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Affirmative action policies are those which aim to increase the demographic representation of underrepresented groups in a variety of contexts. One such context which has seen its day in court several times since 1978 is within college undergraduate admissions. The U.S. Supreme Court interprets the constitutionality of such policies under the framework of strict scrutiny, and most policies have been upheld because they were determined to serve a compelling government interest in their specific use. With so many upheld policies, it is curious that we still see new cases developing, but this is due largely to a single term found in affirmative action litigation: critical mass.

Colloquially, critical mass is an ideal number or proportion of students which, when attained, will account for equal representation and opportunity among different subgroups. Since schools cannot legally use strict enrollment quotas, these cases tend to avoid any definitions pertaining to critical mass, while paradoxically still advocating for their constitutionality on the basis of moving demographics towards this idealized enrollment total. Such avoidance can be seen in the most recent case which states that critical mass is neither some number nor percentage of students in the general population.

Figure 1. Diagram of Markov chain model for college admissions pipeline.

The vagueness of critical mass has been an asset to colleges and universities implementing affirmative action policies in their admissions. However, it is only a matter of time until this will eventually become a detriment for them. It only takes one decision to overturn decades of precedence, which is entirely possible due to the lack of understanding on what critical mass means. For this reason, many schools have put together task groups whose sole goal is to analyze the role of affirmative action in their school admissions process and provide feedback to admissions offices on how to move towards critical mass. This feedback loop is entirely legal as it bypasses the admissions offices themselves from determining a course of action, avoiding any miscategorization as producing a racial quota.

These task groups are usually part of a university’s department of inclusion, diversity, equity, and the like. As of now, their main process of analysis has been through retroactive data studies. While straightforward, retroactive data studies are very costly in terms of time and money. Single studies need to wait anywhere from five to fifteen years before data analysis can begin, and the infrastructure needed to continue projects for that long is not trivial. It is due to this inconvenience that we propose a new way of analyzing affirmative action policies.

Figure 2. Future enrollment demographics given no intervention.

Rather than relying on the past to shape current policy, we can look to the future. We have developed a Markov Chain model for the college admissions pipeline which can be used to predict demographic enrollment totals at colleges year to year. To show that such a process is feasible, we conducted a case study at the University of California, Berkeley (UC Berkeley).

The first step in this process was to determine a null model based on a quantification of critical mass. This was done by using demographic projections directly from the State of California’s Department of Finance and the U.S. Census Bureau according to year and race/ethnicity. We only used the racial/ethnic subgroups of White/Caucasian, Hispanic/Latino, African-American, and Asian-American since they make up the vast majority of domestic college applicants. From the demographic projections above, we found critical mass enrollment values for each racial/ethnic subgroup by year using the following formula:

$(Null \enspace Model)_{ij} = CA_{ij} × UCB_{i, in-state} + US_{ij} × UCB_{i, out-of-state}$

where $$i$$ is the year and $$j$$ is the subgroup. This formulation accounts for the fact that UC Berkeley is a public institution and therefore enrolls California residents disproportionately, as well as that the demographics of California are not reflective of the U.S. as a whole.

Next, we project what we would anticipate as the enrollment demographics given the actual trends seen in recent data. We look to the National Center for Education Statistics, the University of California, and UC Berkeley’s Office of Planning & Analysis to predict future application, admission, enrollment, retention, and graduation rates for high school students matriculating to UC Berkeley in a given year. This allows us to parameterize our absorbing Markov chain model (see Figure 1). To follow this model, all individuals begin in 0A (high school senior, fall term) and continue on blue arrows between states until they eventually take a red arrow to DNF (does not finish their time at UC Berkeley) or a green arrow to graduating in four, five, or six years.

Now that we have a null model and projections based on current enrollment trends, we can compare the two enrollment projections to give us insight into what future classes at UC Berkeley will look like. In Figure 2—with years as the input—the output is

$n_{ij} = \frac{(Markov \enspace Chain \enspace Prediction)_{ij}}{(Null \enspace Model)_{ij}} ×100,$

where $$i$$ is the year and $$j$$ is the racial/ethnic subgroup. This means that for every 100 students from group $$j$$ that our null model suggests would enroll in year $$i$$, we would actually see $$n_{ij}$$ students enroll. According to this metric, we see that Asian-American students are the only overrepresented group (by about 3.5-4 times the anticipated rate) while all other groups are underrepresented (with Hispanic/Latino and African-American students doing the worst).

Figure 3. Future enrollment demographics given strong intervention.

As we see in Figure 2, parity will not be achieved if there is no intervention implemented. We may not know exactly what the root of this inequality is, but we know it is due in part to disproportionate application, acceptance, and enrollment rates between the subgroups, in relation to their demographic makeup in the general population. Moving on to policy suggestions, we know that to intervene in enrollment figures we would need to influence application, acceptance, and enrollment rates for incoming classes.

Due to California Proposition 209, affirmative action policies in undergraduate admissions are unconstitutional, which means that admissions offices are not allowed to directly change the acceptance rates on a racial/ethnic basis. Therefore, rather than changing acceptance rates, admissions offices can only directly affect application or enrollment rates. This can be done by recruiting underrepresented groups more (to increase application rates) or by increasing on-campus resources or financial aid for underrepresented groups (to increase enrollment rates).

As an example of what level of effect we could create, Figure 3 shows what the enrollment would be if the admissions office were to increase the application and enrollment rates for White/Caucasian, Hispanic/Latino, and African-American students by three times (in total) over the next ten years while leaving the rates for Asian-American students uninfluenced. We notice in Figure 3 that inequality still exists, but this enrollment projection is much more equal than in Figure 2, with the only underrepresented groups (African American and white/caucasian students) better off than without any intervention.

Overall, we find that by using a null model based on racial/ethnic demographics in the general population, running a Markov chain model to predict student enrollment given current trends, and combining the results of both models into a single metric, we can help admissions offices understand where to allocate their resources in order to create more equal incoming classes. By furthering equity among the enrollment figures at colleges and universities, we can take a step forward in creating a more equal and just society.

Daniel Maes presented this work during a minisymposium on modeling female and minority representation in society at the 2019 SIAM Conference on Applications of Dynamical Systems, which took place in May in Snowbird, Utah. A much more in-depth paper of this work can be found here

 Daniel P. Maes is a master’s ​student at the University of Michi​gan-Ann Arbor in applied and interdisciplinary mathematics. He is an associated member of the Institute for the Quantitative Study of Inclusion, Diversity, and Equity. In addition to his research on increasing diversity and equity, he also works on ecological questions regarding population and community dynamics.