Proportional Subset Voting: Difference between revisions

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Proportional Subset Voting (PSV) is a Single-Winner and Multi-Winner, Cardinal voting systems proposed by Aldo Tragni.

Procedure

Ballot uses range [0%,100%]. $N$ is the number of winners.

For each vote, and for each subset of $N$ candidates:

create a list of $G_{i}$ groups, 1 for each percentage, multiple of ${\frac {100\%}{N}}$ (eg for N = 4 the groups are 0%, 25%, 50%, 75%, 100%).

for each $G_{i}$ , the voters with a total approval $=G_{i}$ are counted getting $C_{i}$ $(\ \mathrm {total\ approval\ of\ the\ N\ candidates} ={\frac {\mathrm {ratings\ sum} }{N}}\ )$ . Each $C_{i}$ is then divided by the number of voters.

find the group with the lowest percentage $G_{min}$ containing $C_{min}>0$ , and associate $S_{min}=\{G_{min},C_{min}\}$ to the subset.

find the group with the highest percentage $G_{max}$ containing $C_{max}>0$ , and associate $S_{max}=\{G_{max},C_{max}\}$ to the subset.

Sort the subsets from major to minor based on the $S_{min}$ value; if there are tied subsets then sort them from minor to major based on the $S_{max}$ value. The subset that is first after sorting wins.

If there are any tied subsets, $P$ is calculated which is the sum of the products between $G_{i}$ and $C_{i}$ . The subset that has highest $P$ wins.

Procedure with range

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 only difference with the procedure that uses multiple-choice ballots is that:

the total approval of a voter for N candidates must be somehow rounded up to a multiple of ${\frac {100\%}{N}}$ , in order to be counted in the list of $G_{i}$ groups.

Example

Ballot uses 2 ratings, that is: {0,1} = {0%, 100%}. There are 6 winners and the following votes:

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 $\{S_{min},S_{max}\}$ 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)

N is the number of winners.

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-Winner system) is used, on the converted votes, to obtain the winning subset.

The size of the range, procedure p1, and procedure p2 chosen, determine the variant of Subset Voting.

@@ Line 3: / Line 3: @@
 ==Procedure==
-Ballot use range [-MAX,MAX], also without 0. N is the number of winners.
+Ballot uses range [0%,100%]. <math>N</math> is the number of winners.
-For each vote, and for each subset of N candidates, the following procedure is applied, considering only the original ratings of the N candidates in the vote:
+For each vote, and for each subset of <math>N</math> candidates:
+* 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, with range [-4,4]:
+===Procedure with range===
-   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:
-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:
+%[] 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)==
@@ Line 55: / Line 72: @@
 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.
+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]]