Unique Stable Matchings

We provide necessary and sufficient conditions on the preferences of market participants for a unique stable matching in models of two-sided matching with non-transferable utility. We use the process of iterated deletion of unattractive alternatives (IDUA), a formalisation of the reduction procedure in Balinski and Ratier (1997), and we show that an instance of the matching problem possesses a unique stable matching if and only if IDUA collapses each participant preference list to a singleton. (This is in a sense the matching problem analog of a strategic game being dominance solvable.)