# Difference between revisions of "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 $\begin{equation}2^0\end{equation}$ , the 2nd highest rating is divided by $\begin{equation}2^1\end{equation}$ , ... , the N-th highest rating (which is the lowest) is divided by $\begin{equation}2^{N-1}\end{equation}$ .
• 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 B[-4] C D
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 AC AB CD BD BC[-2]

Original vote, with range [-4,4]:
A B[-4] C D
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 ABD ABC BCD


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

3 converted votes, with 2 winners:
AD  AC  AB  CD  BD  BC[-2]
AD  AC  AB[-2] CD  BD  BC
AD  AC  AB  CD  BD[-2] BC
Sums for each subset:
AD AC AB  CD BD[-2] BC
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 $\begin{equation}i\end{equation}$ with $\begin{equation}i=1,...,N\end{equation}$ .

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