# Proportional Subset Voting

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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:

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.