The Role of Precedence Order in Matching with Multi-Criteria Admissions

Admission systems often fill seats by applying criterion-specific rankings sequentially. We derive criterion-wise comparative statics with respect to precedence order when rankings may be arbitrarily nonaligned, and each criterion evaluates the entire admitted set through a responsive preference. For a fixed applicant set, reversing the final two blocks makes each criterion weakly prefer the outcome in which its block is later. This comparison can reverse if two blocks are followed by another criterion, even under reserve-type rankings. Under student-proposing deferred acceptance, it can also reverse with two criteria because rejection chains change the focal college's applicant set; such feedback is the only possible source of failure. In regular large markets, failures vanish for almost all colleges and, under sufficient thickness, uniformly across colleges. Under the Boston mechanism, final acceptance prevents feedback; thus the comparison holds in every finite market for fixed submitted rank-order lists and criterion-independent acceptability.