Group satisfaction rateStudents 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 = 1The 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 ) = 1So what could the definition of function f ( m , p i j ) be?

Question

Group satisfaction rateStudents 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&#039;s ordering.I&#039;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&#039;s group contains one of his preferences ranked at for example       p          i      j        =  3 he wouldn&#039;t be satisfied as much as a preference ranked at       p          i      j        =  1The 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        )  =  1So what could the definition of function   f  (  m  ,      p          i      j        ) be?

Teagan Zamora · Accepted Answer

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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