The Distortion of Stable Matching

TL;DR

The study of distortion in stable matching is initiated, and algorithms that have limited access to the agents'cardinal preferences and compute stable matchings of high quality with respect to some aggregate objective, e.g., the social welfare are designed.

Abstract

We initiate the study of distortion in stable matching. Concretely, we aim to design algorithms that have limited access to the agents' cardinal preferences and compute stable matchings of high quality with respect to some aggregate objective, e.g., the social welfare. Our first result is a strong impossibility: the classic Deferred Acceptance (DA) algorithm of Gale and Shapley [1962], as well as any deterministic algorithm that relies solely on ordinal information about the agents' preferences, has unbounded distortion. To circumvent this impossibility, we consider algorithms that either (a) use randomization or (b) perform a small number of value queries to the agents' cardinal preferences. In the former case, we prove that a simple randomized version of the DA algorithm achieves a distortion of $2$, and that this is optimal among all randomized stable matching algorithms. For the latter case, we prove that the same bound of $2$ can be achieved with only $1$ query per agent, and improving upon this bound requires $Ω(\log n)$ queries per agent. We further show that this query bound is asymptotically optimal for any constant approximation: for any $\varepsilon >0$, there exists an algorithm which uses $O(\log n /\varepsilon^2)$ queries, and achieves a distortion of $1+\varepsilon$. Moreover, under natural structural restrictions on the instances of the problem, we provide improved upper bounds on the number of queries required for a $(1+\varepsilon)$-approximation. We complement our main findings above with theoretical and empirical results on the average-case performance of stable matching algorithms, when the preferences of the agents are drawn i.i.d. from a given distribution.

Figures (6)

• Figure 1: An example of a preference profile in which no agent is matched with their top choice, taken from gale1962college. The entry $(m_i, w_j)$ consists of a pair of numbers; the first indicates the rank of $w_j$ for $m_i$ and the second indicates the rank of $m_i$ for $w_j$. For example, the entry $(m_2,w_3)$ is $(3,3)$ which indicates that $m_2$ ranks $w_3$ third among the women, and $w_3$ ranks $m_2$ third among the men. The unique stable matching is indicated by the entries circled in blue.
• Figure 2: An example on which any ordinal stable algorithm has unbounded distortion. The entry $(m_i, w_j)$ consists of a pair of numbers; the first indicates the rank of $w_j$ for $m_i$ and the second indicates the rank of $m_i$ for $w_j$. For example, the entry $(m_2,w_3)$ is $(1,3)$ which indicates that $m_2$ ranks $w_3$ first among the women, and $w_3$ ranks $m_2$ third among the men. In this preference profile there are only two stable matchings, the man-optimal matching, consisting of entries circled in blue with solid lines, and the woman-optimal matching, consisting of entries circled in red with dashed lines.
• Figure 3: The reverse cyclic shift profile used in the proof of \ref{['thm:lower-bound-ordinal-randomized']}. The entry $(m_i, w_j)$ consists of a pair of numbers; the first indicates the rank of $w_j$ for $m_i$ and the second indicates the rank of $m_i$ for $w_j$. For example, the entry $(m_2,w_1)$ is $(n,n-1)$ which indicates that $m_2$ ranks $w_1$ last among the women, and $w_1$ ranks $m_2$ second to last among the men. Notice that the preference of each man $m_i$ (in the rows of the table) is formed by placing $w_i$ first, $w_{i+1}$ second, and so on, with the preference cycling around once the index reaches $n$. The preference of each woman is formed similarly, but in a shifted and reverse way: woman $w_i$ places $m_{i-1}$ first, $m_{i-2}$ second and so on, with preference cycling around once the index reaches $1$.
• Figure 4: The cyclic shift profile used in the proof of \ref{['thm:lower-bound-ordinal-randomized']}. The entry $(m_i, w_j)$ consists of a pair of numbers; the first indicates the rank of $w_j$ for $m_i$ and the second indicates the rank of $m_i$ for $w_j$. For example, the entry $(m_2,w_1)$ is $(n,2)$ which indicates that $m_2$ ranks $w_1$ last among the women, and $w_1$ ranks $m_2$ second among the men. Notice that the preference of each man $m_i$ (in the rows of the table) is formed by placing $w_i$ first, $w_{i+1}$ second, and so on, with the preference cycling around once the index reaches $n$. The preference of each woman is formed similarly, but in a shifted way: woman $w_i$ places $m_{i+1}$ first, $m_{i+2}$ second and so on, with preference cycling around once the index reaches $n$. The set of stable matchings on this instance consists of the "cyclic diagonals". For example, referring to the statement of \ref{['lem:cyclic-shift-profile-stable-matchings']}, $\mu_0$ corresponds to the main diagonal (shown in solid blue lines), $\mu_1$ corresponds to the first superdiagonal, together with the $n$-th subdiagonal (shown in dashed red lines), $\mu_2$ corresponds to the second superdiagonal, together with the $(n-1)$-th subdiagonal (shown in dotted green lines), etc.
• Figure 5: The argument used in the proof of \ref{['thm:lower-bound-queries-adaptive']}. The top part of the figure depicts a situation in which $t-1$ queries have been asked to man $m_1$ and $\ell$ queries have been asked to woman $w_1$; the corresponding uninformed regions $U_{m_1}^{t-1}$ and $U_{w_1}^\ell$ are shaded in red. Observe also the one-to-one correspondence between stable partners of agent $a \in \{m_1,w_1\}$ and the possible stable matchings for this instance, established by \ref{['lem:cyclic-shift-profile-stable-matchings']}. In the figure, we have that $\hat{U}_{m_1}^{t-1} = \hat{U}_{w_1}^\ell=\{\mu_2, \mu_3, \mu_4, \ldots, \mu_{n-4}, \mu_{n-3}\}$. The bottom part of the picture shows the situation after query $t$ has been asked to man $m_1$ for woman $w_4$. This partitions the previous uninformed region $U_{m_1}^{t-1}$ into the new uninformed region $U_{m_1}^t$, and the region of the values learned via this query $L_{m_1}^t$. As we mention in the proof, we assume that this query also updates the uninformed region $U_{w_1}^\ell$ - note that the index in the superscript does not change because there was no query for woman $w_1$ in this round. $U_{w_1}^\ell$ is updated by removing from it the men that are matched with $w_1$ in the matchings in $\hat{L}_{m_1}^t$, namely $m_{n-2} = \mu_3(w_1)$ and $m_{n-1} = \mu_2(w_1)$.

Theorems & Definitions (49)

• Remark 1
• Definition 2.1: Favorites and Suitors
• Definition 2.2: Blocking Pair
• Definition 2.3: Stable Matching
• Definition 2.4: Optimal Stable Matching
• Theorem 2.1
• Definition 2.5: Man-Optimal and Woman-Optimal Stable Matching
• Definition 2.6: (Ex-post) randomized stable matching
• proof
• Theorem 3.1