Mathematical formula for equal distribution of amount among different group

Question

Please let me know if you think i should edit my question or description. Problem statement: lets say i have spent $x on a sports material which needs to shared among total y memeber of the team. But the condition here is that some of the team members are not regular and they also don't know the number of days they have played so putting an equal amount on all of them would not be fair. However they can be categorized in three groups like (least participants, average participants & regular players). What would be the best mathematical formula which can be applied to solve this problem? Any help will be really appreciated. Thanks

Answer 1

That is entirely subjective and depends on how much you think each member of each category should get. For example, if you feel each regular player should get three times as much funding as someone who rarely comes, and each "average" player should get twice as much as someone who rarely comes we can come up with the following:

Let $y_1$ be the number of players who show up infrequently, $y_2$ be the players who show up an average number of times, and $y_3$ be the regular members of the team.

Letting $a$ be the amount used by a player who shows up rarely, the total amount of budget used is $a(y_1) + 2a(y_2)+3a(y_3)=a(y_1+2y_2+3y_3)$. We know the total budget was $x$. We can then solve for $a$ by setting these two equal to one another and dividing.

We find then $a=\frac{x}{y_1+2y_2+3y_3}$

For example, with a budget of $\$1000$, five infrequent players, ten average players, and five regular players, following the rule that regular players receive three times the amount of money that an infrequent player does and average players receive twice the amount of money that an infrequent player does, we see that each infrequent player will receive $\frac{1000}{5+2\cdot 10+3\cdot 5} = \frac{1000}{40}=25$ dollars, each average player will receive $50$ dollars, and each regular player will receive $75$ dollars.

Checking that it adds up correctly, $5\cdot 25 + 10\cdot 50 + 5\cdot 75=1000$

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The choice of having the regular players get three times the amount of funding as the infrequent players was entirely arbitrary. You can come up with all sorts of other ways to distribute the money. The major point being, whatever scheme you decide to use, try to describe it using unknowns and equate that to the amount of money you have available. Then, solve for the unknowns with algebraic techniques.