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We study the strategic simplicity of stable matching mechanisms where oneside has fixed preferences, termed priorities. Specifically, we ask whichpriorities are such that the strategyproofness of deferred acceptance (DA) canbe recognized by agents unable to perform contingency reasoning, that is,\emph{when is DA obviously strategyproof} (Li, 2017)?  We answer this question by completely characterizing those priorities whichmake DA obviously strategyproof (OSP). This solves an open problem of Ashlagiand Gonczarowski, 2018. We find that when DA is OSP, priorities are eitheracyclic (Ergin, 2002), a restrictive condition which allows priorities to onlydiffer on only two agents at a time, or contain an extremely limited cyclicpattern where all priority lists are identical except for exactly two. Weconclude that, for stable matching mechanisms, the tension betweenunderstandability (in the sense of OSP) and expressiveness of priorities isvery high.

Classification of Priorities Such That Deferred Acceptance is Obviously Strategyproof