Efficient Interview Scheduling for Stable Matching

Abstract

The study of stable matchings usually relies on the assumption that agents' preferences over the opposite side are complete and known. In many real markets, however, preferences might be uncertain and revealed only through costly interactions such as interviews. We show how to reach interim-stable matchings, under which all matched pairs must have interviewed and agents use expected utilities whenever true values remain unknown, while minimizing both the expected number of interviews and the expected number of interview rounds. We introduce two adaptive algorithms that produce interim-stable matchings: one operates sequentially, and another is a hybrid algorithm that begins by scheduling some interviews in parallel and continues sequentially. Focusing on cases where agents are ex-ante indifferent between agents on the other side, we show that the sequential algorithm performs 2 interviews per agent in expectation. We complement this by showing that any algorithm that performs less than 2 interviews per agent, does not always guarantee interim-stability. We also demonstrate that the hybrid algorithm requires only polylogarithmic expected number of rounds, while still performing only about 2 interviews per agent in expectation. Additionally, the interviews scheduled by our algorithms guarantee an interim-stable matching when Deferred-Acceptance is run after all interviews are completed.

Figures (4)

• Figure 1: Simulation of \ref{['algorithm:serial-adaptive']}: the average number of interviews per applicant in a bilaterally ex-ante equivalent instance with $n$ applicants and $n$ positions. For all $i,j \in [n]$, the match values are drawn independently from $F_{i,j}=G_{j,i}=\mathcal{U}[0,1]$. Results are averaged over 100 trials. We note that the simulation outcomes are very similar when the distributions are replaced by distributions supported on two points.
• Figure 2: Illustration of the value distributions $G_{j,i}$ for $n=3$ described in \ref{['ex:exp-dist']}. Notice that the lower realizations increase linearly while the upper realizations decrease exponentially. Two nice properties are visible: (1) For every $i' > i$, the entire support of $G_{j,i'}$ lies strictly below the expected value $U_{j,i}$. (2) For every $i'$, the lower realization $\frac{i}{n+1}$, is strictly below the expected value $U_{j,i'}$.
• Figure 3: Simulation results for \ref{['algorithm:serial-adaptive']} with $n$ applicants and $n$ positions. Here, agents on both sides ex-ante agree on the ranking of the other side. Match values for all $i,j \in [n]$ are drawn independently from the distribution described in \ref{['ex:exp-dist']}. The plot shows the average number of interviews per applicant.
• Figure 4: Simulation of a fully parallel algorithm showing the number of interview rounds per applicant. The instance is a bilaterally ex-ante equivalent with $n$ applicants and $n$ positions. For all $i,j \in [n]$, the match values are drawn independently from $F_{i,j}=G_{j,i}=\mathcal{U}[0,1]$. Results are averaged over 100 trials. The dashed line denotes the least-squares linear fit. The horizontal axis is plotted on an exponential scale, consequently, linear growth in the plot corresponds to logarithmic growth in $n$.

Theorems & Definitions (157)

• Theorem 1
• Theorem 2
• Definition 2.1: Interim Utility
• Definition 2.2: Interim Preference
• Definition 2.3: Blocking Pair
• Definition 2.4: Interim-Stable Matching
• Remark
• Lemma 2
• Definition 3.1: Interim Like
• Theorem 3