# 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 ${\displaystyle $$2^0$$}$, the 2nd highest rating is divided by ${\displaystyle $$2^1$$}$, ... , the N-th highest rating (which is the lowest) is divided by ${\displaystyle $$2^{N-1}$$}$.
• 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 ${\d$$i$$splaystyle \beg$$i$$n{equat$$i$$on}$$i$$\end{equat$$i$$on}}$ with ${\displaystyle $$i=1,...,N$$}$.

PSV uses range [-MAX,MAX] and in p1 divides the values by ${\displaystyle $$2^i$$}$ with ${\displaystyle $$i=0,...,N-1$$}$.