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Strategyproofness-Exposing Descriptions of Matching Mechanisms

Yannai A. Gonczarowski, Ori Heffetz, Clayton Thomas

TL;DR

This paper introduces menu descriptions as ex ante explanations of static, direct-revelation matching mechanisms to expose strategyproofness, focusing on Deferred Acceptance (DA) and Top Trading Cycles (TTC). It delivers three core results: (i) a novel menu description of DA that computes each applicant's menu via institution-proposing DA on the market without that applicant and then selects the top menu item, (ii) a menu-in-outcome description of TTC that both describes the menu and yields a simple proof that TTC's traditional description is strategyproof, and (iii) a strong impossibility showing that no similar, minimal tweak can expose DA's strategyproofness through a menu description without incurring quadratic memory. The work highlights a trichotomy among SD, TTC, and DA in terms of description-based simplicity and exposes practical considerations for teaching and implementing strategyproof mechanisms. Overall, menu descriptions offer a principled way to convey strategyproofness to participants, with tradeoffs in describing the full outcome and potential empirical implications for real-world matching markets.

Abstract

A menu description exposes strategyproofness by presenting a mechanism to player $i$ in two steps. Step (1) uses others' reports to describe $i$'s menu of potential outcomes. Step (2) uses $i$'s report to select $i$'s favorite outcome from her menu. We provide novel menu descriptions of the Deferred Acceptance (DA) and Top Trading Cycles (TTC) matching mechanisms. For TTC, our description additionally yields a proof of the strategyproofness of TTC's traditional description, in a way that we prove is impossible for DA.

Strategyproofness-Exposing Descriptions of Matching Mechanisms

TL;DR

This paper introduces menu descriptions as ex ante explanations of static, direct-revelation matching mechanisms to expose strategyproofness, focusing on Deferred Acceptance (DA) and Top Trading Cycles (TTC). It delivers three core results: (i) a novel menu description of DA that computes each applicant's menu via institution-proposing DA on the market without that applicant and then selects the top menu item, (ii) a menu-in-outcome description of TTC that both describes the menu and yields a simple proof that TTC's traditional description is strategyproof, and (iii) a strong impossibility showing that no similar, minimal tweak can expose DA's strategyproofness through a menu description without incurring quadratic memory. The work highlights a trichotomy among SD, TTC, and DA in terms of description-based simplicity and exposes practical considerations for teaching and implementing strategyproof mechanisms. Overall, menu descriptions offer a principled way to convey strategyproofness to participants, with tradeoffs in describing the full outcome and potential empirical implications for real-world matching markets.

Abstract

A menu description exposes strategyproofness by presenting a mechanism to player in two steps. Step (1) uses others' reports to describe 's menu of potential outcomes. Step (2) uses 's report to select 's favorite outcome from her menu. We provide novel menu descriptions of the Deferred Acceptance (DA) and Top Trading Cycles (TTC) matching mechanisms. For TTC, our description additionally yields a proof of the strategyproofness of TTC's traditional description, in a way that we prove is impossible for DA.
Paper Structure (25 sections, 25 theorems, 12 equations, 9 figures, 3 tables, 6 algorithms)

This paper contains 25 sections, 25 theorems, 12 equations, 9 figures, 3 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mechanisms
  4. Descriptions
  5. Menus and Menu Descriptions
  6. Menu-in-Outcome Descriptions
  7. Uses of Menu Descriptions
  8. A Menu Description of DA
  9. A Menu-in-Outcome Description of TTC
  10. An Impossibility Result for Menu-in-Outcome Descriptions of DA
  11. Properties of Slight Tweaks of Traditional Descriptions
  12. Impossibility Theorem
  13. Related work
  14. Discussion
  15. Proof of Main Impossibility Theorem
  16. ...and 10 more sections

Key Result

theorem 2.5

A matching mechanism $f$ is strategyproof if and only if each applicant $i$ always receives her favorite institution from her menu. That is, for every $\succ_{-i}\in\mathcal{T}_{-i}$ and $\succ_i \in \mathcal{T}_i$, it holds that $f_i(\succ_i, \succ_{-i}) \succeq_i h$ for all $h \in \mathcal{M}_{\su

Figures (9)

  • Figure 1: Illustration of trichotomy for traditional descriptions of SD, TTC, and DA
  • Figure 2: SD
  • Figure 3: TTC
  • Figure 4: DA
  • Figure 5: Menu calculation in \ref{['alg:individ-dict-ttc']} Notes: Circles represent applicants; squares represent institutions; each institution (resp. each applicant except $i$) points to her favorite remaining applicant (resp. institution). Cycles not involving $i$ were already eliminated, so wherever $i$ points will form a cycle.
  • ...and 4 more figures

Theorems & Definitions (61)

  • definition 2.1: SD
  • definition 2.2: TTC
  • definition 2.3: DA
  • definition 2.4: Menu
  • theorem 2.5: Hammond79
  • proof
  • definition 2.6: Menu Description
  • definition 2.7: Menu-in-Outcome Description
  • example 2.8: A "brute force" menu description
  • example 3.1
  • ...and 51 more