From signaling to interviews in random matching markets

TL;DR

The paper analyzes signaling-based interview formation in centralized two-sided matching markets, introducing interim stability to assess stability when interview refinements shape preferences. It shows a fundamental trade-off: almost interim stability can be achieved with $d=\omega(1)$ signals, while perfect interim stability generally requires $d=\Omega(\log^2 n)$, with short-side signaling becoming crucial in imbalanced markets. A Bayes-Nash incentive analysis demonstrates truthful signaling is approximately optimal, and the authors develop a local neighborhood–driven, message-passing algorithm to determine interim stability efficiently in sparse, locally tree-like interviews graphs. Extending to multi-tiered markets, the results reveal how hierarchical structure and general imbalances affect signaling requirements and stability, offering practical guidance for congestion reduction in residency-like markets and other large-scale matching settings.

Abstract

In many two-sided labor markets, interviews are conducted before matches are formed. The growing number of interviews in medical residency markets has increased demand for signaling mechanisms, where applicants send a limited number of signals to communicate interest. We study the role of signaling mechanisms to reduce interviews in centralized random matching markets where initial preferences are refined through interviews. Agents can only match with those they interview. For the market to clear, we focus on perfect interim stability: no pair of agents-even if they never interviewed each other-prefers each other to their assigned partners under their interim preferences. A matching is almost interim stable if it is perfect interim stable after removing a vanishingly small fraction of agents. We analyze signaling mechanisms in random matching markets with $n$ agents where agents on the short side, long side, or both sides signal their top $d$ preferred partners. The interview graph connects pairs where at least one party signaled the other. We reveal a fundamental trade-off between almost and perfect interim stability. For almost interim stability, $d=ω(1)$ signals suffice: short-side signaling is always effective, whereas long-side signaling is effective only when the market is weakly imbalanced, i.e., when any size difference between the two sides becomes negligible as the market grows. For perfect interim stability, at least $d=Ω(\log^2 n)$ signals are necessary, and short-side signaling becomes crucial in any imbalanced market. We establish that truthful signaling is a Bayes-Nash equilibrium and extend our analysis to markets with hierarchical structure. As a technical contribution, we develop a message-passing algorithm that efficiently determines interim stability by leveraging local neighborhood structures.

Figures (6)

• Figure 1: Example of the multi-tiered signaling mechanism with the applicant tiers $(\mathcal{A}_1, \mathcal{A}_2, \mathcal{A}_3)$ and firm tiers $(\mathcal{J}_1, \mathcal{J}_2, \mathcal{J}_3)$ arranged in order of desirability. The length of each box represents the size of the respective tier, with $\mathcal{A}_3$ being the highest-ranked applicant tier and $\mathcal{J}_3$ being the highest-ranked firm tier. Arrows indicate target tiers.
• Figure 2: Given $H$ as a two-sided market with applicants ${\mathcal{A}} =\{a_1,a_2,\cdots, a_{14}\}$ and firms ${\mathcal{J}} =\{j_1,j_2,\cdots, j_{13}\}$, $H_2(a_1)$ and $H_3(a_1)$ are the $2$-hop and $3$-hop neighborhoods of $a_1$ on $H$, respectively. In $H_2(a_1)$, $a_1$ is weakly worse off compared to $H$, while in $H_3(a_1)$, $a_1$ is weakly better off compared to $H$. For example, if $j_1$ is available to $a_1$ on $H_2(a_1)$, it must also be available to $a_1$ in $H$, and if $j_1$ is not available to $a_1$ in $H_3(a_1)$, $j_1$ must not be available to $a_1$ in $H$.
• Figure 3: An illustration of the hierarchical proposal-passing algorithm on the $3$-hop neighborhood of $a_1$ on $H$, denoted as $H_3(a_1)$ as shown in Figure \ref{['fig:every']}, with truncated preferences shown in tab:preferences_updated. (a) The proposing phase (from bottom to top): all proposals are indicated by red arrows. (b) The clean-up matching phase (from top to bottom): all accepted proposals are indicated by blue arrows.
• Figure 4: The number of applicants and the number of firms involved in at least one interim blocking pair in the applicant-optimal stable matching for $d=10, 20$ respectively, with $800 \leq n_{\mathcal{A}} \leq 1200$, $n_{\mathcal{J}} = 1000$, $\mathbb{B} = {\mathcal{N}}(0,1)$, and $\mathbb{A} = {\mathcal{U}}[-1,1]$, where the interview graph is constructed by the applicant-signaling mechanism. Each data point represents the average over $10$ trials of simulations.
• Figure 5: The number of applicants and the number of firms involved in at least one interim blocking pair in the applicant-optimal stable matching, $n_{\mathcal{A}}=n_{\mathcal{J}}=1000$, $1\leq d\leq 50$, $\mathbb{B} = {\mathcal{N}}(0,1)$, and $\mathbb{A}={\mathcal{U}}[-1,1]$, where the interview graph is constructed by the applicant-signaling mechanism. Each data point represents the average over $10$ trials of simulations.

Theorems & Definitions (104)

• Definition 1: Interview Graph
• Definition 2: Stable matching
• Definition 3: Interim blocking pair
• Definition 4: Interim stability
• Theorem 1: Effectiveness of one-side signaling for almost interim stability
• Remark 1
• Definition 5: Pre-interview scores outweighing post-interview scores
• Theorem 2: Failure of one-side signaling for almost interim stability
• Corollary 1: Failure on both-side signaling for almost interim stability
• Corollary 2: Effectiveness on both-side signaling for almost interim stability