Having access to quality higher education is considered by many as an effective way to improve the chances of moving up the social ladder. As a consequence, many governments around the world have implemented affirmative action policies which expand the access that ethnic minorities and students from low-income families have to universities. One of the largest such programs is currently being implemented in Brazil, where a law enacted in 2012 determines that students who are black, low-income or studied in public high-schools should have higher priority in the access to 50% of more than 180.000 seats offered at federal public universities each year.

More specifically, the law establishes that in order to take advantage of those seats, the student must have studied in a public high-school, and when filling half of those seats, priority should be given to students from low-income families. Similarly, also when filling a proportion of those seats equivalent to the share of blacks in the overall population in the state at which the University is, priority should be given to black students. In a state where blacks constitute 25% of the population, for example, a university with capacity of 80 will have 40 seats giving higher priority for students applying as having studied in a public high-school. At least 20 of those should give higher priority for those applying as low-income, and 10 for those applying as black.

But students may have different combinations of such characteristics. For example, a student may be black and low-income, or white and low-income. How to implement those policies? The method proposed by the Ministry of Education and used by the universities splits the seats in each university into five different sets: seats for low-income black students from public high-schools, for high-income black students from public high-schools, for low-income white students from public high-schools, for high-income white students from public high-schools and a set of "open access" seats. In the example given above, the number of seats in each of these sets would be 5, 5, 15, 15 and 40, respectively. With few exceptions, the criterion for admittance to federal institutions is grades obtained in a national exam, which students take and learn their grade before applying to the universities. Students who are eligible to some of these combinations, therefore, simply apply to those seats and those with the highest national exam grades are then selected.

As shown in a paper by Inácio Bó (WZB Berlin Social Science Center) and Orhan Aygün (Bogazigi University), however, this method leads to unintended incentives for students to choose strategically to which set of seats they apply to.[1] For example, a low-income black student may not be accepted to a certain university if she applies to the seats reserved for low-income black students, but could be accepted if she applies as a high-income black student when her exam grade is higher than a sufficient number of students who apply to those seats. This is possible since students have to prove their status as low-income students, but may choose not to. A similar argument holds for those applying as black. As a result, it is possible for a student who is the target of the affirmative action policy not to be accepted into a university while another student with a lower exam grade and who is not target of that affirmative action to be accepted instead.

Consider again the numerical example given above. Say student A applied as black and low income from public high-school and has a grade of 80 in the national exam, and student B applied as non-black, non-low-income from public high-school for the same university and has a grade of 70 (inferior) in the national exam. Imagine that since the number of seats reserved for black and low income from public high-school is only five, student A's grade is not high enough to be accepted, but since the number of seats reserved for students applying simply as public high-school is 15, a grade of 70 is enough for student B to be among the top 15 students applying to those seats. As a result, student B will be accepted into that university while student A will not. This observation is not purely theoretical. The authors analyzed the "cut-off values" for acceptance at more than 3.000 university programs in Brazil in the year of 2013. The cut-off value is the lowest exam grade among those who were accepted to each set of seats in each university program. If the cut-off grade for seats reserved for students applying with a certain combination of characteristics (for example, black and low-Income from public high-school) is higher than the cut-off for seats reserved for students applying with a subset of those characteristics (for example, non-black and low-Income from public high-school) then there is a chance that some student applied to the first set of seats and was not accepted, but would have been accepted if she had applied to the second set of seats. Strikingly, an analysis of the university admissions in 2013 shows that in more than 49% of the universities those conditions for distortions were observed.

After pointing out those problems, the authors of the study propose a new method for choosing students, which eliminates the problems above. The method works by guaranteeing that, while selecting the students to be accepted at a school, a student who claims belonging to one of the demographic types established in the policy is always considered for seats that prioritizes those types. Therefore, students don't compete only with those who applied with the same set of characteristics, but with all those who apply with some subset of the characteristics provided. For example, a student who applies as black from public high-school competes also with students from public high-school who don't apply as black or low-Income and also with those who are not from public high-schools and therefore can only be accepted to open access seats.

Since the proposed method for choosing students satisfies two technical conditions, namely substitutability and Law of Aggregate demand, the authors also show that it can be used in a centralized clearinghouse for matching students to universities in a mechanism which is strategy-proof and satisfies a desired fairness condition based on students' preferences and characteristics.

[1] Inácio Bó  and Orhan Aygün,"College Admissions with Multidimensional Reserves: The Brazillian Affirmative Action Case’’, Boston College Working Paper 2014.