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%]. Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}N\end{equation}} is the number of winners.
For each vote, and for each subset of Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}N\end{equation}} candidates:
- create a list of Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}G_i\end{equation}} groups, 1 for each percentage, multiple of Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}\frac{100\%}{N}\end{equation}} (eg for N = 4 the groups are 0%, 25%, 50%, 75%, 100%).
- for each Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}G_i\end{equation}} , the voters with a total approval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{equation}=G_i\end{equation}} are counted getting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{equation}C_i\end{equation}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{equation}(\ \mathrm{ total\ approval\ of\ the\ N\ candidates}=\frac{\mathrm{ratings\ sum}}{N}\ )\end{equation}} . Each Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}C_i\end{equation}} is then divided by the number of voters.
- find the group with the lowest percentage Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}G_{min}\end{equation}} containing Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}C_{min}>0\end{equation}} , and associate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{equation}S_{min}=\{G_{min},C_{min}\}\end{equation}} to the subset.
- find the group with the highest percentage Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}G_{max}\end{equation}} containing Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}C_{max}>0\end{equation}} , and associate Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}S_{max}=\{G_{max},C_{max}\}\end{equation}} to the subset.
Sort the subsets from major to minor based on the Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}S_{min}\end{equation}} value; if there are tied subsets then sort them from minor to major based on the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{equation}S_{max}\end{equation}} value. The subset that is first after sorting wins.
If there are any tied subsets, Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}P\end{equation}} is calculated which is the sum of the products between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{equation}G_i\end{equation}} and Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}C_i\end{equation}} . The subset that has highest Failed to parse (unknown function "\begin{equation}"): {\displaystyle \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 Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}\frac{100\%}{N}\end{equation}} , in order to be counted in the list of Failed to parse (unknown function "\begin{equation}"): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.