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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. 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:
* <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
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>
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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