One-sided version of Gale-Shapley proposal algorithm and its likely behavior under random preferences

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 $0.5n^{3\mathrm{/}2}$.