# Students are partitioned into groups of equal size m similar to Stable roommates problem except m can be any natural number greater than 1. Each student ranks the other students in strict order of preferences. For each person i in {1,…,n}, i can rank all other students from the most preferred to the least preferred. These preferences can be thought of as p_(ij), where p_(ij) is rank (1 highest) of j in i's ordering. I'm trying to calculate the weighted rate of satisfaction a person would get mathematically from being in a group that contains his preference ranked at n. The weighted rate could be calculated with a function f(m,p_(ij))=? The satisfaction rate is a number between 0 and 1 that multiplied by 100 could be converted to percentages. Each preference should be weighted differently.

Ayanna Jarvis 2022-10-14 Answered
Group satisfaction rate
Students are partitioned into groups of equal size $m$ similar to Stable roommates problem except $m$ can be any natural number greater than 1. Each student ranks the other students in strict order of preferences.
For each person $i\in \left\{1,\dots ,n\right\}$ can rank all other students from the most preferred to the least preferred. These preferences can be thought of as ${p}_{ij}$, where ${p}_{ij}$ is rank (1 highest) of $j$ in $i$'s ordering.
I'm trying to calculate the weighted rate of satisfaction a person would get mathematically from being in a group that contains his preference ranked at n. The weighted rate could be calculated with a function $f\left(m,{p}_{ij}\right)=?$
The satisfaction rate is a number between 0 and 1 that multiplied by 100 could be converted to percentages.
Each preference should be weighted differently. If the student's group contains one of his preferences ranked at for example ${p}_{ij}=3$ he wouldn't be satisfied as much as a preference ranked at ${p}_{ij}=1$
The ideal group for a person would consist of his preferences ranked at ${p}_{ij}=1$ to ${p}_{ij}=m-1$ (amount of students in a group except for $i$ himself) this would give him a satisfaction rate of 1 (meaning fully satisfied), or written mathematically:
$\sum _{{p}_{ij}=1}^{m-1}f\left(m,{p}_{ij}\right)=1$
So what could the definition of function $f\left(m,{p}_{ij}\right)$ be?
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Teagan Zamora
Let me give an example. There are $n=4$ people and group of size $m=3$. The rankings are as follows (row is $i$, column is $j$, that is, those are the ${p}_{ij}$):
$\begin{array}{cccc}0& 1& 2& 3\\ 3& 0& 2& 1\\ 3& 2& 0& 1\\ 1& 2& 3& 0\end{array}$
That is, person $1$ wants to be with $2$ most and $4$ least. Person $3$, say, wants to be with $4$ most and $1$ least.
Suppose the group is $G=\left\{2,3,4\right\}$. Then the satisfaction for $i=2$ is $\sum _{j\in G}{p}_{2j}=1+2=3$. The worst satisfaction a player can get from $G$ is $\sum _{j=1}^{m-1}m+1-i=3+2=5$. The best satisfaction a player can get from $G$ is $\sum _{i=1}^{m-1}i=1+2=3$. Denoting these by $S$, $W$ and $B$ respectively, $\frac{W-S}{B-S}$ is a measure that has higher numbers for happier people and ranges on $\left[0,1\right]$