Proportional Subset Voting

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

Procedure

Ballot use range [0,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 1, the 2nd highest rating is divided by 2, ... , the N-th highest rating (which is the lowest) is divided by N.
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 [0,5]:
  A[5] B[0] C[2] D[4]
Subsets
  A,B: 5/1 + 0/2 = 5
  A,C: 5/1 + 2/2 = 6 
  A,D: 5/1 + 4/2 = 7
  B,C: 2/1 + 0/2 = 2
  B,D: 4/1 + 0/2 = 4
  C,D: 4/1 + 2/2 = 5
Converted vote:
  A,D[7] A,C[6] A,B[5] C,D[5] B,D[4] B,C[2]

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

3 converted votes:
  A,D[7]  A,C[6]  A,B[5]  C,D[5]  B,D[4]  B,C[2]
  A,D[5]  A,C[6]  A,B[2]  C,D[7]  B,D[4]  B,C[5]
  A,D[6]  A,C[7]  A,B[5]  C,D[5]  B,D[2]  B,C[4]
Sums for each subset:
  A,D[18] A,C[19] A,B[12] C,D[17] B,D[10] B,C[11]
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 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)

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.