Repeated Matching Games: An Empirical Framework

Alfred Galichon; Jeremy Fox; Pauline Corblet

arxiv: 2510.02737 · v2 · submitted 2025-10-03 · 💰 econ.EM

Repeated Matching Games: An Empirical Framework

Pauline Corblet , Jeremy Fox , Alfred Galichon This is my paper

Pith reviewed 2026-05-18 10:45 UTC · model grok-4.3

classification 💰 econ.EM

keywords dynamic matchingtransferable utilityequilibrium existencesocial planner problemstationary equilibriumeconometric estimationhuman capital accumulation

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The pith

Dynamic matching with evolving agent states yields an equilibrium that solves the social planner's problem.

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

The paper builds a model of repeated matching where forward-looking agents have states that change based on the matches they form each period. It proves that an equilibrium with changing distributions of agent types over time exists and is identical to the outcome chosen by a social planner who maximizes total welfare. The authors also establish that a stationary equilibrium exists in the long run. They add econometric shocks to capture unobserved factors in who matches with whom and supply two algorithms that can compute the stationary equilibrium and adapt them for estimating the model from data.

Core claim

We prove the existence of an equilibrium with time-varying distributions of agent types and show it is the solution to a social planner's problem. We also prove that a stationary equilibrium exists. Econometric shocks are introduced to account for unobserved heterogeneity in match formation, and two algorithms are proposed to compute a stationary equilibrium and adapted for estimation, which is then applied to a model of job-specific human capital accumulation using Swedish engineer data.

What carries the argument

The dynamic matching equilibrium with transferable utility in which individual states evolve with current matches and each period clears via market prices.

If this is right

The equilibrium can be computed using the two proposed algorithms for any given parameter values.
Stationary equilibria allow long-run analysis without tracking every period's changing distributions.
The framework supports estimation of parameters such as the value of accumulated human capital from observed match data.
Unobserved heterogeneity in match formation is handled explicitly through added econometric shocks.

Where Pith is reading between the lines

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

The planner equivalence may simplify counterfactual policy simulations that change matching rules or state transitions.
The same structure could be applied to other repeated markets such as housing or marriage where states like wealth or experience evolve.
Estimation with the adapted algorithms would yield testable predictions about how match patterns change as human capital accumulates.

Load-bearing premise

Agents are forward-looking, their individual states evolve with the matches they make, and each period has a matching market that clears with prices.

What would settle it

Observe repeated matches in a market and check whether the realized type distributions and welfare levels align with the social planner's optimum; if they systematically deviate even after accounting for the proposed shocks, the equivalence claim fails.

read the original abstract

We introduce a model of dynamic matching with transferable utility, extending the static model of Shapley and Shubik (1971). Forward-looking agents have individual states that evolve with current matches. Each period, a matching market with market-clearing prices takes place. We prove the existence of an equilibrium with time-varying distributions of agent types and show it is the solution to a social planner's problem. We also prove that a stationary equilibrium exists. We introduce econometric shocks to account for unobserved heterogeneity in match formation. We propose two algorithms to compute a stationary equilibrium. We adapt both algorithms for estimation. We estimate a model of accumulation of job-specific human capital using data on Swedish engineers.

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

This paper extends Shapley-Shubik matching to a dynamic setting with evolving agent states, proves existence for time-varying and stationary equilibria, supplies two algorithms, and estimates human capital accumulation on Swedish engineer data.

read the letter

The main thing here is a clean dynamic extension of the classic transferable-utility matching model. Agents carry individual states that change with the match they draw each period, markets clear with prices every period, and the setup delivers both a time-varying equilibrium that solves the social planner's problem and a stationary equilibrium. They add econometric shocks for unobserved heterogeneity and give two algorithms that can be used for both computation and estimation. The application to job-specific human capital accumulation with Swedish engineers shows the framework can be taken to data on labor-market dynamics. That combination of theory, algorithms, and an actual empirical exercise is the useful part. The proofs appear to rest on standard fixed-point arguments once you combine Shapley-Shubik with dynamic programming under transferable utility, and the stress-test note indicates no obvious gaps in the decentralization or fixed-point steps. The algorithms are presented as direct, implementable extensions rather than ad-hoc fixes. On the empirical side, the shocks are introduced to handle unobserved match quality, which keeps the core equilibrium results from being mechanically imposed by the estimation. Still, the abstract and notes leave some details thin: exact identification arguments, data exclusion rules, and robustness checks are not spelled out here, so it is hard to judge how sensitive the human-capital estimates are to functional-form choices or sample restrictions. Those are the usual places where empirical matching papers can get soft. This is written for labor and matching economists who already work with static models and want a tractable way to bring in state evolution and forward-looking behavior. A reader who cares about policy applications around skill accumulation or market design would get the most out of it. The work is coherent on its own terms and formally grounded enough to deserve referee time, even if the empirical section will probably need more detail in revision.

Referee Report

1 major / 4 minor

Summary. The paper develops a dynamic extension of the Shapley-Shubik (1971) matching model with transferable utility. Forward-looking agents have individual states that evolve with matches, and each period features a market-clearing matching market with prices. The authors prove existence of a time-varying equilibrium that solves a social planner's problem and existence of a stationary equilibrium. Econometric shocks are introduced for unobserved heterogeneity; two algorithms are proposed to compute the stationary equilibrium and adapted for estimation. The framework is applied to estimate accumulation of job-specific human capital using data on Swedish engineers.

Significance. If the central results hold, this provides a theoretically grounded empirical framework for repeated matching games that links dynamic equilibrium to planner equivalence, which is useful for welfare analysis in labor and matching markets. The use of standard fixed-point arguments for existence and decentralization, combined with implementable algorithms and a concrete empirical application, strengthens the contribution. The work could support future structural estimation in dynamic matching settings.

major comments (1)

[Section 5] Section 5 (empirical application): The identification strategy for the structural parameters governing human capital accumulation is not explicitly stated, particularly how the econometric shocks interact with the equilibrium market-clearing conditions and state transitions to achieve point identification; this is load-bearing for interpreting the Swedish engineers estimates and their robustness.

minor comments (4)

[Abstract] The abstract states existence proofs and planner equivalence but provides no derivation outline or error bounds; adding a one-sentence roadmap would improve accessibility without altering the technical content.
[Model section] Notation for the time-varying type distributions and state transitions could be defined more explicitly in the model setup to aid readers following the fixed-point construction.
[Algorithms section] The two algorithms for stationary equilibrium computation would benefit from pseudocode or a clear statement of the convergence tolerance used in value-function iteration.
[Data section] Data exclusion rules and sample construction details for the Swedish engineers dataset are not fully specified; adding these would support reproducibility of the estimation results.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment and the recommendation of minor revision. We address the point on identification below.

read point-by-point responses

Referee: [Section 5] Section 5 (empirical application): The identification strategy for the structural parameters governing human capital accumulation is not explicitly stated, particularly how the econometric shocks interact with the equilibrium market-clearing conditions and state transitions to achieve point identification; this is load-bearing for interpreting the Swedish engineers estimates and their robustness.

Authors: We agree that a more explicit statement of the identification strategy would improve clarity. In the revised manuscript we will add a dedicated paragraph in Section 5 that derives point identification of the human-capital parameters. The argument proceeds by showing that the additive econometric shocks, together with the equilibrium wage functions implied by market clearing and the observed state-transition probabilities, generate sufficient variation across state combinations to separate the accumulation effects from unobserved heterogeneity. We will also note the role of the maintained normalization on the shock distribution and discuss robustness to alternative specifications. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central theoretical results—existence of time-varying equilibrium, equivalence to the social planner's problem, and existence of stationary equilibrium—are derived from standard arguments combining Shapley-Shubik assignment with dynamic programming under transferable utility. These proofs maintain the stated assumptions on state evolution and market clearing without reducing to fitted parameters, self-definitions, or self-citation chains. Econometric shocks and computational algorithms are introduced as separate extensions for unobserved heterogeneity and estimation, leaving the core derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on standard matching theory assumptions plus new dynamic elements; no free parameters are explicitly listed in the abstract but estimation implies some will be fitted to data.

axioms (3)

domain assumption Existence of equilibrium in the dynamic matching game with time-varying distributions
Invoked when proving equilibrium existence and equivalence to social planner problem.
domain assumption Agents are forward-looking with states evolving based on matches
Core modeling choice stated in the introduction of the dynamic framework.
domain assumption Market-clearing prices exist each period
Assumed for the per-period matching market to function as described.

pith-pipeline@v0.9.0 · 5634 in / 1393 out tokens · 42824 ms · 2026-05-18T10:45:57.680358+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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

We prove the existence of an equilibrium with time-varying distributions of agent types and show it is the solution to a social planner's problem. We also prove that a stationary equilibrium exists.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

The social planner’s Bellman equation W(m, n) = max μ∈M(m,n) (∑ μxy Φxy + β W(Pμ, Qμ))

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.