Who Gets to Be in Mankiw's Seminar: The Optimal Market Design

By Vivian Zhang

In light of the popularity of his freshman seminar, Harvard economics professor Gregory Mankiw begins his class with a deceptively simple question: what is the best method of allocating the 12 seats available given a large number of applicants? I really enjoyed participating in the class debate that ensued.

His question points to the broader debate on the optimal market design for course allocation. In a perfect world, he would select the students who gain the most utility from the course and have the greatest ability to contribute ideas that benefit their peers; students who value the course at a similar level should have equal chances of receiving an offer. Whether universities choose to adopt a bidding system, ask students to rank their choices or set up a simple lottery there is a necessary trade-off between the metrics of ideal market design. This is especially true in larger lecture-style classes unlike seminars; professors do not have the luxury of reading student applications to gauge each individual’s interest in their course. Particularly in those cases, the question is which system of course allocation would be preferable on the net.

The bidding system is not uncommon. At Northwestern University’s Kellogg School of Management, students compete for the most popular cases using a multi-round bidding system. First-years receive 2000 points over two quarters, while second-years receive 3000. These points are like virtual money. Students allocate points towards various classes to signal how much relative utility they expect to get out of each course. If the bid was successful, the student’s bid minus the minimum successful bid for a given class is returned to the student’s account at the end of the bidding period. The Kellogg School’s bidding system is considered “multi-round” because it occurs in three key phases. Students bid for seats allocated to their specific program in the first phase; in the second and third phases, they bid for seats that are still open or for waitlist positions. Other bidding models such as the single round of bidding at the Harvard Kennedy School of Government involve fewer complicated steps.

The main rationale for this market design is that students can express their cardinal utility, the specific utility value they would assign to a given course. Furthermore, the system is fairer for students. Whereas a pure lottery system could mean that an unlucky student receives only one of their most valued courses and a lucky one receives five throughout their degree, a bidding system is set up with the basic assumption that each student is guaranteed a certain level of utility. Bidding is an imperfect system; some students may still receive the shorter end of the stick. However, it significantly mitigates the randomness of pure lottery systems that do not give preference to students who have not received slots in their top-choice classes in the past.

There are two theories of price instability that opponents of the bidding design generally cite. Firstly, if the prices from the previous semester are published online, students will use the data to speculate, leading to unpredictable fluctuations in price for the most popular classes. If the minimum successful bid in the last semester was almost the full budget, for instance, students may opt out of bidding for that class. They may favor securing two classes that were less popular in the past to spending all their points on one. In the subsequent semester, students would notice that the course has a lower price and begin bidding for it again. The lack of stability, in this case, prevents students from making a sufficiently informed decision. This prevents them from expressing their true, cardinal preferences and defeats the purpose of the bidding system. Moreover, in the long run, inflation will likely be in a positive trajectory as students want to be safe and bid slightly above previous prices. As such, the administration would have to increase the point allocation per semester. Since points carry through semesters, this adjustment to maintain fairness could be complex and time-consuming.

Some students at Harvard Kennedy School have also complained that they cannot purchase books and properly focus on classes until bidding results return a few weeks after they return to campus. These fluctuating numbers also assign a relatively arbitrary value to each course. A student consulting the list of prices from past semesters is nudged by the point system to follow mob mentality. They may spend their points on courses that others have historically valued more, forgetting about the value of developing a unique, individualized set of academic interests.

Meanwhile, Harvard Business School has embraced a different mechanism of allocation. Students are asked to rank a set of elective courses. Subsequently, each student is assigned a different number. Going in ascending order, students are assigned to their highest ranking class that has not reached capacity. During the next round, courses are allocated in descending order based on the number the student was assigned.

With a ranked system, students are not incentivized to engage in strategic behavior; they will genuinely list out their preferences. The rankings are more accessible to non-economics concentrators as well. However, this user-friendly setup comes at the cost of knowing the cardinal preferences students have. The administration cannot tell if a student values their second preference at a thousand utils above their third. The lack of precision means that some allocations are not optimal.

The even simpler method of course allocation is a separate lottery for each popular class, similar to Harvard College’s General Education course allocation system before Spring 2020. We can think of this like a roll of a dice. Winning a spot in a popular class does not make someone less likely to receive an offering for another prized course. This old system did not consider ordinal utility, let alone cardinal utility. There were no stakes to entering as many lotteries as possible, so students who were more or less indifferent about certain classes received offers for seats that others were desperate to have.

If students are confused by multi-stage bidding and there is value in tracking a student’s relative preference for a course, a ranked lottery may be one of the more viable designs.