# 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]. 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 a vote:

N = 2 (winners)
Original vote, with range [-4,4]:
A[4] B[-4] C[0] D[2]
Subsets
A,B: 4/1 + -4/2 = 2
A,C: 4/1 + 0/2 = 4
A,D: 4/1 + 2/2 = 5
B,C: -4/1 + 0/2 = -4
B,D: -4/1 + 2/2 = -3
C,D: 0/1 + 2/2 = 1
Converted vote:
A,D[5] A,C[4] A,B[2] C,D[1] B,D[-3] B,C[-4]


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

3 converted votes:
A,D[5]  A,C[4]  A,B[2]  C,D[1]  B,D[-3]  B,C[-4]
A,D[2]  A,C[4]  A,B[-4] C,D[7]  B,D[-3]  B,C[1]
A,D[4]  A,C[5]  A,B[1]  C,D[2]  B,D[-4]  B,C[-3]
Sums for each subset:
A,D[11] A,C[13] A,B[-1] C,D[10] B,D[-10] B,C[-6]
The winner is {A,C}


## 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.

### Thiele method

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