Classification of Priorities Such That Deferred Acceptance is OSP Implementable

We study the strategic simplicity of stable matching mechanisms where one side has fixed preferences, termed priorities. Specifically, we ask which priorities are such that the strategyproofness of deferred acceptance (DA) can be recognized by agents unable to perform contingency reasoning, that is, when does DA have an obviously strategyproof (OSP) implementation (Li, 2017)? We answer this question by completely characterizing those priorities for which DA is OSP implementable. This solves an open problem of Ashlagi and Gonczarowski, 2018. We find that when DA is OSP implementable, priorities are either acyclic (Ergin, 2002), a restrictive condition which allows priorities to only differ on only two agents at a time, or contain an extremely limited cyclic pattern where all priority lists are identical except for exactly two. We conclude that, for stable matching mechanisms, the tension between understandability (in the sense of OSP) and expressiveness of priorities is very high.