User:Matijaskala/Probabilistic Approval Voting: Difference between revisions

'''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.

== 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

* <math display="inline">V(A \or B)</math> ... number of voters who approve of A or B or both

Let's say that there is a function <math>P:C \times C \to [0, 1]</math> that maps two given candidates to the probability that they 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> (P(A,A) equals 1). 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.

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>

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(A \or X)}}</math>.

<math display="inline">\frac{V(A \and B)}{V(A \or B)} = \frac{29}{30}</math>

<math display="inline">\frac{V(A \and C)}{V(A \or C)} = \frac{0}{43}</math>

<math display="inline">\frac{V(B \and C)}{V(B \or C)} = \frac{0}{44}</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(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}{44}) = 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]]