User:Matijaskala/Probabilistic Approval Voting: Difference between revisions

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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:

${\textstyle C}$ ... the set of all candidates
${\textstyle W}$ ... the set of already elected candidates
${\textstyle V(A)}$ ... number of voters who approve of A
${\textstyle V(A\land B)}$ ... 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 ${\textstyle {\frac {V(A\land B)}{V(B)}}}$ . Expected number of votes that the decision gets equals ${\textstyle \sum _{B\in W}{\frac {V(A\land B)}{V(B)}}}$ or ${\textstyle {\frac {\sum _{B\in W}{\frac {V(A\land B)}{V(B)}}}{V(A)}}}$ per voter. To minimize number of votes per voter we maximize ${\frac {V(A)}{1+\sum _{X\in W}{\frac {V(A\land X)}{V(X)}}}}$ for each newly elected candidate.

Example

29 AB
1 B
14 C

In each step we elect the candidate with the highest ${\textstyle {\frac {V(A)}{1+\sum _{X\in W}{\frac {V(A\land X)}{V(X)}}}}}$ .

${\textstyle {\frac {V(A\land B)}{V(B)}}={\frac {29}{30}}}$

${\textstyle {\frac {V(A\land C)}{V(C)}}={\frac {0}{14}}}$

${\textstyle {\frac {V(B\land C)}{V(B)}}={\frac {0}{30}}}$

First seat:

A: ${\textstyle V(A)/1=29/1=29}$

B: ${\textstyle V(B)/1=30/1=30}$

C: ${\textstyle V(C)/1=14/1=14}$

B is elected

Second seat:

A: ${\textstyle V(A)/(1+{\frac {V(A\land B)}{V(B)}})=29/(1+{\frac {29}{30}})=14,745762711864}$

C: ${\textstyle V(C)/(1+{\frac {V(C\land B)}{V(B)}})=14/(1+{\frac {0}{30}})=14}$

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.

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-'''Probabilistic Approval Voting''' is a sequential [[Proportional representation|proportional voting system]] that uses either [[Approval ballot|approval]] or [[Score voting|score]] ballots. Its winners are found by the use of probabilistic calculations.
+'''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]].
 == Derivation ==
+Given:
-Let's say that there is a function  <math>P:C \times C \to [0, 1]</math>which returns the probability that two given candidates belong to the same political faction. Then for a given candidate ''A'', the expected number of elected candidates belonging to the same faction equals <math display="inline">\sum_{X \in W} P(A,X)</math> and ''A''<nowiki/>'s voter efficiency equals <math display="inline">\frac{\sum_{X \in W} P(A,X)}{V(A)}</math>. If ''A'' is not already elected then we can calculate the ''A''<nowiki/>'s voter efficiency after ''A''<nowiki/>'s election as <math display="inline">\frac{1+\sum_{X \in W} P(A,X)}{V(A)}</math>. Let's call <math display="inline">\frac{V(A)}{1+\sum_{X \in W} P(A,X)}</math> ''A''<nowiki/>'s score and <math display="inline">\frac{1+\sum_{X \in W}P(A,X)}{V(A)}</math> ''A''<nowiki/>'s inverse score. If we keep electing candidates whose inverse score is at least as low as inverse Hare quota then we can expect an outcome where each winner's voter efficiency is below a certain limit aka a proportional outcome.
+* <math display="inline">C</math> ... the set of all candidates
-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 fail-safe approvals which we previously didn't consider or elect the candidate with the lowest inverse score and hope it is low enough.
+* <math display="inline">W</math> ... the set of already elected candidates
+* <math display="inline">V(A)</math> ... number of voters who approve of A
+* <math display="inline">V(A \and B)</math> ... number of voters who approve of both A and B
-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 [[Summability criterion|precinct-summable]] and passes [[universally liked candidate criterion]].
+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.
+== Example ==
+<blockquote>
+AB
+B
+C
+</blockquote>
+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>.
+<math display="inline">\frac{V(A \and B)}{V(B)} = \frac{29}{30}</math>
+<math display="inline">\frac{V(A \and C)}{V(C)} = \frac{0}{14}</math>
+<math display="inline">\frac{V(B \and C)}{V(B)} = \frac{0}{30}</math>
+First seat:
+A: <math display="inline">V(A)/1 = 29/1 = 29</math>
+B: <math display="inline">V(B)/1 = 30/1 = 30</math>
+C: <math display="inline">V(C)/1 = 14/1 = 14</math>
+B is elected
+Second seat:
+A: <math display="inline">V(A)/(1 + \frac{V(A \and B)}{V(B)}) = 29/(1 + \frac{29}{30}) = 14,745762711864</math>
+C: <math display="inline">V(C)/(1 + \frac{V(C \and B)}{V(B)}) = 14/(1 + \frac{0}{30}) = 14</math>
+A is elected
 == Score ballots ==
+[[File:Probabilistic Approval Voting with score ballots.jpg|thumb|402x402px|One possible procedure to elect a candidate using 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.
-[[Category:Proportional voting methods]]
+[[Category:Cardinal PR methods]]
-[[Category:Approval methods]]
+[[Category:Probability]]
-[[Category:Cardinal voting methods]]