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Proportional Subset Voting: Difference between revisions

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Proportional Subset Voting (view source)

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==Procedure==
 
Ballot useuses range [-MAX0%,MAX100%], also without 0. <math>N</math> is the number of winners.
 
For each vote, and for each subset of <math>N</math> candidates, the following procedure is applied, considering only the original ratings of the N candidates in the vote:
* create a list of <math>G_i</math> groups, 1 for each percentage, multiple of <math>\frac{100\%}{N}</math> ''(eg for N = 4 the groups are 0%, 25%, 50%, 75%, 100%)''.
* the highest rating is divided by <math>\begin{equation}2^0\end{equation}</math>, the 2nd highest rating is divided by <math>\begin{equation}2^1\end{equation}</math>, ... , the N-th highest rating (which is the lowest) is divided by <math>\begin{equation}2^{N-1}\end{equation}</math>.
* after this division, the ratings are added to obtain the value S.
 
* for each <math>G_i</math>, the voters with a total approval <math>=G_i</math> are counted getting <math>C_i</math> <math>(\ \mathrm{ total\ approval\ of\ the\ N\ candidates}=\frac{\mathrm{ratings\ sum}}{N}\ )</math>. Each <math>C_i</math> is then divided by the number of voters.
By applying this procedure, in the end, we obtain for each vote a list of scores S, one for each subset.
 
* find the group with the lowest percentage <math>G_{min}</math> containing <math>C_{min}>0</math>, and associate <math>S_{min}=\{G_{min},C_{min}\}</math> to the subset.
The scores S, for each subset, are added together and the subset with the highest sum contains the N winners.
 
* find the group with the highest percentage <math>G_{max}</math> containing <math>C_{max}>0</math>, and associate <math>S_{max}=\{G_{max},C_{max}\}</math> to the subset.
===Example===▼
 
Sort the subsets from major to minor based on the <math>S_{min}</math> value; if there are tied subsets then sort them from minor to major based on the <math>S_{max}</math> value. The subset that is first after sorting wins.
The following example shows how scores S are obtained from one vote:
 
If there are any tied subsets, <math>P</math> is calculated which is the sum of the products between <math>G_i</math> and <math>C_i</math>. The subset that has highest <math>P</math> wins.
Original vote, with range [-4,4]:
A[4] B[-4] C[0] D[2]
Subsets for N = '''2 winners'''
AB: 4/1 + -4/2 = 2
AC: 4/1 + 0/2 = 4
AD: 4/1 + 2/2 = 5
BC: 0/1 + -4/2 = -2
BD: 2/1 + -4/2 = 0
CD: 2/1 + 0/2 = 2
Converted vote:
AD[5] AC[4] AB[2] CD[2] BD[0] BC[-2]
 
Original vote,===Procedure with range [-4,4]:===
A[4] B[-4] C[0] D[2]
Subsets for N = '''3 winners'''
ABC: 4/1 + 0/2 + -4/4 = 3
ACD: 4/1 + 2/2 + 0/4 = 5
ABD: 4/1 + 2/2 + -4/4 = 4
BCD: 2/1 + 0/2 + -4/4 = 1
Converted vote:
ACD[5] ABD[4] ABC[3] BCD[1]
 
The ratings of the range have values between [0%,100%]. For example, if it has 5 ratings {0,1,2,3,4} then the respective values will be {0%,25%,50%,75%,100%}.
The following example shows how the sums for each subset are obtained, given the converted votes:
 
The only difference with the procedure that uses multiple-choice ballots is that:
3 converted votes, with '''2 winners''':
 
AD[5] AC[4] AB[2] CD[2] BD[0] BC[-2]
* the total approval of a voter for N candidates must be somehow rounded up to a multiple of <math>\frac{100\%}{N}</math>, in order to be counted in the list of <math>G_i</math> groups.
AD[2] AC[4] AB[-2] CD[7] BD[0] BC[2]
 
AD[4] AC[5] AB[2] CD[2] BD[-2] BC[0]
▲===Example===
Sums for each subset:
 
AD[11] AC[13] AB[2] CD[11] BD[-2] BC[0]
Ballot uses 2 ratings, that is: {0,1} = {0%, 100%}. There are 6 winners and the following votes:
The winner is AC.
 
A[1] B[1] C[1] D[1] E[1] F[1] G[0] ...
A[1] B[1] C[1] D[1] E[1] F[1] G[0] ...
A[1] B[1] C[1] D[0] E[0] F[0] G[1] ...
 
The list of groups associated with the subset of 6 candidates is:
 
0%[] 16%[] 33%[] 50%[] 66%[] 83%[] 100%[]
 
The total approval of the 3 voters is calculated for the following subsets:
 
ABCDEF: [100%, 100%, 50%]
ABCDEG: [83%, 83%, 66%]
 
Total approvals are counted for each subset, and then divided by the number of voters (3 in this case):
 
ABCDEF
Count: 0%[] 16%[] 33%[] 50%['''1'''] 66%[] 83%[] 100%['''2''']
Division: 0%[] 16%[] 33%[] 50%['''33%'''] 66%[] 83%[] 100%['''66%''']
Short: 50%['''33%'''] 100%['''66%''']
ABCDEG
Count: 0%[] 16%[] 33%[] 50%[] 66%['''1'''] 83%['''2'''] 100%[]
Division: 0%[] 16%[] 33%[] 50%[] 66%['''33%'''] 83%['''66%'''] 100%[]
Short: 66%['''33%'''] 83%['''66%''']
 
Find <math>\{S_{min},S_{max}\}</math> and sort:
 
ABCDEG: {33%,83%}
ABCDEF: {33%,100%}
 
ABCDEG wins.
 
Calculate P, just to show how this is done:
 
ABCDEF: 50%[33%] 100%[66%] --> 50%*33% + 100%*66% = 0,825
ABCDEG: 66%[33%] 83%[66%] --> 66%*33% + 83%*66% = 0,765
 
==Subset Voting (category)==
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For each vote, and for each subset of N candidates, a score S is obtained using procedure p1, finally obtaining the converted votes.
 
Procedure p2 (eg. a [[Single Member system|Single-Winner system]]) is used, on the converted votes, to obtain the winning subset.
 
''In the converted votes, subsets are considered as single candidates with a score.''
 
''The size of the range, procedure p1, and procedure p2 chosen, determine the variant of Subset Voting.''
 
===Thiele method===
 
Thiele method uses range [0,MAX] and in p1 divides the values by <math>\begin{equation}i\end{equation}</math> with <math>\begin{equation}i=1,...,N\end{equation}</math>.
 
PSV uses range [-MAX,MAX] and in p1 divides the values by <math>\begin{equation}2^i\end{equation}</math> with <math>\begin{equation}i=0,...,N-1\end{equation}</math>.
 
[[Category:Single-winner voting methods]]
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