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Discovery program (DISC) is a policy used by the New York City Department of Education (NYC DOE) to increase the number of admissions of students from low socio-economic background to specialized high schools. This policy has been instrumental in increasing the number of disadvantaged students attending these schools. However, assuming that students care more about the school they are assigned to rather than the type of seat they occupy (\emph{school-over-seat hypothesis}), our empirical analysis using data from 12 recent academic years shows that DISC creates about 950 in-group blocking pairs each year amongst disadvantaged students, impacting about 650 disadvantaged students every year. Moreover, we find that this program does not respect improvements, thus unintentionally creating an incentive to under-perform. These experimental results are confirmed by our theoretical analysis. In order to alleviate the concerns caused by DISC, we explore two alternative policies: the minority reserve (MR) and the joint-seat allocation (JSA) mechanisms. As our main theoretical contribution, we introduce a feature of markets, that we term high competitiveness, and we show that under this condition, JSA dominates MR for all disadvantaged students. We give sufficient conditions under which high competitiveness is verified. Data from NYC DOE satisfies the high competitiveness condition, and for this dataset our empirical results corroborate our theoretical predictions, showing the superiority of JSA. Given that JSA can be implemented by a simple modification of the classical deferred acceptance algorithm with responsive preference lists, we believe that, when the school-over-seat hypothesis holds, the discovery program can be changed for the better by implementing the JSA mechanism, leading in particular to aligned incentives for the top-performing disadvantaged students.