Stable Marriage: Loyalty vs. Competition

TL;DR

This work analyzes stable matching in two-sided markets with random preferences under a Deferred Acceptance (DA) framework augmented by an endowment-style loyalty parameter $k$ on the accepting side. In balanced markets, the authors prove a tight $Θ(n\log n)$ bound on the total number of proposals, yielding an average doctor rank of $Θ(\log n)$ for any consistent DA variant. In unbalanced markets, they reveal a phase transition driven by loyalty: for $k \ge n-\sqrt{n}$ the doctor rank remains $Θ(\log n)$, but at $k= n-\sqrt{n}\log n$ the doctor rank jumps to $\tilde{Θ}(\sqrt{n})$, showing that competition can be dramatically more harmful than the benefit of proposing. Simulations corroborate the theoretical thresholds and highlight the practical implications for mechanism design where endowment effects and loyalty can shape stability and efficiency in large markets.

Abstract

We consider the stable matching problem (e.g. between doctors and hospitals) in a one-to-one matching setting, where preferences are drawn uniformly at random. It is known that when doctors propose and the number of doctors equals the number of hospitals, then the expected rank of doctors for their match is $Θ(\log n)$, while the expected rank of the hospitals for their match is $Θ(n/\log n)$, where $n$ is the size of each side of the market. However, when adding even a single doctor, [Ashlagi, Kanoria and Leshno, 2017] show that the tables have turned: doctors have expected rank of $Θ(n/\log n)$ while hospitals have expected rank of $Θ(\log n)$. That is, (slight) competition has a much more dramatically harmful effect than the benefit of being on the proposing side. Motivated by settings where agents inflate their value for an item if it is already allocated to them (termed endowment effect), we study the case where hospitals exhibit ``loyalty". We model loyalty as a parameter $k$, where a hospital currently matched to their $\ell$th most preferred doctor accepts proposals from their $\ell-k-1$th most preferred doctors. Hospital loyalty should help doctors mitigate the harmful effect of competition, as many more outcomes are now stable. However, we show that the effect of competition is so dramatic that, even in settings with extremely high loyalty, in unbalanced markets, the expected rank of doctors already becomes $\tildeΘ(\sqrt{n})$ for loyalty $k=n-\sqrt{n}\log n=n(1-o(1))$.

Figures (8)

• Figure 1: Average doctors' rank in unbalanced market with varying level of loyalty $k$.
• Figure 2: Simulations illustrating the average rank of doctors in balanced and unbalanced markets under varying levels of loyalty. On Figure \ref{['fig:main_balanced']}, the horizontal dashed gray line marks the $\ln(1000)\approx 6.9$ height. On Figure \ref{['fig:main_unbalanced']}, the two vertical dashed gray lines represent the two loyalties between which the phase transition occurs: $n-\sqrt{n}\ln n$ and $n-\sqrt{n}$.
• Figure 3: Doctors' average rank for varying amounts of loyalty.
• Figure 4: Total number of proposals across the Balanced Phase and Unbalanced Phase for varying amounts of loyalty.
• Figure 5: Hospital rank distribution for $k=0$.

Theorems & Definitions (43)

• Definition 2.1: Stable Matching
• Definition 2.2: Consistency
• Lemma 2.1
• proof
• Lemma 2.2
• Claim 2.1
• Lemma 2.3: Multiplicative Chernoff Bound
• Theorem 3.1
• Corollary 3.1
• Lemma 3.1