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I'm working with percentage splits and I'm trying to add extra players into the split but I'm coming unstuck...

Scenario:

I have agreements with a number of parties that set out what type of split we get from a pot of cash. With some it's 50/50, with others it may be 60/40 in favour of either party or perhaps even 70/30. In a scenario where there are only 2 players, myself and someone else, this is simple. What happens however when I add a third party?

So lets say I am already working with someone that gets an equal cut (50/50) with me, and we have to bring someone else on board whom I have a 70/30 agreement. So now that adjusts the equation, two parties must make equal amounts and the third party must make some factor less to balance the pot to 100%.

I can figure out that this is about ratios - 30%/70% = 0.428571. By trial and error, I can figure out that each of the parties getting an equal cut now get 41.2% of the pot and the remaining party (30%) gets 17.6% of the pot. I'd like to figure out a formula that gives me the values but I can only seem to get so far...

Assuming that the splits are as follows:

Me     Other    Ratio
--     -----    -----
70%    30%      0.428571
50%    50%      1

X  = First player   (Me/Player 1)
Y  = Second player  (Player 2)
Z  = Third player   (Player 3)
R1 = Ratio split between X and Y
R2 = Ratio split between X and Z

Y + Z = X / 0.428571

Given that Y = Z in this scenario, we could say that:

  2Y = X / 0.428571 or 
  Y (or Z) = X / 0.428571 / 2

Or

  Y (or Z) = X / (2 x 0.428571)
           = X / 0.857143

It seems like X / R1 = Y, which divided by R2 = Z, the sum of which adds up to 1. But I come unstuck when I calculate different splits using this kind of algorithm:

Me    Other    Ratio
--    -----    -----
60%   40%      0.666667
30%   70%      2.333333

When I calculate using the second ratio, I should be multiplying(?) In the first scenario, R1 is 1, so it makes no difference whether I divide or multiply, so I guess I should be multiplying in this case.

Perhaps I've take completely the wrong path. Can anyone show me either where I'm going wrong or show me the next leap I need to make to get to a result? Given the original splits, is there a formula to quickly tell me what the adjusted split would be when the extra party is added to the equation?

Any help would be greatly appreciated, I've been out of school so long, this kind of theoretical math is way outside my normal range of expertise.

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1 Answer 1

up vote 3 down vote accepted

If you have a two way deal with all the other players, you can just normalize them all. Say you have a 70/30 split with A and a 40/60 split with B. Then if you get 1, A gets 7/3 and B gets 4/6. That would call for a total of 7/3+1+2/3=4, so divide everybody by 4. A gets 7/12, you get 1/4=3/12, B gets 1/6=2/12.

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Thank you so much, that totally simplifies things. –  BobTheBuilder Dec 23 '10 at 18:43

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