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.

Taking this as the criterion, it can be expanded upon a bit. If the number of voters in each faction can be “cancelled down”, then the proportionality criterion must apply on the cancelled down ballots. (A faction is a group of voters that all vote in exactly the same way.)

For example, if there are 100 voters, then the criterion as defined above really only kicks in if there are 100 candidates. But we might have:

Faction A: 50 voters
Faction B: 25 voters
Faction C: 25 voters

In this case, the proportionality criterion must work as if it was:

Faction A: 2 voters
Faction B: 1 voter
Faction C: 1 voter

Also, where exact cancelling isn’t possible, adding voters to factions must act in a way consistent with the participation criterion. For example:

Faction A: 8 voters
Faction B: 4 voters

If there are three seats, it must be possible to assign two candidates to faction A and one to faction B. However, if you added a voter to faction B, you’d have:

Faction A: 8 voters
Faction B: 5 voters

An exact cancelling isn’t possible, but you wouldn’t then say the criterion doesn’t apply unless there are at least 13 voters. Faction B must still get a candidate.

Also, the candidates must be elected in such a way that is consistent with this proportionality being maintained if more candidates are then added and the previous candidates remain elected.

Of course, voters won’t necessarily vote in such a neat way so these factions might not exist at all anyway. But another part we can add to the criterion is that (assuming the voters have approved enough candidates) if k candidates are elected, then k voters must be able to have a candidate that they approved uniquely assigned to them. If k is greater than the number of voters (v), then from v+1 onwards, voters would start getting a second unique candidate each.

Obviously this would still have gaps (some of which may be fillable), but I see this as necessary but not sufficient to proportionality. Also, unless a method is contrived to only pass in the specific cases defined, then it’s probably going to be OK from this point of view.

Another one is that a voter is considered to be in a faction with all other voters that have voted for some of the same candidates as them and none that they haven’t, but not vice versa. So:

Voter 1: ABC
Voter 2: A

Voter 1 is not “punished” for the extra approvals of B and C, and so is in a faction of two. However, Voter 2 is not considered to be in a faction with Voter 1 for these purposes, so could lose out. For example:

100 voters: ABC
100 voters: ABD
1 voter: C
1 voter: D

If there are two candidates to elect, then it is fine to elect A and B. The single C and single D voters are not considered to be in the larger factions that partially agree with them. If A, B, C and D were parties, there would be no requirement to elect a C or a D candidate until seats 201 and 202.

Anyway, this is looking less and less like a criterion, and more like lots of criteria stuck together but it might be possible to make this all a bit neater.