User:Matijaskala/Probabilistic Approval Voting: Difference between revisions

Let's say that there is a function <math>P:C C×C\times →C \to [0, 1] </math>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''<nowiki/>'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''<nowiki/>'s voters after ''A''<nowiki/>'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''<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 voters' expected average load is below a certain limit aka a proportional outcome.