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 { 1 , , n } can rank all other students from the most preferred to the least preferred. These preferences can be thought of as p i j , where p i j 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 i j ) = ?
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 i j = 3 he wouldn't be satisfied as much as a preference ranked at p i j = 1
The ideal group for a person would consist of his preferences ranked at p i j = 1 to p i j = 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:
p i j = 1 m 1 f ( m , p i j ) = 1
So what could the definition of function f ( m , p i j ) be?
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Answers (1)

Teagan Zamora
Answered 2022-10-15 Author has 18 answers
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 i j ):
0 1 2 3 3 0 2 1 3 2 0 1 1 2 3 0
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 = { 2 , 3 , 4 }. Then the satisfaction for i = 2 is j G p 2 j = 1 + 2 = 3. The worst satisfaction a player can get from G is j = 1 m 1 m + 1 i = 3 + 2 = 5. The best satisfaction a player can get from G is i = 1 m 1 i = 1 + 2 = 3. Denoting these by S, W and B respectively, W S B S is a measure that has higher numbers for happier people and ranges on [ 0 , 1 ]
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