# 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 (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 N}**
is the number of winners.

For each vote, and for each subset of **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 N}**
candidates:

- create a list of
**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 G_i}**groups, 1 for each percentage, multiple of**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 \frac{100\%}{N}}***(eg for N = 4 the groups are 0%, 25%, 50%, 75%, 100%)*.

- for each
**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 G_i}**, 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 =G_i}**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 C_i}****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 (\ \mathrm{ total\ approval\ of\ the\ N\ candidates}=\frac{\mathrm{ratings\ sum}}{N}\ )}**. Each**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 C_i}**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 G_{min}}**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 C_{min}>0}**, 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 S_{min}=\{G_{min},C_{min}\}}**to the subset.

- find the group with the highest 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 G_{max}}**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 C_{max}>0}**, 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 S_{max}=\{G_{max},C_{max}\}}**to the subset.

Sort the subsets from major to minor 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 S_{min}}**
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 S_{max}}**
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 P}**
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 G_i}**
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 C_i}**
. The subset that has highest **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 P}**
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 (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 \frac{100\%}{N}}**, in order to be counted in the list of

## 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 \{S_{min},S_{max}\}}**
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.*