For a two-sided ($n$ men/$n$ women) stable matching problem) Gale and Shapley studied a proposal algorithm (men propose/women select, or the other way around), that determines a matching, not blocked by any unmatched pair. Irving used this algorithm as a first phase of his algorithm for one-sided (stable roommates) matching problem with $n$ agents. We analyze a fully extended version of Irving's proposal algorithm that runs all the way until either each agent holds a proposal or an agent gets rejected by everybody on the agent's preference list. It is shown that the terminal, directed, partnerships form a stable permutation with matched pairs remaining matched in any other stable permutation. A likely behavior of the proposal algorithm is studied under assumption that all $n$ rankings are independently uniform. It is proved that with high probability (w.h.p.) every agent has a partner, and that both the number of agents in cycles of length $\ge 3$ and the total number of stable matchings are bounded in probability. W.h.p. the total number of proposals is asymptotic to $0.5 n^{3/2}$.

中文翻译：

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Gale-Shapley 提议算法的片面版本及其在随机偏好下的可能行为

对于双边（$n$ 男性/$n$ 女性）稳定匹配问题）Gale 和 Shapley 研究了一种提议算法（男性提议/女性选择，或相反），该算法确定匹配，不会被任何无与伦比的一对。欧文将此算法用作其算法的第一阶段，用于解决 $n$ 代理的单边（稳定室友）匹配问题。我们分析了欧文提案算法的完全扩展版本，该算法一直运行，直到每个代理持有提案或代理被代理偏好列表中的每个人拒绝。结果表明，终端、定向、伙伴关系形成稳定的排列，匹配对在任何其他稳定排列中保持匹配。在假设所有 $n$ 排名都是独立统一的情况下研究提议算法的可能行为。证明了每个智能体都有一个合作伙伴的概率很高（whp），并且长度为 $\ge 3$ 的循环中的智能体数量和稳定匹配的总数都在概率上是有界的。Whp 提案总数渐近到 $0.5 n^{3/2}$。