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Two-Stage Stochastic Capacity Expansion in Stable Matching under Truthful or Strategic Preference Uncertainty

Maria Bazotte, Margarida Carvalho, Thibaut Vidal

Abstract

Recent studies on many-to-one matching markets have explored agents with flexible capacity and truthful preference reporting, focusing on mechanisms that jointly design capacities and select a matching. However, in real-world applications such as school choice and residency matching, preferences are revealed after capacity decisions are made, with matching occurring afterward; uncertainty about agents' preferences must be considered during capacity planning. Moreover, even under strategy-proof mechanisms, agents may strategically misreport preferences based on beliefs about admission chances. We introduce a two-stage stochastic matching problem with uncertain preferences, using school choice as a case study. In the first stage, the clearinghouse expands schools' capacities before observing students' reported preferences. Students either report their true preferences, producing exogenous uncertainty, or act strategically, submitting reported preferences based on their true preferences and admission chances (which depend on capacities), introducing endogenous uncertainty. In the second stage, the clearinghouse computes the student-optimal stable matching based on schools' priorities and students' reported preferences. In strategic cases, endogenous reported preferences are utility-maximizing transformations of capacity decisions and exogenous true preferences; we handle uncertainty using sample average approximation(SAA). We develop behavior-based mathematical formulations and, due to problem complexity, propose Lagrangian- and local-search-based behavior-specific heuristics for near-optimal solutions. Our SAA-based approaches outperform the average scenario approach on students' matching preferences and admission outcomes, emphasizing the impact of stochastic preferences on capacity decisions. Student behavior notably influences capacity design, stressing the need to consider misreports.

Two-Stage Stochastic Capacity Expansion in Stable Matching under Truthful or Strategic Preference Uncertainty

Abstract

Recent studies on many-to-one matching markets have explored agents with flexible capacity and truthful preference reporting, focusing on mechanisms that jointly design capacities and select a matching. However, in real-world applications such as school choice and residency matching, preferences are revealed after capacity decisions are made, with matching occurring afterward; uncertainty about agents' preferences must be considered during capacity planning. Moreover, even under strategy-proof mechanisms, agents may strategically misreport preferences based on beliefs about admission chances. We introduce a two-stage stochastic matching problem with uncertain preferences, using school choice as a case study. In the first stage, the clearinghouse expands schools' capacities before observing students' reported preferences. Students either report their true preferences, producing exogenous uncertainty, or act strategically, submitting reported preferences based on their true preferences and admission chances (which depend on capacities), introducing endogenous uncertainty. In the second stage, the clearinghouse computes the student-optimal stable matching based on schools' priorities and students' reported preferences. In strategic cases, endogenous reported preferences are utility-maximizing transformations of capacity decisions and exogenous true preferences; we handle uncertainty using sample average approximation(SAA). We develop behavior-based mathematical formulations and, due to problem complexity, propose Lagrangian- and local-search-based behavior-specific heuristics for near-optimal solutions. Our SAA-based approaches outperform the average scenario approach on students' matching preferences and admission outcomes, emphasizing the impact of stochastic preferences on capacity decisions. Student behavior notably influences capacity design, stressing the need to consider misreports.

Paper Structure

This paper contains 52 sections, 5 theorems, 23 equations, 11 figures, 6 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Works
  4. Matching Markets
  5. Mechanisms and Mathematical Formulations
  6. Capacity expansion
  7. Stochastic setting
  8. Agents' behavior
  9. Two-Stage Stochastic Programs with Exogenous or Endogenous Uncertainty
  10. Problem Statement
  11. Problem Formulation
  12. Mathematical Formulation
  13. Reported Students' Preferences
  14. Utility Maximization ($a ={{\texttt{UM}}}$)
  15. Conjoint Expected Utility Maximization ($a = {{\texttt{CEUM}}}$)
  16. ...and 37 more sections

Key Result

Lemma 1

Constraints constr:safety-strategy ensure that Assumption assump:safety-strat is satisfied by the formulation.

Figures (11)

  • Figure 1: Timeline of the 2-STMMP
  • Figure 2: Average $\overline{{{\texttt{VSS}}}}$, $\overline{{{\texttt{VSS-E}}}}$, and $\overline{{{\texttt{VSS-I}}}}$ per budget $B$ for large instances
  • Figure 3: Average $\overline{{\texttt{DIFF-RANK}}}^k$ per list size limit $K$ for large instances with budget $B=60$
  • Figure 4: Stochastic behavior gaps per list size limit $K$ and budget $B$ for large instances.
  • Figure 5: Deterministic behavior gaps per list size limit $K$ and budget $B$ for large instances.
  • ...and 6 more figures

Theorems & Definitions (6)

  • Definition 1: Stable matching
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5