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Probabilistic Voting is a sequential [[Proportional representation|proportional voting system]] that uses either [[Approval ballot|approval]] or score ballots. |
'''Probabilistic Approval Voting''' is a sequential [[Proportional representation|proportional voting system]] that uses either [[Approval ballot|approval]] or [[Score voting|score]] ballots. Probabilistic calculations are used as a tool to predict potential power balance of elected political factions. The method itself is deterministic. The system is 2-level [[Summability criterion|precinct-summable]] and passes the [[universally liked candidate criterion]]. |
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== Derivation == |
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Given: |
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* <math display="inline">C</math> ... the set of all candidates |
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* <math display="inline">W</math> ... the set of already elected candidates |
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* <math display="inline">V(A)</math> ... number of voters who approve of A |
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* <math display="inline">V(A \and B)</math> ... number of voters who approve of both A and B |
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Let's say that |
Let's say that a candidate A proposes a decision. The probability that a candidate supports the decision equals <math display="inline">\frac{V(A \and B)}{V(B)}</math>. Expected number of votes that the decision gets equals <math display="inline">\sum_{B \in W}\frac{V(A \and B)}{V(B)}</math> or <math display="inline">\frac{\sum_{B \in W}\frac{V(A \and B)}{V(B)}}{V(A)}</math> per voter. To minimize number of votes per voter we maximize <math>\frac{V(A)}{1+\sum_{X \in W} \frac{V(A \and X)}{V(X)}}</math> for each newly elected candidate. |
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== Example == |
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In practice, electing a candidate whose inverse score is at least as low as inverse hare quota will not always be possible. In that case we need to either add failsafe approvals which we previously didn't consider or elect the candidate with the lowest inverse score and hope it is low enough. |
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<blockquote> |
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29 AB |
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A reasonable choice for P(A,B) would be <math display="inline">\frac{V(A \and B)}{V(A \or B)}</math> which would give <math>\frac{V(A)}{1+\sum_{X \in W} \frac{V(A \and X)}{V(A \or X)}}</math> as the formula for the score of candidate A. In each step we elect the candidate with the highest score. This version of the system is 2-level precint-summable and passes universal liking condition. |
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1 B |
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14 C |
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</blockquote> |
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In each step we elect the candidate with the highest <math display="inline">\frac{V(A)}{1+\sum_{X \in W} \frac{V(A \and X)}{V(X)}}</math>. |
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<math display="inline">\frac{V(A \and B)}{V(B)} = \frac{29}{30}</math> |
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<math display="inline">\frac{V(A \and C)}{V(C)} = \frac{0}{14}</math> |
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<math display="inline">\frac{V(B \and C)}{V(B)} = \frac{0}{30}</math> |
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First seat: |
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A: <math display="inline">V(A)/1 = 29/1 = 29</math> |
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B: <math display="inline">V(B)/1 = 30/1 = 30</math> |
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C: <math display="inline">V(C)/1 = 14/1 = 14</math> |
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B is elected |
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Second seat: |
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A: <math display="inline">V(A)/(1 + \frac{V(A \and B)}{V(B)}) = 29/(1 + \frac{29}{30}) = 14,745762711864</math> |
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C: <math display="inline">V(C)/(1 + \frac{V(C \and B)}{V(B)}) = 14/(1 + \frac{0}{30}) = 14</math> |
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A is elected |
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== Score ballots == |
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[[File:Probabilistic Approval Voting with score ballots.jpg|thumb|402x402px|One possible procedure to elect a candidate using score ballots]] |
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[[Category:Cardinal PR methods]] |
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[[Category:Probability]] |
Latest revision as of 12:22, 26 February 2024
Probabilistic Approval Voting is a sequential proportional voting system that uses either approval or score ballots. Probabilistic calculations are used as a tool to predict potential power balance of elected political factions. The method itself is deterministic. The system is 2-level precinct-summable and passes the universally liked candidate criterion.
Derivation
Given:
- ... the set of all candidates
- ... the set of already elected candidates
- ... number of voters who approve of A
- ... number of voters who approve of both A and B
Let's say that a candidate A proposes a decision. The probability that a candidate supports the decision equals . Expected number of votes that the decision gets equals or per voter. To minimize number of votes per voter we maximize for each newly elected candidate.
Example
29 AB
1 B
14 C
In each step we elect the candidate with the highest .
First seat:
A:
B:
C:
B is elected
Second seat:
A:
C:
A is elected
Score ballots
Probabilistic voting can be done with score ballots. We start by treating maximum score as approval. Once every candidate's score falls below Hare quota we progressively add lower scores.