# 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 , 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 winnersAB: 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 winnersABC: 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, with2 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 .