Difference between revisions of "Proportional Subset Voting"

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==Procedure==
 
==Procedure==
   
Ballot use range [-MAX,MAX]. N is the number of winners.
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Ballot uses range [0%,100%]. <math>\begin{equation}N\end{equation}</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:
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For each vote, and for each subset of <math>\begin{equation}N\end{equation}</math> candidates:
* 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>.
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* 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%)''.
* after this division, the ratings are added to obtain the value S.
 
   
  +
* 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.
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>\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.
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>\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.
===Example===
 
   
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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.
The following example shows how scores S are obtained from a vote:
 
   
  +
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.
N = 2 (winners)
 
Original vote, with range [-4,4]:
 
A[4] B[-4] C[0] D[2]
 
Subsets
 
A,B: 4/1 + -4/2 = 2
 
A,C: 4/1 + 0/2 = 4
 
A,D: 4/1 + 2/2 = 5
 
B,C: -4/1 + 0/2 = -4
 
B,D: -4/1 + 2/2 = -3
 
C,D: 0/1 + 2/2 = 1
 
Converted vote:
 
A,D[5] A,C[4] A,B[2] C,D[1] B,D[-3] B,C[-4]
 
   
  +
===Procedure with range===
The following example shows how the sums for each subset are obtained, given the converted votes:
 
   
  +
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%}.
3 converted votes:
 
A,D[5] A,C[4] A,B[2] C,D[1] B,D[-3] B,C[-4]
 
A,D[2] A,C[4] A,B[-4] C,D[7] B,D[-3] B,C[1]
 
A,D[4] A,C[5] A,B[1] C,D[2] B,D[-4] B,C[-3]
 
Sums for each subset:
 
A,D[11] A,C[13] A,B[-1] C,D[10] B,D[-10] B,C[-6]
 
The winner is {A,C}
 
   
  +
The only difference with the procedure that uses multiple-choice ballots is that:
==Subset Voting (category)==
 
   
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* 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.
N is the number of winners.
 
   
 
==Example==
Considering one vote, a score S is obtained for each subset of N candidates. This procedure is applied to all votes and returns converted votes.
 
   
  +
Ballot uses 2 ratings, that is: {0,1} = {0%, 100%}. There are 6 winners and the following votes:
Considering the subsets as single candidates, a Single-Winner system applicable on the converted votes, is used to obtain the winning subset.
 
  +
  +
A[1] B[1] C[1] D[1] E[1] F[1] G[0] ...
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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%[]
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  +
The total approval of the 3 voters is calculated for the following subsets:
  +
  +
ABCDEF: [100%, 100%, 50%]
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ABCDEG: [83%, 83%, 66%]
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  +
Total approvals are counted for each subset, and then divided by the number of voters (3 in this case):
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  +
ABCDEF
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Count: 0%[] 16%[] 33%[] 50%['''1'''] 66%[] 83%[] 100%['''2''']
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Division: 0%[] 16%[] 33%[] 50%['''33%'''] 66%[] 83%[] 100%['''66%''']
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Short: 50%['''33%'''] 100%['''66%''']
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ABCDEG
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Count: 0%[] 16%[] 33%[] 50%[] 66%['''1'''] 83%['''2'''] 100%[]
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Division: 0%[] 16%[] 33%[] 50%[] 66%['''33%'''] 83%['''66%'''] 100%[]
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Short: 66%['''33%'''] 83%['''66%''']
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  +
Find <math>\begin{equation}\{S_{min},S_{max}\}\end{equation}</math> and sort:
  +
  +
ABCDEG: {33%,83%}
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ABCDEF: {33%,100%}
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ABCDEG wins.
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  +
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
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==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.
''The way of calculating S and the [[Single Member system|Single-Winner system]] chosen, determines the variant of Subset Voting.''
 
   
 
Procedure p2 (eg a [[Single Member system|Single-Winner system]]) is used, on the converted votes, to obtain the winning subset.
===Thiele method===
 
   
Thiele method uses range [0,MAX] and divides the values by <math>\begin{equation}i\end{equation}</math> with <math>\begin{equation}i=1,...,N\end{equation}</math>.
 
   
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''The size of the range, procedure p1, and procedure p2 chosen, determine the variant of Subset Voting.''
PSV uses range [-MAX,MAX] and 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]]
 
[[Category:Single-winner voting methods]]

Latest revision as of 16:12, 7 October 2020

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

Procedure[edit | edit source]

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

For each vote, and for each subset of candidates:

  • create a list of groups, 1 for each percentage, multiple of (eg for N = 4 the groups are 0%, 25%, 50%, 75%, 100%).
  • for each , the voters with a total approval are counted getting . Each is then divided by the number of voters.
  • find the group with the lowest percentage containing , and associate to the subset.
  • find the group with the highest percentage containing , and associate to the subset.

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

If there are any tied subsets, is calculated which is the sum of the products between and . The subset that has highest wins.

Procedure with range[edit | edit source]

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 , in order to be counted in the list of groups.

Example[edit | edit source]

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 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)[edit | edit source]

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.