Some things that came out of the paper:
If A, B, C and D are parties fielding multiple candidates, then with the following ballots, what proportion of the seats should each party win?
COWPEA would say that A and B would each win just under 1/4, and C and D would each win just over 1/4.
However, Optimised PAV would elect C and D with half the weight each, and A and B would not get any weight. The multi-winner Pareto efficiency criterion assumes that a voter's satisfaction with a result can be measured purely by the number of elected candidates they have approved. And going by that, any result containing any A or B candidates is Pareto dominated by CD.
This can be seen as a two-dimensional voting space with an AB axis and a CD axis. PAV would ignore the AB axis. Arguably though COWPEA makes better use of the voting space.
That was an example where any proportions are allowed, but here is another where 2 candidates are to be elected.
Basically any deterministic would elect either AB or CD. CD Pareto dominates AB in this multi-winner sense as all 502 voters have an approved candidate under CD, compared with 500 under AB (no-one has two approved candidates). However, AB is more proportional. 250 have approved A and 250 have approved B. 291 have approved C and 211 D. So under the CD result, the 211 D voters would wield a disproportionate amount of power. So what do you think? Does that matter or is it all about number of approved candidates?
If you go for AB, then you are rejecting the multi-winner Pareto efficiency criterion, but also consistency as a by-product. This is because you can have a set of ballots where the C and D approvals are just swapped round. So it's 211 for C and 291 for D but otherwise the same. Then combining the two sets of ballots together you get:
If there are 2 to elect here, it must be CD. So if you went for AB in the other examples, then you are rejecting the consistency criterion in multi-winner elections.