There are $n*q$ students and there are $n$ schools, each with $q$ seats. Students have a strict preference over schools and schools have a strict preference over students. An assignment of students to schools is stable if there is no student that is not matched to a school that has either empty seats or seats allocated to less preferred students.
Assume now the preferences for both sides are uniform at random. We are concerned with the expected ranking of the school to which the average student is assigned in the student optimal stable assignment. A rank of 1 means he is assigned to the best school, 2 to the second best, $n$ means the worst.
It is known that if $q=1$, the average rank of assigned schools is $r \approx log(n)$, see https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-856j-randomized-algorithms-fall-2002/lecture-notes/n4.pdf.
The question is what is the corresponding ranking in terms of $n$ and $q$? Note that as $q$ grows, it is more likely that a student attends his best school because it has more seats (i.e. when $q\to \infty$, $r \approx 1$).
Simulations: Consider $n*q=500$.
- $q=1$ and $n=500$ -> $r\approx 6.2$
- $q=2$ and $n=250$ -> $r\approx 4.07$
- $q=5$ and $n=100$ -> $r\approx 2.47$
- $q=10$ and $n=50$ -> $r\approx 1.79$
A conjecture that works pretty well is that $r\approx ceiling(log(n*q)/q)$. So that
- $q=1$ and $n=500$ -> $r\approx 6.2$ -> 7
- $q=2$ and $n=250$ -> $r\approx 4.07$ -> 4
- $q=5$ and $n=100$ -> $r\approx 2.47$ -> 2
- $q=10$ and $n=50$ -> $r\approx 1.79$ -> 1
But I wonder if there is a general formula that can be derived from the original coupon collector problem. Edit 1: $r \approx (log(n*q)/q)+1$ is a really good approximation.