I'm looking for a formula for the following problem. Hopefully I can explain this clearly and it all makes sense. No, it's not my homework, it's part of a competition I'm involved with managing and we're trying to figure out the fairest way to go about this. The competition is not exactly as described below, this is just the simplest way I can think of explaining the problem without boring you all with the details of the competition.
There are $340$ numbers in a hat ($1-340$). ($340$ is specific to this).
Someone draws $5$ numbers from the hat, calculates their score then puts the numbers back in the hat. Then the next person does the same, and so on... so each person has $340$ numbers in the hat ($1 – 340$) to draw upon.
We can say that the lowest score wins... i.e.: Person $1$ draws numbers $1, 2, 3, 4$ and $5$, their score will be $15$. ($1+2+3+4+5 = 15$). Person $2$ draws numbers $15, 28, 74, 156$ and $328$, their score will be $601$. ($15+28+74+156+328 = 601$). Person $1$ wins with the lowest score of $15$.
But let's say we want to do it this way...
Person $1$ draws $1, 2, 3, 4$ and $340$. Their score is $350$. Person $2$ draws $50, 51, 52, 53$ and $54$. Their score is $260$.
What if we want to decide who has drawn the "MOST NUMBER OF LOWEST NUMBERS"? If we go with "lowest score wins", in the above case, person $2$ will win with the lowest score, but person $1$ has drawn the MOST NUMBER OF LOWEST NUMBERS, so we want them to win.
We're looking for a formula to give a score to each person based on the "MOST NUMBER OF LOWEST NUMBERS DRAWN", with this score working for any total of $5$ numbers drawn out of the $340$ numbers available, then we want to be able to decide the winner based on this score (whether that score be the highest or the lowest).
Thanks for reading, I hope it makes sense, and I hope you can help.
There is no one the best criteria to select the winner, but I can show one based on statistical ranking.
Let's say we have $n$ people and for person $i$ we've got $5$ numbers: $X_{i1}<\ldots<X_{i5}.$
We can create vector $Y=(X_{11},\ldots,X_{15},X_{21},\ldots,X_{25},\ldots,X_{n1},\ldots,X_{n5})$ with all results.
Then let $V=sort(Y)$ be the sorted vector and let $W=(W_{11},\ldots,W_{15},\ldots,W_{n1},\ldots,W_{n5}),$ where $W_{ij}$ is the rank of $X_{ij}$ in vector $V.$
We compute $S_i=W_{i1}+\ldots+W_{i5}$ - it's the score of person $i$ and then person with the lowest score wins.
In case of ties in our scores we can take $4$ (or smaller number) lowest numbers $X_{i1},\ldots,X_{i4}$ and then do these steps once more.
For example we have: $X_1=(1,4,10,11,12),$ $X_2=(2,3,10,11,12),$ $X_3=(4,5,9,10,11),$ $X_4=(2,5,8,11,12).$
Then $Y=(1,4,10,11,12,2,3,10,11,12,4,5,9,10,11,2,5,8,11,12),$ $V=(\underbrace{1}_{X_1,rank=1},\underbrace{2,2}_{X_2,X_4, rank=2.5},\underbrace{3}_{X_2,rank=4},\underbrace{4,4}_{X_1,X_3,rank=5.5},\underbrace{5,5}_{X_3,X_4,rank=7.5},\underbrace{8}_{X_4,rank=9},\underbrace{9}_{X_3,rank=10},\underbrace{10,10,10}_{X_1,X_2,X_3,rank=12},\underbrace{11,11,11,11}_{X_1,X_2,X_3,X_4,rank=15.5},\underbrace{12,12,12}_{X_1,X_2,X_4,rank=19})$
and
$W=(1,5.5,12,15.5,19,2.5,4,12,15.5,19,5.5,7.5,10,12,15.5,2.5,7.5,9,15.5,19).$
Hence:
$S_1=1+5.5+12+15.5+19=53,$
$S_2=2.5+4+12+15.5+19=53,$
$S_3=5.5+7.5+10+12+15.5=50.5,$
$S_4=2.5+7.5+9+15.5+19=53.5.$
So here person with number $3$ wins.