# Proportional Subset Voting

Proportional Subset Voting (PSV) is a Single-Winner and Multi-Winner, Cardinal voting systems proposed by Aldo Tragni.

## Procedure[edit | edit source]

Ballot uses range [0%,100%]. is the number of winners.

For each vote, and for each subset of candidates:

- create a list of groups, 1 for each percentage, multiple of
*(eg for N = 4 the groups are 0%, 25%, 50%, 75%, 100%)*.

- for each , the voters with a total approval are counted getting . Each is then divided by the number of voters.

- find the group with the lowest percentage containing , and associate to the subset.

- find the group with the highest percentage containing , and associate to the subset.

Sort the subsets from major to minor based on the value; if there are tied subsets then sort them from minor to major based on the value. The subset that is first after sorting wins.

If there are any tied subsets, is calculated which is the sum of the products between and . The subset that has highest wins.

### Procedure with range[edit | edit source]

The ratings of the range have values between [0%,100%]. For example, if it has 5 ratings {0,1,2,3,4} then the respective values will be {0%,25%,50%,75%,100%}.

The only difference with the procedure that uses multiple-choice ballots is that:

- the total approval of a voter for N candidates must be somehow rounded up to a multiple of , in order to be counted in the list of groups.

## Example[edit | edit source]

Ballot uses 2 ratings, that is: {0,1} = {0%, 100%}. There are 6 winners and the following votes:

A[1] B[1] C[1] D[1] E[1] F[1] G[0] ... A[1] B[1] C[1] D[1] E[1] F[1] G[0] ... A[1] B[1] C[1] D[0] E[0] F[0] G[1] ...

The list of groups associated with the subset of 6 candidates is:

0%[] 16%[] 33%[] 50%[] 66%[] 83%[] 100%[]

The total approval of the 3 voters is calculated for the following subsets:

ABCDEF: [100%, 100%, 50%] ABCDEG: [83%, 83%, 66%]

Total approvals are counted for each subset, and then divided by the number of voters (3 in this case):

ABCDEF Count: 0%[] 16%[] 33%[] 50%[1] 66%[] 83%[] 100%[2] Division: 0%[] 16%[] 33%[] 50%[33%] 66%[] 83%[] 100%[66%] Short: 50%[33%] 100%[66%] ABCDEG Count: 0%[] 16%[] 33%[] 50%[] 66%[1] 83%[2] 100%[] Division: 0%[] 16%[] 33%[] 50%[] 66%[33%] 83%[66%] 100%[] Short: 66%[33%] 83%[66%]

Find and sort:

ABCDEG: {33%,83%} ABCDEF: {33%,100%}

ABCDEG wins.

Calculate P, just to show how this is done:

ABCDEF: 50%[33%] 100%[66%] --> 50%*33% + 100%*66% = 0,825 ABCDEG: 66%[33%] 83%[66%] --> 66%*33% + 83%*66% = 0,765

## Subset Voting (category)[edit | edit source]

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.

*The size of the range, procedure p1, and procedure p2 chosen, determine the variant of Subset Voting.*