Multiwinner Approval Voting

We extend approval voting so as to elect multiple candidates, who may be either individuals or members of a political party, in rough proportion to their approval in the electorate. We analyze two divisor methods of apportionment, first proposed by Jefferson and Webster, that iteratively depreciate the approval votes of voters who have one or more of their approved candidates already elected. We compare the usual sequential version of these methods with a nonsequential version, which is computationally complex but feasible for many elections. Whereas Webster apportionments tend to be more representative of the electorate than those of Jefferson, the latter, whose equally spaced vote thresholds for winning seats duplicate those of cumulative voting in 2-party elections, is even-handed or balanced.