Stable Matchings and Their Histories: Historical Reachability under Population Shocks

Sequential-entry procedures need not reach every stable matching. We ask whether full reachability is restored when agents on both sides may enter and exit repeatedly, with proposal-chain restabilization after each population change. It is not: a three-by-three marriage market has a stable matching that is unreachable from the empty active market along any such history restricted to terminal agents. We then allow temporary historical agents who are absent at the terminal date. For every target stable matching, we construct a target-dependent, preference-preserving augmentation and an admissible history that reaches it, with all temporary agents exiting before termination. The construction extends to many-to-one markets with responsive hospital preferences. Thus reachability depends on the admissible history class. Terminal-agent histories may select a proper subset of the stable set, whereas preference-preserving augmented histories recover the full stable set. When such augmented histories are admissible but past participants are unobserved, the terminal primitives alone do not exclude any stable matching.