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College admissions with stable score-limits
A common feature of the Hungarian, Irish, Spanish and Turkish higher education admission systems is that students apply for programmes and are ranked according to their scores. Students who apply for a programme with the same score are tied. Ties are broken by lottery in Ireland, by objective factors in Turkey (such as date of birth) and by other precisely defined rules in Spain. In Hungary, however, an equal treatment policy is used, students applying for a programme with the same score are all accepted or rejected together. In such a situation there is only one decision to make, whether or not to admit the last group of applicants with the same score who are at the boundary of the quota. Both concepts can be described in terms of stable score-limits. The strict rejection of the last group with whom a quota would be violated corresponds to the concept of H-stable (i.e. higher-stable) score-limits that is currently used in Hungary. We call the other solutions based on the less strict admission policy as L-stable (i.e. lower-stable) score-limits. We show that the natural extensions of the Gale–Shapley algorithms produce stable score-limits, moreover, the applicant-oriented versions result in the lowest score-limits (thus optimal for students) and the college-oriented versions result in the highest score-limits with regard to each concept. When comparing the applicant-optimal H-stable and L-stable score-limits we prove that the former limits are always higher for every college. Furthermore, these two solutions provide upper and lower boundaries for any solution arising from a tie-breaking strategy. Finally we show that both the H-stable and the L-stable applicant-proposing score-limit algorithms are manipulable.