Let us assume that there are only these three rules limiting the possible combinations out of the 2^32:
1. If everyone votes nay, then the result is nay.
2. If a combination of votes is nay, then the opposite combination, where everyone votes in the opposite direction, is yea (and vice versa).
3. If a combination of votes is yea, then it will still be yea if anyone who voted nay changes to yea.
Because of the second rule, I only need to list 16 of the 32 voting combination. The other 16 combinations simply have the opposite results. "Y" stands for "yea" and "N" stands for "nay."
YYNNN YNNNN NNNNN
YNYNN NYNNN
YNNYN NNYNN
YNNNY NNNYN
NYYNN NNNNY
NYNYN
NYNNY
NNYYN
NNYNY
NNNYY
By rule 1, the result of the third column must be against. By rules 2 and 3 together, no more than one combination in column 2 can result yea. If one combination in column 2 did result in favor, then the voting power is distributed in such a way that one person's vote has complete control (like a distribution of {5,1,1,1,1}). When this is the case, we have no more freedom to change the results of the 10 combinations in column 1. There are five permutations of this voting power distribution.
The rest of the voting power distributions must result in nay for all of column 2. Following rules 2 and 3, there are only a few basic combinations of results possible.
In this list, each column after the first is numbered and represents one of the possible results of outcomes, excluding the permutations.
1 2 3 4 5 6
YYNNN Y Y Y Y Y N
YNYNN Y Y Y Y N N
YNNYN Y Y N N N N
YNNNY Y N N N N N
NYYNN N N Y N N N
NYNYN N N N N N N
NYNNY N N N N N N
NNYYN N N N N N N
NNYNY N N N N N N
NNNYY N N N N N N
The following list shows an example distribution for each of the 6 possible combinations of outcomes, and the number of permutations.
example permutations
1: 3,1,1,1,1 5
2: 4,2,2,2,1 20
3: 3,3,3,1,1 10
4: 3,2,2,1,1 30
5: 2,2,1,1,1 10
6: 1,1,1,1,1 1
Adding up all the permutations as well as the first 5 distributions counted, there are 81 unique voting power distributions. Since I found an example for each, my original assumption is correct, at least for the 5-owner case. |
