We consider a model for repeated stochastic matching where compatibility is probabilistic, is realized the first time agents are matched, and persists in the future. Such a model has applications in the gig economy, kidney exchange, and mentorship matching. We ask whether a $decentralized$ matching process can approximate the optimal online algorithm. In particular, we consider a decentralized $stable$ $matching$ process where agents match with the most compatible partner who does not prefer matching with someone else, and known compatible pairs continue matching in all future rounds. We demonstrate that the above process provides a 0.316-approximation to the optimal online algorithm for matching on general graphs. We also provide a $\frac{1}{7}$-approximation for many-to-one bipartite matching, a $\frac{1}{11}$-approximation for capacitated matching on general graphs, and a $\frac{1}{2k}$-approximation for forming teams of up to $k$ agents. Our results rely on a novel coupling argument that decomposes the successful edges of the optimal online algorithm in terms of their round-by-round comparison with stable matching.

中文翻译：

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概率环境中的分散匹配

我们考虑了一种重复随机匹配模型，其中兼容性是概率性的，在第一次匹配代理时实现，并在未来持续存在。这样的模型在零工经济、肾脏交换和导师匹配中都有应用。我们询问 $decentralized$ 匹配过程是否可以近似最优在线算法。特别是，我们考虑了一个分散的 $stable$$matching$ 过程，其中代理与不喜欢与其他人匹配的最兼容的伙伴匹配，并且已知的兼容对在所有未来轮次中继续匹配。我们证明了上述过程为一般图匹配的最佳在线算法提供了 0.316 的近似值。我们还为多对一二分匹配提供了 $\frac{1}{7}$-approximation，一个 $\frac{1}{11}$-近似用于一般图上的容量匹配，以及一个 $\frac{1}{2k}$-近似用于形成多达 $k$ 个代理的团队。我们的结果依赖于一种新颖的耦合参数，该参数根据与稳定匹配的逐轮比较来分解最佳在线算法的成功边缘。