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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. ItsProbabilistic winnerscalculations are foundused byas thea usetool to predict potential power balance of probabilisticelected calculationspolitical 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▼
 
ALet's reasonablesay choicethat fora candidate P(A,B) wouldproposes a decision. The probability that a candidate supports the decision beequals <math display="inline">\frac{V(A \and B)}{V(A \or B)}</math>. whichExpected wouldnumber giveof votes that the decision gets equals <math display="inline">\sum_{B \in W}\frac{V(A \and B)}{1+V(B)}</math> or <math display="inline">\frac{\sum_{XB \in W} \frac{V(A \and XB)}{V(A \or XB)}}{V(A)}</math> asper thevoter. formulaTo forminimize the scorenumber of candidatevotes A.per In each stepvoter we electmaximize the<math>\frac{V(A)}{1+\sum_{X candidate\in withW} the highest score. This version of the system is 2-level [[Summability criterion|precinct-summable]]\frac{V(A \and passesX)}{V(X)}}</math> thefor [[universallyeach likednewly candidateelected criterion]]candidate.
 
== Example ==
▲<math display="inline">C</math> ... the set of all candidates
<blockquote>
 
29 AB
▲<math display="inline">W</math> ... the set of already elected candidates
 
1 B
▲<math display="inline">V(A)</math> ... number of voters who approve A
 
14 C
▲<math display="inline">V(A \and B)</math> ... number of voters who approve both A and B
 
</blockquote>
<math display="inline">V(A \or B)</math> ... number of voters who approve A or B or both
 
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:ProportionalCardinal votingPR methods]]
[[Category:Approval methodsProbability]]
[[Category:Cardinal voting methods]]
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