A possible proportionality criterion for approval voting

If there are c elected candidates and v voters in total, then if you pick any n voters, then you should be able to assign to each of these n voters c/v candidates (rounded down) with no overlap, as long as each voter has approved enough candidates to make this possible.

First of all, in real life elections, c/v will generally be less than 1, so rounding down gives 0, so it might seem like a pointless criterion. However, the criterion is about how the method would behave in these situations and we can devise any scenario we want, and also it does become relevant when we have a factions of voters that effectively vote as one.

But it’s basically saying that a certain proportion of voters should have that same proportion of candidates, evenly split between them. If you take a random sample of half the voters, then you should be able to find a way of splitting half the candidates equally among them.

This proportionality criterion does not make any reference to parties or factions, which proportionality criteria often do, but since voters might not vote in factions, these criteria are insufficient. This criterion or something similar/equivalent might have come up somewhere before, but I’m not aware of it.

Thiele PAV is known to fail this as seen here and here are examples showing failures for Thiele, Phragmen and Ebert.

Edit - I suppose it could be simpler to just say that if there are equal numbers of voters and candidates, each voter should be able to be assigned their own unique candidate that they approved.