Dynamic Matching Bandit For Two-Sided Online Markets

TL;DR

Dynamic Matching Bandit For Two-Sided Online Markets addresses learning dynamic preferences in two-sided matching by framing the problem as a Dynamic Matching Problem (DMP) and solving it with a dynamic matching bandit algorithm that combines online ridge regression with the Deferred Acceptance (DA) procedure. The method jointly learns context-dependent preferences and computes agent-optimal stable matchings with high probability, achieving a logarithmic regret in the time horizon $T$ and providing an instance-dependent lower bound to establish optimality. Theoretical guarantees include a regret bound of $\tilde{O}\left(\dfrac{d^{2}\sigma^{2}}{\rho^{2}}\log(NKT)\right)$ and stability properties, supported by a proof framework that bounds invalid ranking probabilities via online concentration results. Empirical results on synthetic and real data (including a LinkedIn-like job-market) show robustness to varying context dimensions, noise levels, and dynamic changes, with the dynamic algorithm consistently reducing the social welfare gap compared to baselines. Overall, the work advances online decision-making in dynamic two-sided markets by delivering stable, context-aware matching decisions under bandit feedback with provable performance guarantees.

Abstract

Two-sided online matching platforms are employed in various markets. However, agents' preferences in the current market are usually implicit and unknown, thus needing to be learned from data. With the growing availability of dynamic side information involved in the decision process, modern online matching methodology demands the capability to track shifting preferences for agents based on contextual information. This motivates us to propose a novel framework for this dynamic online matching problem with contextual information, which allows for dynamic preferences in matching decisions. Existing works focus on online matching with static preferences, but this is insufficient: the two-sided preference changes as soon as one side's contextual information updates, resulting in non-static matching. In this paper, we propose a dynamic matching bandit algorithm to adapt to this problem. The key component of the proposed dynamic matching algorithm is an online estimation of the preference ranking with a statistical guarantee. Theoretically, we show that the proposed dynamic matching algorithm delivers an agent-optimal stable matching result with high probability. In particular, we prove a logarithmic regret upper bound $\mathcal{O}(\log(T))$ and construct a corresponding instance-dependent matching regret lower bound. In the experiments, we demonstrate that dynamic matching algorithm is robust to various preference schemes, dimensions of contexts, reward noise levels, and context variation levels, and its application to a job-seeking market further demonstrates the practical usage of the proposed method.

Figures (14)

• Figure 1: Arm $a_{1}$'s profile changes with an angular velocity, which results in different optimal matching results. Phase 1's optimal matching: (company 1, $a_{1}$), (company 2, $a_{2}$), Phase 2's optimal matching: (company 1, $a_{2}$), (company 1, $a_{1}$), and Phase 3's optimal matching: (company 1, $a_{2}$), (company 2, $a_{1}$).
• Figure 2: Left: UCB algorithm, Right: our algorithm. Incapable exploration of UCB method.
• Figure 3: A generic design of dynamic matching platform.
• Figure 4: Flow of sufficient conditions for optimal matching.
• Figure 5: The corresponding matching results for $p_{1}$ and $p_{2}$ if $p_{1}$ has valid ranking $a_{2} > a_{3} > a_{1}$ and $p_{2}$ has six possible rankings. Valid ranking for both and optimal matching: Case 1, 2, and 3. Single invalid ranking and non-optimal matching: Case 4, 5, and 6.

Theorems & Definitions (27)

• Definition 2.1: Blocking
• Definition 2.2: Stable Matching
• Definition 2.3: Valid Match
• Definition 2.4: Agent-Optimal Match
• Remark 3.1
• Remark 4.1: Doubling Trick for Unknown $T$ for Dynamic Matching Algorithm
• Remark 4.2: Computational Complexity
• Remark 4.3: Comparison with $\epsilon$-greedy Algorithm
• Example 5.1
• Claim 5.1