Suppose a nation is electing k individuals, to populate a senate or something. Now approval voting has some good properties, so suppose the protocol is to let each of the m citizens vote for as many candidates as they want. We then award a seat to the candidate with the most votes, and deduct m/k from their vote tally, or thereabouts. And then we repeat the process using the new tallies, until we have k elected candidates. It seems like a problem with this approach is that if the most popular candidate could copy themselves k-1 times, then their copies would all have the same score as themselves, and so they would completely monopolise the senate. In a very real, practical sense, this protocol would mean that one political party could dominate the senate. This would undercut the ability of the senate to combine disparate views, and to represent the diversity of opinions of the population, as well as reducing its robustness to takeover by one group.
My questions are: 1) is this how approval voting is proposed to be used for multi-individual body, and 2) is there any suitable proposal that would retain desirable properties of approval voting (simple interface, and attaining high utility) while avoiding this problem?