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Structural Complexities of Matching Mechanisms

Yannai A. Gonczarowski, Clayton Thomas

TL;DR

This work develops a unified framework to quantify non-computational (structural) complexity measures for two canonical matching mechanisms, TTC and DA, by bounding the bits needed to perform four core tasks: outcome effects, options effects, representation, and verification. It establishes a nuanced landscape: TTC generally exhibits higher complexity than DA on outcome- and options-based questions, with Omega($n^{2}$) lower bounds in several settings, but DA can match TTC in verification while maintaining lower representation costs under known-priorities. The authors introduce novel constructs (e.g., $ extsf{SD}^{ ext{rot}}$ and the un-rejection graph) and leverage existing results (GHT, Leshno-Leshno-Leshno) to derive tight bounds and connect to prior literature on menus and cutoffs. Collectively, the results formalize intuitive policy concerns about transparency and explainability, clarifying when TTC’s greater complexity translates into practical differences and when DA’s structure yields tangible simplicity. The framework provides precise, quantifiable benchmarks for evaluating the tradeoffs between predictability, explainability, and trust in matching mechanisms, with implications for policy design and mechanism choice in school-choice-like settings.

Abstract

We study various novel complexity measures for two-sided matching mechanisms, applied to the two canonical strategyproof matching mechanisms, Deferred Acceptance (DA) and Top Trading Cycles (TTC). Our metrics are designed to capture the complexity of various structural (rather than computational) concerns, in particular ones of recent interest within economics. We consider a unified, flexible approach to formalizing our questions: Define a protocol or data structure performing some task, and bound the number of bits that it requires. Our main results apply this approach to four questions of general interest; for mechanisms matching applicants to institutions, our questions are: (1) How can one applicant affect the outcome matching? (2) How can one applicant affect another applicant's set of options? (3) How can the outcome matching be represented / communicated? (4) How can the outcome matching be verified? Holistically, our results show that TTC is more complex than DA, formalizing previous intuitions that DA has a simpler structure than TTC. For question (2), our result gives a new combinatorial characterization of which institutions are removed from each applicant's set of options when a new applicant is added in DA; this characterization may be of independent interest. For question (3), our result gives new tight lower bounds proving that the relationship between the matching and the priorities is more complex in TTC than in DA. We nonetheless showcase that this higher complexity of TTC is nuanced: By constructing new tight lower-bound instances and new verification protocols, we prove that DA and TTC are comparable in complexity under questions (1) and (4). This more precisely delineates the ways in which TTC is more complex than DA, and emphasizes that diverse considerations must factor into gauging the complexity of matching mechanisms.

Structural Complexities of Matching Mechanisms

TL;DR

This work develops a unified framework to quantify non-computational (structural) complexity measures for two canonical matching mechanisms, TTC and DA, by bounding the bits needed to perform four core tasks: outcome effects, options effects, representation, and verification. It establishes a nuanced landscape: TTC generally exhibits higher complexity than DA on outcome- and options-based questions, with Omega() lower bounds in several settings, but DA can match TTC in verification while maintaining lower representation costs under known-priorities. The authors introduce novel constructs (e.g., and the un-rejection graph) and leverage existing results (GHT, Leshno-Leshno-Leshno) to derive tight bounds and connect to prior literature on menus and cutoffs. Collectively, the results formalize intuitive policy concerns about transparency and explainability, clarifying when TTC’s greater complexity translates into practical differences and when DA’s structure yields tangible simplicity. The framework provides precise, quantifiable benchmarks for evaluating the tradeoffs between predictability, explainability, and trust in matching mechanisms, with implications for policy design and mechanism choice in school-choice-like settings.

Abstract

We study various novel complexity measures for two-sided matching mechanisms, applied to the two canonical strategyproof matching mechanisms, Deferred Acceptance (DA) and Top Trading Cycles (TTC). Our metrics are designed to capture the complexity of various structural (rather than computational) concerns, in particular ones of recent interest within economics. We consider a unified, flexible approach to formalizing our questions: Define a protocol or data structure performing some task, and bound the number of bits that it requires. Our main results apply this approach to four questions of general interest; for mechanisms matching applicants to institutions, our questions are: (1) How can one applicant affect the outcome matching? (2) How can one applicant affect another applicant's set of options? (3) How can the outcome matching be represented / communicated? (4) How can the outcome matching be verified? Holistically, our results show that TTC is more complex than DA, formalizing previous intuitions that DA has a simpler structure than TTC. For question (2), our result gives a new combinatorial characterization of which institutions are removed from each applicant's set of options when a new applicant is added in DA; this characterization may be of independent interest. For question (3), our result gives new tight lower bounds proving that the relationship between the matching and the priorities is more complex in TTC than in DA. We nonetheless showcase that this higher complexity of TTC is nuanced: By constructing new tight lower-bound instances and new verification protocols, we prove that DA and TTC are comparable in complexity under questions (1) and (4). This more precisely delineates the ways in which TTC is more complex than DA, and emphasizes that diverse considerations must factor into gauging the complexity of matching mechanisms.
Paper Structure (45 sections, 51 theorems, 36 equations, 11 figures, 2 tables)

This paper contains 45 sections, 51 theorems, 36 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Outcome-Effect
  3. Options-Effect
  4. Representation
  5. Verification
  6. Additional Questions.
  7. Discussion.
  8. Paper Organization.
  9. Related work
  10. Preliminaries
  11. Strategyproofness and menus
  12. Outcome-Effect Complexity
  13. Warmup: SD
  14. TTC
  15. Building up to our construction.
  16. ...and 30 more sections

Key Result

theorem 1.1

For both ${{\mathsf{TTC}}}$ and $\mathsf{DA}$, the complexity of how one applicant can affect the outcome matching is (nearly) as high as possible.$\mathsf{DA}$ comes in two variants: applicant-proposing $\mathsf{DA}$ ($\mathsf{APDA}$), and institution-proposing $\mathsf{DA}$ ($\mathsf{IPDA}$). This

Figures (11)

  • Figure 1: Illustration of ${{\mathsf{SD}}}(\cdot, P_{-1})$, as given by \ref{['thrm:gtc-SD-UB']}, on the preferences profile below.
  • Figure 2: Illustration of the proof of \ref{['thrm:gtc-SDrot-LB']}.
  • Figure 3: Illustration of the set of preferences showing that the outcome-effect complexity of $TTC$ is $\Omega(n^2)$ (\ref{['thrm:gtc-TTC-LB']}).
  • Figure 4: Illustration of the set of preferences showing that the outcome-effect complexity of $\mathsf{IPDA}$ is $\Omega(n^2)$ (\ref{['thrm:gtc-IPDA']}).
  • Figure 5: Illustration of the selection gadget used to bound the options-effect complexity of ${{\mathsf{TTC}}}$ (\ref{['thrm:type-to-menu-TTC']}).
  • ...and 6 more figures

Theorems & Definitions (104)

  • theorem 1.1: Informal; Combination of \ref{['thrm:gtc-TTC-LB']}, \ref{['thrm:gtc-DA-LB']}, and \ref{['thrm:gtc-IPDA']}
  • theorem 1.2: Informal; Combination of \ref{['thrm:type-to-menu-TTC']} and \ref{['thrm:type-to-menu-DA']}
  • theorem 1.3: Informal; Combination of \ref{['thrm:representing-DA']} and \ref{['thrm:representing-TTC']}
  • theorem 1.4: Informal; Combination of \ref{['thrm:verif-complexity-ttc']} and \ref{['thrm:verif-complexity-da']}
  • definition 2.1
  • definition 2.2
  • definition 2.3
  • definition 2.4
  • theorem 2.5: Hammond79
  • definition 3.1
  • ...and 94 more