# Optimizing for Students

## 2020/08/26

Increasingly, priorities viewed as design lever. Only two problems: not clear what objective is, and not clear how to achieve it.

This post will go through a very stylized set of results.

# Designing DA Priorities

Consider a market with $$n$$ students and $$n$$ schools, each with a single seat. We are going to assign students using the student-proposing deferred acceptance algorithm. The student preference profile $$P$$ is drawn from known distribution $$D$$. We get to design the priorities $$L$$ before observing $$P$$. Our goal is to choose $$L$$ to minimize (or maximize) the expected sum of students' ranks for their assignment, which we denote $$R(P,L)$$.

$$$\overline{R}(L) = \mathbb{E}_{P \sim D} [R(P,L)]$$$ $$$\min_{L}\,\,\overline{R}(L) = \min_{L} \mathbb{E}_{P \sim D} [R(P,L)].$$$

Some possibilities:

• All schools share a common ranking of students.
• Schools hold independent lotteries.
• Each school has a different first choice, and school rankings are all cyclic permutations of each other. (One school ranks $$A > B > C ... > Z$$, another ranks $$B > C > ... > Z > A$$, a third ranks $$C > ... > Z > A > B$$, and so on).
• Half of schools share a ranking of students, and half reverse this ranking.

We know student preferences are uniformly random. We want to design complete priorities to help students.

With $$n$$ students and $$n$$ schools, Knuth (1976,1997) shows that for any priorities, students average at most $$H_n - 1 \leq \log(n)$$ rejections. He conjectures that average rank is minimized by having identical rankings across all schools. In this case, students still average more than $$H_n - 2$$ rejections. If identical priorities are optimal with fewer students than seats, then they are also optimal with more -- can always choose so that additional students don't hurt the existing ones. (Wait, I might be able to prove this by induction!)

Knoblauch (2009) partially proves this -- in particular, for any

In other words, carefully choosing priorities can change students' average rank by at most $$1$$ (priorities don't matter much).

# Moving to Other Mechanisms

Equivalence between TTC and RSD. Holds for any symmetric preference profile. Suggests (oddly) that TTC weakly worse than DA, but also that RSD is optimal (in line with Knuth's conjecture).

# Peeking at Preferences

Mention Nikzad result, conjecture. Identify open question: can this be done in IC way?

e #Varying Number of Students

The preceding analysis assumed an equal number of students and schools (with one seat per school). Extending it to the case where schools exceed students is straightforward: we get that with $$m < n$$ students with uniform random preferences, for any priority rule, the average rank is at most $\frac{1}{m}\sum_{i =0}^{m-1} \frac{n}{n-i} = \frac{n}{m}( H_n-H_{n-m}) \approx \frac{n}{m} \log\left(\frac{n}{n-m}\,\,\right).$

Meanwhile, for RSD, the total number of proposals is $\frac{1}{m}\sum_{i =0}^{m-1} \frac{n+1}{n-i+1} = \frac{n+1}{m}( H_{n+1}-H_{n-m+1}).$ This looks like Bulow-Klemperer! Adding one school is better than optimizing priorities. Also implies difference is at most one.as

More students (once we define) -- get some striking result!

Then we can force our desired matching. This results in those who are matched getting (n+1)/2 rank on average: (n-1)/2 rejections, where n = number of schools.