In a matching market you cannot simply choose what you want by paying the asking price; you also have to be chosen, because the other side has preferences over you too. Jobs, college admissions, and marriage are matching markets: price does not clear them, so both sides must select each other rather than one side just buying.
A marketplace works well when it is thick (enough participants present at once to find good matches), uncongested (participants have time to evaluate options and make offers before opportunities vanish), and safe (participants can act on their true preferences without being penalized for honesty). Market failures can be diagnosed and repaired by checking which of these three conditions is broken.
When participants fear losing scarce partners, each tries to make or accept offers earlier than rivals, and the hiring or matching date creeps progressively earlier until deals are struck long before information is available. Roth documents this in markets for new doctors and law clerks, where appointments were being made years ahead of when the work would begin.
In the deferred-acceptance algorithm, one side proposes and the other tentatively holds its best offer while remaining open to better ones, rejecting lower offers as improvements arrive, until no one wants to break away. This yields a stable matching where no unmatched pair would both prefer each other, which is what keeps a clearinghouse from being undermined by side deals.
Transactions many people find morally repugnant (selling organs, certain forms of payment) get banned even when willing buyers and sellers exist, so these moral objections function as genuine constraints that market designers must work around rather than dismiss. What counts as repugnant varies across time and culture, so the boundary of acceptable markets shifts.