# Difference between revisions of "Proportional Subset Voting"

Aldo Tragni (talk | contribs) (Created page with "Proportional Subset Voting (PSV) is a Single-Winner and Multi-Winner, Cardinal voting systems developed by User:Aldo Tra...") |
Aldo Tragni (talk | contribs) |
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− | Proportional Subset Voting (PSV) is a [[Single Member system|Single-Winner]] and [[Multi-Member System|Multi-Winner]], [[Cardinal voting systems]] |
+ | Proportional Subset Voting (PSV) is a [[Single Member system|Single-Winner]] and [[Multi-Member System|Multi-Winner]], [[Cardinal voting systems]] proposed by [[User:Aldo Tragni|Aldo Tragni]]. |

==Procedure== |
==Procedure== |
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− | Ballot use range [ |
+ | Ballot use range [-MAX,MAX]. N is the number of winners. |

For each vote, and for each subset of N candidates, the following procedure is applied, considering only the original ratings of the N candidates in the vote: |
For each vote, and for each subset of N candidates, the following procedure is applied, considering only the original ratings of the N candidates in the vote: |
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− | * the highest rating is divided by |
+ | * the highest rating is divided by <math>\begin{equation}2^0\end{equation}</math>, the 2nd highest rating is divided by <math>\begin{equation}2^1\end{equation}</math>, ... , the N-th highest rating (which is the lowest) is divided by <math>\begin{equation}2^{N-1}\end{equation}</math>. |

* after this division, the ratings are added to obtain the value S. |
* after this division, the ratings are added to obtain the value S. |
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N = 2 (winners) |
N = 2 (winners) |
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− | Original vote, with range [ |
+ | Original vote, with range [-4,4]: |

− | A[ |
+ | A[4] B[-4] C[0] D[2] |

Subsets |
Subsets |
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− | A,B: |
+ | A,B: 4/1 + -4/2 = 2 |

− | A,C: |
+ | A,C: 4/1 + 0/2 = 4 |

− | A,D: |
+ | A,D: 4/1 + 2/2 = 5 |

− | B,C: |
+ | B,C: -4/1 + 0/2 = -4 |

− | B,D: 4/1 + |
+ | B,D: -4/1 + 2/2 = -3 |

− | C,D: |
+ | C,D: 0/1 + 2/2 = 1 |

Converted vote: |
Converted vote: |
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− | A,D[ |
+ | A,D[5] A,C[4] A,B[2] C,D[1] B,D[-3] B,C[-4] |

The following example shows how the sums for each subset are obtained, given the converted votes: |
The following example shows how the sums for each subset are obtained, given the converted votes: |
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3 converted votes: |
3 converted votes: |
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− | A,D[ |
+ | A,D[5] A,C[4] A,B[2] C,D[1] B,D[-3] B,C[-4] |

− | A,D[ |
+ | A,D[2] A,C[4] A,B[-4] C,D[7] B,D[-3] B,C[1] |

− | A,D[ |
+ | A,D[4] A,C[5] A,B[1] C,D[2] B,D[-4] B,C[-3] |

Sums for each subset: |
Sums for each subset: |
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− | A,D[ |
+ | A,D[11] A,C[13] A,B[-1] C,D[10] B,D[-10] B,C[-6] |

The winner is {A,C} |
The winner is {A,C} |
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− | |||

− | ===Name derivation=== |
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− | |||

− | Proportional Subset Voting (PSV): |
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− | |||

− | * "Proportional" (P): describes how the scores S are calculated from the vote (reference to [[Proportional approval voting|PAV]]). |
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− | * "Subset" (S): because the initial vote is converted into a cardinal vote in which the options evaluated are the subsets (instead of the individual candidates), on which [[Range Voting]] is then applied. |
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− | * "[[Range Voting]]" (V): for each candidate, the sum of the values obtained in the votes is calculated, and the highest one wins. |
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==Subset Voting (category)== |
==Subset Voting (category)== |
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Considering one vote, a score S is obtained for each subset of N candidates. This procedure is applied to all votes and returns converted votes. |
Considering one vote, a score S is obtained for each subset of N candidates. This procedure is applied to all votes and returns converted votes. |
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− | Considering the subsets as single candidates, a |
+ | Considering the subsets as single candidates, a Single-Winner system applicable on the converted votes, is used to obtain the winning subset. |

''The way of calculating S and the [[Single Member system|Single-Winner system]] chosen, determines the variant of Subset Voting.'' |
''The way of calculating S and the [[Single Member system|Single-Winner system]] chosen, determines the variant of Subset Voting.'' |
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+ | |||

+ | ===Thiele method=== |
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+ | |||

+ | Thiele method uses range [0,MAX] and divides the values by <math>\begin{equation}i\end{equation}</math> with <math>\begin{equation}i=1,...,N\end{equation}</math>. |
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+ | |||

+ | PSV uses range [-MAX,MAX] and divides the values by <math>\begin{equation}2^i\end{equation}</math> with <math>\begin{equation}i=0,...,N-1\end{equation}</math>. |
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+ | |||

+ | [[Category:Single-winner voting methods]] |
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+ | [[Category:Multi-winner voting methods]] |
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+ | [[Category:Cardinal voting methods]] |

## Revision as of 14:26, 29 September 2020

Proportional Subset Voting (PSV) is a Single-Winner and Multi-Winner, Cardinal voting systems proposed by Aldo Tragni.

## Procedure

Ballot use range [-MAX,MAX]. N is the number of winners.

For each vote, and for each subset of N candidates, the following procedure is applied, considering only the original ratings of the N candidates in the vote:

- the highest rating is divided by , the 2nd highest rating is divided by , ... , the N-th highest rating (which is the lowest) is divided by .
- after this division, the ratings are added to obtain the value S.

By applying this procedure, in the end, we obtain for each vote a list of scores S, one for each subset.

The scores S, for each subset, are added together and the subset with the highest sum contains the N winners.

### Example

The following example shows how scores S are obtained from a vote:

N = 2 (winners) Original vote, with range [-4,4]: A[4] B[-4] C[0] D[2] Subsets A,B: 4/1 + -4/2 = 2 A,C: 4/1 + 0/2 = 4 A,D: 4/1 + 2/2 = 5 B,C: -4/1 + 0/2 = -4 B,D: -4/1 + 2/2 = -3 C,D: 0/1 + 2/2 = 1 Converted vote: A,D[5] A,C[4] A,B[2] C,D[1] B,D[-3] B,C[-4]

The following example shows how the sums for each subset are obtained, given the converted votes:

3 converted votes: A,D[5] A,C[4] A,B[2] C,D[1] B,D[-3] B,C[-4] A,D[2] A,C[4] A,B[-4] C,D[7] B,D[-3] B,C[1] A,D[4] A,C[5] A,B[1] C,D[2] B,D[-4] B,C[-3] Sums for each subset: A,D[11] A,C[13] A,B[-1] C,D[10] B,D[-10] B,C[-6] The winner is {A,C}

## Subset Voting (category)

N is the number of winners.

Considering one vote, a score S is obtained for each subset of N candidates. This procedure is applied to all votes and returns converted votes.

Considering the subsets as single candidates, a Single-Winner system applicable on the converted votes, is used to obtain the winning subset.

*The way of calculating S and the Single-Winner system chosen, determines the variant of Subset Voting.*

### Thiele method

Thiele method uses range [0,MAX] and divides the values by with .

PSV uses range [-MAX,MAX] and divides the values by with .