Difference between revisions of "Proportional Subset Voting"
Aldo Tragni (talk  contribs) 
Aldo Tragni (talk  contribs) (Corrected the example.) 

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==Procedure== 
==Procedure== 

−  Ballot use range [MAX,MAX]. N is the number of winners. 
+  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 
+  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 
+  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: 
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: 
The following example shows how the sums for each subset are obtained, given the converted votes: 

−  3 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: 
Sums for each subset: 

−  +  AD[11] AC[13] AB[2] CD[11] BD[2] BC[0] 

−  The winner is 
+  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. 

−  +  For each vote, and for each subset of N candidates, a score S is obtained using procedure p1, finally obtaining the converted votes. 

+  
⚫  
+  ''In the converted votes, subsets are considered as single candidates with a score.'' 

⚫  
−  ''The 
+  ''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,...,N1\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,...,N1\end{equation}</math>. 
[[Category:Singlewinner voting methods]] 
[[Category:Singlewinner voting methods]] 
Revision as of 12:16, 30 September 2020
Proportional Subset Voting (PSV) is a SingleWinner and MultiWinner, 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 Nth 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 SingleWinner 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 .