Proportional Subset Voting: Difference between revisions

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

For each vote, and for each subset of <math>\begin{equation}N\end{equation}</math> candidates:

* create a list of <math>\begin{equation}G_i\end{equation}</math> groups, 1 for each percentage, multiple of <math>\begin{equation}\frac{100\%}{N}\end{equation}</math> ''(eg for N = 4 the groups are 0%, 25%, 50%, 75%, 100%)''.

* for each <math>\begin{equation}G_i\end{equation}</math>, the voters with a total approval <math>\begin{equation}=G_i\end{equation}</math> are counted getting <math>\begin{equation}C_i\end{equation}</math> <math>\begin{equation}(\ \mathrm{ total\ approval\ of\ the\ N\ candidates}=\frac{\mathrm{ratings\ sum}}{N}\ )\end{equation}</math>. Each <math>\begin{equation}C_i\end{equation}</math> is then divided by the number of voters.

* find the group with the lowest percentage <math>\begin{equation}G_{min}\end{equation}</math> containing <math>\begin{equation}C_{min}>0\end{equation}</math>, and associate <math>\begin{equation}S_{min}=\{G_{min},C_{min}\}\end{equation}</math> to the subset.

* find the group with the highest percentage <math>\begin{equation}G_{max}\end{equation}</math> containing <math>\begin{equation}C_{max}>0\end{equation}</math>, and associate <math>\begin{equation}S_{max}=\{G_{max},C_{max}\}\end{equation}</math> to the subset.

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

If there are any tied subsets, <math>\begin{equation}P\end{equation}</math> is calculated which is the sum of the products between <math>\begin{equation}G_i\end{equation}</math> and <math>\begin{equation}C_i\end{equation}</math>. The subset that has highest <math>\begin{equation}P\end{equation}</math> 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 <math>\begin{equation}\frac{100\%}{N}\end{equation}</math>, in order to be counted in the list of <math>\begin{equation}G_i\end{equation}</math> 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 <math>\begin{equation}\{S_{min},S_{max}\}\end{equation}</math> 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

Procedure p2 (eg a [[Single Member system|Single-Winner system]]) is used, on the converted votes, to obtain the winning subset.