Students are partitioned into groups of equal size similar to Stable roommates problem except can be any natural number greater than 1. Each student ranks the other students in strict order of preferences.
For each person can rank all other students from the most preferred to the least preferred. These preferences can be thought of as , where is rank (1 highest) of in '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
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 he wouldn't be satisfied as much as a preference ranked at
The ideal group for a person would consist of his preferences ranked at to (amount of students in a group except for himself) this would give him a satisfaction rate of 1 (meaning fully satisfied), or written mathematically:
So what could the definition of function be?