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 uses range [0%,100%]. <math>\begin{equation}N\end{equation}</math> 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 <math>\begin{equation}N\end{equation}</math> candidates:
+For each vote, and for each subset of <math>N</math> candidates:
-* create a list of <math>\begin{equation}G_i\end{equation}</math> groups, 1 for each percentage, multiple of <math>\begin{equation}\frac{100\%}{N}\end{equation}</math> ''(eg for N = 4 the groups are 0%, 25%, 50%, 75%, 100%)''.
+* 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%)''.
-* for each <math>\begin{equation}G_i\end{equation}</math>, the voters with a total approval <math>\begin{equation}=G_i\end{equation}</math> are counted getting <math>\begin{equation}C_i\end{equation}</math> <math>\begin{equation}(\ \mathrm{ total\ approval\ of\ the\ N\ candidates}=\frac{\mathrm{ratings\ sum}}{N}\ )\end{equation}</math>. Each <math>\begin{equation}C_i\end{equation}</math> is then divided by the number of voters.
+* 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.
-* find the group with the lowest percentage <math>\begin{equation}G_{min}\end{equation}</math> containing <math>\begin{equation}C_{min}>0\end{equation}</math>, and associate <math>\begin{equation}S_{min}=\{G_{min},C_{min}\}\end{equation}</math> to the 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.
-* find the group with the highest percentage <math>\begin{equation}G_{max}\end{equation}</math> containing <math>\begin{equation}C_{max}>0\end{equation}</math>, and associate <math>\begin{equation}S_{max}=\{G_{max},C_{max}\}\end{equation}</math> to the subset.
+* 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.
-Sort the subsets from major to minor based on the <math>\begin{equation}S_{min}\end{equation}</math> value; if there are tied subsets then sort them from minor to major based on the <math>\begin{equation}S_{max}\end{equation}</math> value. The subset that is first after sorting wins.
+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.
-If there are any tied subsets, <math>\begin{equation}P\end{equation}</math> is calculated which is the sum of the products between <math>\begin{equation}G_i\end{equation}</math> and <math>\begin{equation}C_i\end{equation}</math>. The subset that has highest <math>\begin{equation}P\end{equation}</math> wins.
+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.
 ===Procedure with range===
@@ Line 24: / Line 24: @@
 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 <math>\begin{equation}\frac{100\%}{N}\end{equation}</math>, in order to be counted in the list of <math>\begin{equation}G_i\end{equation}</math> groups.
+* 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.
 ==Example==
@@ Line 54: / Line 54: @@
   Short:    66%['''33%'''] 83%['''66%''']
-Find <math>\begin{equation}\{S_{min},S_{max}\}\end{equation}</math> and sort:
+Find <math>\{S_{min},S_{max}\}</math> and sort:
  ABCDEG: {33%,83%}