Difference between revisions of "Proportional Subset Voting"

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(Corrected the example.)
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==Procedure==
 
==Procedure==
   
Ballot use range [-MAX,MAX]. N is the number of winners.
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Ballot use range [-MAX,MAX], also without 0. 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:
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===Example===
 
===Example===
   
The following example shows how scores S are obtained from a vote:
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The following example shows how scores S are obtained from one vote:
   
N = 2 (winners)
 
 
Original vote, with range [-4,4]:
 
Original vote, with range [-4,4]:
 
A[4] B[-4] C[0] D[2]
 
A[4] B[-4] C[0] D[2]
Subsets
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Subsets for N = '''2 winners'''
A,B: 4/1 + -4/2 = 2
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AB: 4/1 + -4/2 = 2
A,C: 4/1 + 0/2 = 4
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AC: 4/1 + 0/2 = 4
A,D: 4/1 + 2/2 = 5
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AD: 4/1 + 2/2 = 5
B,C: -4/1 + 0/2 = -4
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BC: 0/1 + -4/2 = -2
B,D: -4/1 + 2/2 = -3
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BD: 2/1 + -4/2 = 0
C,D: 0/1 + 2/2 = 1
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CD: 2/1 + 0/2 = 2
 
Converted vote:
 
Converted vote:
A,D[5] A,C[4] A,B[2] C,D[1] B,D[-3] B,C[-4]
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AD[5] AC[4] AB[2] CD[2] BD[0] BC[-2]
  +
  +
Original vote, 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 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:
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3 converted votes, with '''2 winners''':
A,D[5] A,C[4] A,B[2] C,D[1] B,D[-3] B,C[-4]
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AD[5] AC[4] AB[2] CD[2] BD[0] BC[-2]
A,D[2] A,C[4] A,B[-4] C,D[7] B,D[-3] B,C[1]
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AD[2] AC[4] AB[-2] CD[7] BD[0] BC[2]
A,D[4] A,C[5] A,B[1] C,D[2] B,D[-4] B,C[-3]
+
AD[4] AC[5] AB[2] CD[2] BD[-2] BC[0]
 
Sums for each subset:
 
Sums for each subset:
A,D[11] A,C[13] A,B[-1] C,D[10] B,D[-10] B,C[-6]
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AD[11] AC[13] AB[2] CD[11] BD[-2] BC[0]
The winner is {A,C}
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The winner is AC.
   
 
==Subset Voting (category)==
 
==Subset Voting (category)==
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N is the number of winners.
 
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.
+
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.''
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.''
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''The size of the range, procedure p1, and procedure p2 chosen, determine the variant of Subset Voting.''
   
 
===Thiele method===
 
===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>.
+
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 divides the values by <math>\begin{equation}2^i\end{equation}</math> with <math>\begin{equation}i=0,...,N-1\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]]
 
[[Category:Single-winner voting methods]]

Revision as of 12:16, 30 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], also without 0. 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 one vote:

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

3 converted votes, with 2 winners:
  AD[5]  AC[4]  AB[2]  CD[2]  BD[0]  BC[-2]
  AD[2]  AC[4]  AB[-2] CD[7]  BD[0]  BC[2]
  AD[4]  AC[5]  AB[2]  CD[2]  BD[-2] BC[0]
Sums for each subset:
  AD[11] AC[13] AB[2]  CD[11] BD[-2] BC[0]
The winner is AC.

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.

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 with .

PSV uses range [-MAX,MAX] and in p1 divides the values by with .