Proportional Subset Voting

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[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 $$\begin{equation}\{S_{min},S_{max}\}\end{equation}$$ 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.