User:Matijaskala/Probabilistic Approval Voting: Difference between revisions

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 theythe belongsecond one belongs to the samefirst one's 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.

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 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}{4314}</math>

<math display="inline">\frac{V(B \and C)}{V(B \or C)} = \frac{0}{4430}</math>

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>