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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 ==
Let's say that there is a function P: C×C → [0,1] 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 the expected average load of A's voters 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 expected average load of A's voters after A'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's score and <math display="inline">\frac{1+\sum_{X \in W}P(A,X)}{V(A)}</math> A'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 voters' expected average load is below a certain limit aka a proportional outcome.
Given:
 
* <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 for theminimize scorenumber of candidatevotes A.per In each stepvoter we electmaximize the<math>\frac{V(A)}{1+\sum_{X candidate\in withW} the\frac{V(A highest\and score.X)}{V(X)}}</math> Thisfor versioneach ofnewly theelected system is 2-level [[Summability criterion|precinct-summable]] and passes universal liking conditioncandidate.
 
== Example ==
<blockquote>
 
29 AB
 
1 B
 
14 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:Cardinal PR methods]]
[[Category:Probability]]
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