Difference between revisions of "Proportional Subset Voting"

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Proportional Subset Voting (PSV) is a [[Single Member system|Single-Winner]] and [[Multi-Member System|Multi-Winner]], [[Cardinal voting systems]] developed by [[User:Aldo Tragni|Aldo Tragni]].
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Proportional Subset Voting (PSV) is a [[Single Member system|Single-Winner]] and [[Multi-Member System|Multi-Winner]], [[Cardinal voting systems]] proposed by [[User:Aldo Tragni|Aldo Tragni]].
   
 
==Procedure==
 
==Procedure==
   
Ballot use range [0,MAX]. N is the number of winners.
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Ballot use range [-MAX,MAX]. N 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 N candidates, the following procedure is applied, considering only the original ratings of the N candidates in the vote:
* the highest rating is divided by 1, the 2nd highest rating is divided by 2, ... , the N-th highest rating (which is the lowest) is divided by N.
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* 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.
 
* after this division, the ratings are added to obtain the value S.
   
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N = 2 (winners)
 
N = 2 (winners)
Original vote, with range [0,5]:
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Original vote, with range [-4,4]:
A[5] B[0] C[2] D[4]
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A[4] B[-4] C[0] D[2]
 
Subsets
 
Subsets
A,B: 5/1 + 0/2 = 5
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A,B: 4/1 + -4/2 = 2
A,C: 5/1 + 2/2 = 6
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A,C: 4/1 + 0/2 = 4
A,D: 5/1 + 4/2 = 7
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A,D: 4/1 + 2/2 = 5
B,C: 2/1 + 0/2 = 2
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B,C: -4/1 + 0/2 = -4
B,D: 4/1 + 0/2 = 4
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B,D: -4/1 + 2/2 = -3
C,D: 4/1 + 2/2 = 5
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C,D: 0/1 + 2/2 = 1
 
Converted vote:
 
Converted vote:
A,D[7] A,C[6] A,B[5] C,D[5] B,D[4] B,C[2]
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A,D[5] A,C[4] A,B[2] C,D[1] B,D[-3] B,C[-4]
   
 
The following example shows how the sums for each subset are obtained, given the converted votes:
 
The following example shows how the sums for each subset are obtained, given the converted votes:
   
 
3 converted votes:
 
3 converted votes:
A,D[7] A,C[6] A,B[5] C,D[5] B,D[4] B,C[2]
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A,D[5] A,C[4] A,B[2] C,D[1] B,D[-3] B,C[-4]
A,D[5] A,C[6] A,B[2] C,D[7] B,D[4] B,C[5]
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A,D[2] A,C[4] A,B[-4] C,D[7] B,D[-3] B,C[1]
A,D[6] A,C[7] A,B[5] C,D[5] B,D[2] B,C[4]
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A,D[4] A,C[5] A,B[1] C,D[2] B,D[-4] B,C[-3]
 
Sums for each subset:
 
Sums for each subset:
A,D[18] A,C[19] A,B[12] C,D[17] B,D[10] B,C[11]
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A,D[11] A,C[13] A,B[-1] C,D[10] B,D[-10] B,C[-6]
 
The winner is {A,C}
 
The winner is {A,C}
 
===Name derivation===
 
 
Proportional Subset Voting (PSV):
 
 
* "Proportional" (P): describes how the scores S are calculated from the vote (reference to [[Proportional approval voting|PAV]]).
 
* "Subset" (S): because the initial vote is converted into a cardinal vote in which the options evaluated are the subsets (instead of the individual candidates), on which [[Range Voting]] is then applied.
 
* "[[Range Voting]]" (V): for each candidate, the sum of the values ​​obtained in the votes is calculated, and the highest one wins.
 
   
 
==Subset Voting (category)==
 
==Subset Voting (category)==
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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.
 
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.
   
Considering the subsets as single candidates, a [[Single Member system|Single-Winner system]] applicable on the converted votes, is used to obtain the winning subset.
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Considering the subsets as single candidates, a Single-Winner system applicable on the converted votes, is used to obtain the winning subset.
   
 
''The way of calculating S and the [[Single Member system|Single-Winner system]] chosen, determines the variant of Subset Voting.''
 
''The way of calculating S and the [[Single Member system|Single-Winner system]] chosen, determines the variant of Subset Voting.''
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===Thiele method===
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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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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>.
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[[Category:Single-winner voting methods]]
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[[Category:Multi-winner voting methods]]
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[[Category:Cardinal voting methods]]

Revision as of 14:26, 29 September 2020

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

Procedure

Ballot use range [-MAX,MAX]. N 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:

  • the highest rating is divided by , the 2nd highest rating is divided by , ... , the N-th highest rating (which is the lowest) is divided by .
  • after this division, the ratings are added to obtain the value S.

By applying this procedure, in the end, we obtain for each vote a list of scores S, one for each subset.

The scores S, for each subset, are added together and the subset with the highest sum contains the N winners.

Example

The following example shows how scores S are obtained from a vote:

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]

The following example shows how the sums for each subset are obtained, given the converted votes:

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}

Subset Voting (category)

N is the number of winners.

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.

Considering the subsets as single candidates, a Single-Winner system applicable on the converted votes, is used to obtain the winning subset.

The way of calculating S and the Single-Winner system chosen, determines the variant of Subset Voting.

Thiele method

Thiele method uses range [0,MAX] and divides the values by with .

PSV uses range [-MAX,MAX] and divides the values by with .