Real life problem: How many finalists can participate per school?

Question

I have to organize a tournament with following numbers:

I have a total of 1338 participants from different schools, and we need to distribute the 16 finalist's places as fair as possible.

The students are distributed as follwing:

School and number of participiants a 254 b 211 c 150 d 186 e 158 f 85 g 66 h 105 i 123

Every school gets at least one finalist. How do I distribute the 16 final places to the different school?

Thank you so much for your help!

Answer 1

It seems natural to assign the places roughly proportional to the number of students. Then again, this might easily leave school with $0$ finalists. And we have to deal with rounding. So the next natural thing is to assign $$ \lfloor \alpha n+\beta\rfloor$$ finalists to a school with $n$ students. Remains to pick $\alpha$ and $\beta$. Typically, one fixes one of them and adjusts the other (e.g., by trial and error) until $$ \sum_{i=1}^9\lfloor \alpha n_i+\beta\rfloor =16.$$ Different strategies are possible (and some correspond to vote counting methods commonly in use to distribute parliamentary seats according to vote counts):

• Set $\beta=0$ and adjust $\alpha$. This method tends to favour large "parties" and may not lead to a "seat" for the smallest school here
• Set $\alpha=\frac{16}{\sum n_i}$ and adjust $\beta$. This method assigns the naive average as "cost" of a seat and adjusts the offset $\beta$ to make the first seats cheaper until the sum is right
• Set $\beta=1$ and adjust $\alpha$. This is essentially like the first, except that every school gets one seat for free to start with.
• Set $\beta=\frac12$ and adjust $\alpha$. This is somewhere between first and third option
• ...

Answer 2

The assignment $(3,3,2,2,2,1,1,1,1)$ is optimal for each of the following three objectives:

• Minimize sum of absolute differences between number of seats and quota
• Minimize maximum of absolute differences between number of seats and quota
• Minimize sum of absolute differences between people per seat and $1338/16$