the couples application are concrete strengths; the former supplies behavioral content and the latter shows practical payoff where existing notions fail.

minor comments (3)
  1. [§3.2] §3.2: the precise termination condition for the iterated elimination process (finite vs. transfinite) is stated only informally; an explicit inductive definition or reference to a standard lemma would remove ambiguity.
  2. [Theorem 5.1] Theorem 5.1: the proof that conjecture-rationalizable stability implies rationalizability is sketched at a high level; adding a short diagram or step-by-step outline of the belief-correction argument would improve readability.
  3. [§6] The couples example in §6 could usefully include a small numerical instance showing both the emptiness of standard stability and the non-emptiness of the new concept.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and constructive report, which accurately summarizes the paper's contributions and recommends minor revision. No specific major comments are listed in the report, so our response below addresses the overall assessment. We will incorporate any minor editorial changes in the revised version.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines new concepts (rationalizable conjectures via iterated elimination of never-best-response beliefs, conjecture-rationalizable stability, and rationalizable matchings) and derives their properties (existence, extension of Gale-Shapley stability, coincidence without externalities, and epistemic foundation from pairwise rationality plus common belief) directly from those definitions and standard rationalizability logic. No step reduces by construction to a fitted parameter, renames a known result, or relies on a load-bearing self-citation whose content is unverified; the central claims are independent of the inputs and do not collapse into tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review limited to abstract; ledger reflects only assumptions explicitly invoked or implied by the abstract text.

axioms (2)
  • domain assumption Agents form conjectures about post-deviation matchings that can be subjected to iterated elimination of unreasonable beliefs.
    Central to the definition of rationalizable conjectures stated in the abstract.
  • domain assumption Pairwise rationality and common belief in pairwise rationality are the behavioral primitives for the epistemic foundation.
    Invoked in the final sentence of the abstract for the epistemic justification.

pith-pipeline@v0.9.1-grok · 5677 in / 1396 out tokens · 57342 ms · 2026-06-29T02:00:35.530381+00:00 · methodology

discussion (0)

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Reference graph

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