# Proportional Subset Voting

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

## Procedure

Ballot uses range [0%,100%]. $\begin{equation}N\end{equation}$ is the number of winners.

For each vote, and for each subset of $\begin{equation}N\end{equation}$ candidates:

• create a list of $\begin{equation}G_i\end{equation}$ groups, 1 for each percentage, multiple of $\begin{equation}\frac{100\%}{N}\end{equation}$ (eg for N = 4 the groups are 0%, 25%, 50%, 75%, 100%).
• for each $\begin{equation}G_i\end{equation}$ , the voters with a total approval $\begin{equation}=G_i\end{equation}$ are counted getting $\begin{equation}C_i\end{equation}$ $\begin{equation}(\ \mathrm{ total\ approval\ of\ the\ N\ candidates}=\frac{\mathrm{ratings\ sum}}{N}\ )\end{equation}$ . Each $\begin{equation}C_i\end{equation}$ is then divided by the number of voters.
• find the group with the lowest percentage $\begin{equation}G_{min}\end{equation}$ containing $\begin{equation}C_{min}>0\end{equation}$ , and associate $\begin{equation}S_{min}=\{G_{min},C_{min}\}\end{equation}$ to the subset.
• find the group with the highest percentage $\begin{equation}G_{max}\end{equation}$ containing $\begin{equation}C_{max}>0\end{equation}$ , and associate $\begin{equation}S_{max}=\{G_{max},C_{max}\}\end{equation}$ to the subset.

Sort the subsets from major to minor based on the $\begin{equation}S_{min}\end{equation}$ value; if there are tied subsets then sort them from minor to major based on the $\begin{equation}S_{max}\end{equation}$ value. The subset that is first after sorting wins.

If there are any tied subsets, $\begin{equation}P\end{equation}$ is calculated which is the sum of the products between $\begin{equation}G_i\end{equation}$ and $\begin{equation}C_i\end{equation}$ . The subset that has highest $\begin{equation}P\end{equation}$ wins.

### Procedure with range

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 $\begin{equation}\frac{100\%}{N}\end{equation}$ , in order to be counted in the list of $\begin{equation}G_i\end{equation}$ groups.

## Example

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

A B C D E F G ...
A B C D E F G ...
A B C D E F G ...


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


Find $\begin{equation}\{S_{min},S_{max}\}\end{equation}$ 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)

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.