User:Matijaskala/Probabilistic Approval Voting: Difference between revisions

▲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.

▲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 failsafe 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 precint-summable and passes universal liking condition.

▲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.