Proportional Subset Voting

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{equation}2^0\end{equation}} , the 2nd highest rating is divided by Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}2^1\end{equation}} , ... , the N-th highest rating (which is the lowest) is divided by Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}2^{N-1}\end{equation}} .
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 Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}i\end{equation}} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{equation}i=1,...,N\end{equation}} .

PSV uses range [-MAX,MAX] and in p1 divides the values by Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}2^i\end{equation}} with Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}i=0,...,N-1\end{equation}} .