The two main proposals are sequential proportional approval voting (SPAV) and proportional approval voting (PAV).
SPAV proceeds in rounds. In the first round, the candidate with the most votes wins. In the second round, the ballots are reweighted such that those which have the first winner selected have 1⁄2 weight and all others retain full weight. This is repeated until each seat is filled, and, in each round, a ballot that has voted for n candidates who have already been elected is weighted at 1/(n + 1).
For an election to fill N seats, PAV looks at each possible set of N candidates and elects the set which maximizes the utility function given by 1*(# of ballots with at least one of the candidates selected) + 1/2*(# of ballots with at least two of the candidates selected) … + 1/n*(# of ballots with at least n of the candidates selected).
Both PAV and SPAV yield proportional representation while using approval-style ballots. PAV yields better results, but SPAV is easier to explain the outcomes of. Another option is to use the allocated score algorithm on approval ballots.
Thanks. These algorithms seem like they would be better for passing the independence of clone alternatives criterion.