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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 ==
* <math display="inline">C</math> ... the set of all candidates
* <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
* <math display="inline">V(A \or B)</math> ... number of voters who approve A or B or both
Let's say that there is a functioncandidate A <math>P:Cproposes \timesa Cdecision. \to [0, 1]</math>which returns theThe probability that two given candidates belong to the same political faction. Then for a given candidate ''A'',supports the expected number of elected candidates belonging to the same factiondecision equals <math display="inline">\sum_frac{XV(A \inand WB)} P{V(A,XB)}</math>. andExpected ''A''<nowiki/>'snumber voterof efficiencyvotes that the decision gets equals <math display="inline">\frac{\sum_{XB \in W} P\frac{V(A,X \and B)}{V(AB)}</math>. If ''A'' is not already elected then we can calculate the ''A''<nowiki/>'s voter efficiency after ''A''<nowiki/>'s election asor <math display="inline">\frac{1+\sum_{XB \in W} P\frac{V(A,X \and B)}{V(B)}}{V(A)}</math> per voter. Let'sTo callminimize number of votes per voter we maximize <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_{XV(A \inand W}P(A,X)}{V(AX)}}</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 wherefor each winner'snewly voterelected efficiency is below a certain limit aka a proportional outcomecandidate.
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.
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 the [[universally liked candidate criterion]].
In [[party list case]] P(A,B) equals 1 if A and B belong to the same party and 0 otherwise. Because of that the system decays into [[D'Hondt]].
== Example ==
</blockquote>
In each step we elect the candidate with the highest <math display="inline">\frac{V(A)}{1+\sum_{X \andin B)W} \frac{V(A \orand BX)} = \frac{29V(X)}{30}</math>.
<math display="inline">\frac{V(A \and B)}{V(B)} = \frac{29}{30}</math>
<math display="inline">\frac{V(A \and C)}{V(A \or C)} = \frac{0}{4314}</math>
<math display="inline">\frac{V(B \and C)}{V(B \or C)} = \frac{0}{4430}</math>
First seat:
Second seat:
A: <math display="inline">V(A)/(1 + \frac{V(A \and B)}{V(A \or B)}) = 29/(1 + \frac{29}{30}) = 14,745762711864</math>
C: <math display="inline">V(C)/(1 + \frac{V(C \and B)}{V(C \or B)}) = 14/(1 + \frac{0}{4430}) = 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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