Proportional Subset Voting: Difference between revisions

From electowiki
Content added Content deleted
VisualWikitext
(Corrected the example.)
(Correction of the procedure)
Line 3:
==Procedure==
 
Ballot useuses range [-MAX0%,MAX100%], also without 0. <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, the following procedure is applied, considering only the original ratings of the N candidates in the vote:
* the highest ratingcreate isa dividedlist byof <math>\begin{equation}2^0G_i\end{equation}</math> groups, the1 2ndfor highesteach ratingpercentage, ismultiple divided byof <math>\begin{equation}2^1\frac{100\%}{N}\end{equation}</math>, ...''(eg , thefor N-th highest= rating4 (whichthe isgroups theare lowest)0%, is25%, divided50%, by75%, <math>\begin{equation}2^{N-1}\end{equation}</math>100%)''.
* after this division, the ratings are added to obtain the value S.
 
* 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.
By applying this procedure, in the end, we obtain for each vote a list of scores S, one for each subset.
 
* 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.
The scores S, for each subset, are added together and the subset with the highest sum contains the N winners.
 
* 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.
===Example===▼
 
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.
The following example shows how scores S are obtained from one vote:
 
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.
Original vote, with range [-4,4]:
A[4] B[-4] C[0] D[2]
Subsets for N = '''2 winners'''
AB: 4/1 + -4/2 = 2
AC: 4/1 + 0/2 = 4
AD: 4/1 + 2/2 = 5
BC: 0/1 + -4/2 = -2
BD: 2/1 + -4/2 = 0
CD: 2/1 + 0/2 = 2
Converted vote:
AD[5] AC[4] AB[2] CD[2] BD[0] BC[-2]
 
Original vote,===Procedure with range [-4,4]:===
A[4] B[-4] C[0] D[2]
Subsets for N = '''3 winners'''
ABC: 4/1 + 0/2 + -4/4 = 3
ACD: 4/1 + 2/2 + 0/4 = 5
ABD: 4/1 + 2/2 + -4/4 = 4
BCD: 2/1 + 0/2 + -4/4 = 1
Converted vote:
ACD[5] ABD[4] ABC[3] BCD[1]
 
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 following example shows how the sums for each subset are obtained, given the converted votes:
 
The only difference with the procedure that uses multiple-choice ballots is that:
3 converted votes, with '''2 winners''':
 
AD[5] AC[4] AB[2] CD[2] BD[0] BC[-2]
* 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.
AD[2] AC[4] AB[-2] CD[7] BD[0] BC[2]
 
AD[4] AC[5] AB[2] CD[2] BD[-2] BC[0]
▲===Example===
Sums for each subset:
 
AD[11] AC[13] AB[2] CD[11] BD[-2] BC[0]
Ballot uses 2 ratings, that is: {0,1} = {0%, 100%}. There are 6 winners and the following votes:
The winner is AC.
 
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
 
==Subset Voting (category)==
Line 55 ⟶ 72:
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 Member system|Single-Winner system]]) is used, on the converted votes, to obtain the winning subset.
 
''In the converted votes, subsets are considered as single candidates with a score.''
 
''The size of the range, procedure p1, and procedure p2 chosen, determine the variant of Subset Voting.''
 
===Thiele method===
 
Thiele method uses range [0,MAX] and in p1 divides the values by <math>\begin{equation}i\end{equation}</math> with <math>\begin{equation}i=1,...,N\end{equation}</math>.
 
PSV uses range [-MAX,MAX] and in p1 divides the values by <math>\begin{equation}2^i\end{equation}</math> with <math>\begin{equation}i=0,...,N-1\end{equation}</math>.
 
[[Category:Single-winner voting methods]]

Revision as of 16:12, 7 October 2020

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 (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}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 (unknown function "\begin{equation}"): {\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 (unknown function "\begin{equation}"): {\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 (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_{min}\end{equation}} containing 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_{min}>0\end{equation}} , and associate Failed to parse (unknown function "\begin{equation}"): {\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 (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}=\{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 (unknown function "\begin{equation}"): {\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 (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}P\end{equation}} is calculated which is the sum of the products between Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation}G_i\end{equation}} and 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}} . 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 (unknown function "\begin{equation}"): {\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.

Retrieved from "https://electowiki.org/wiki/Proportional_Subset_Voting?oldid=12980"