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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!

RobPratt
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math2hard
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    Why not the best? – Hagen von Eitzen Sep 18 '20 at 20:07
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    what do you mean "as fair as possible"? Is it just ensuring there is at least one finalist from each school or any other consideration? – Math Lover Sep 18 '20 at 20:10
  • Sorry, maybe i was unclear. The schools will be allocated starting places at the tournament. The more students, the more starting places. This is not yet a matter of performance. I have 16 starting places at the tournament and would like to distribute them among the schools as fairly as possible. – math2hard Sep 18 '20 at 20:11
  • Math Lover - there's just one rule, every school get's at least one starting place. School g is very small, so by normal distribution they would'nt get a starting place. – math2hard Sep 18 '20 at 20:12
  • How is having one finalist from each school fair? If school $A$ has 1000 students and school $B$ has 100 students, there is an unfair bias toward school $B$. If you really want one finalist from each school then each school should have a competition to see which delegate they send. – John Douma Sep 18 '20 at 20:19
  • Fair enough. You give one starting position to every school as a good gesture and then remaining of $16$ in ratio of their size because you want to be fair. But how is it a probability question? I am just looking at the question tag. – Math Lover Sep 18 '20 at 20:24
  • John, it is not fair and I am aware of that. But that's just one rule I must abide. Every school does send their best students - the question is, which school can send how many of their best students – math2hard Sep 18 '20 at 20:24

2 Answers2

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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
  • ...
Hagen von Eitzen
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  • Thank you so much, Hagen, that did confirm what I was suspecting! I just was'nt sure, If I was missing something, if there was THE formula to calculate it - but there is several ways of looking at it. Vielen Dank! – math2hard Sep 18 '20 at 20:31
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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 $1338/16$
RobPratt
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