Multi Member Voting Systems


Please allow me to correct my errors with the form of argumentation I should have presented from the beginning:

Which of the following voters contributes the most to A being seated?

  1. A5, B5, C5, D5
  2. A4, B3, C3, D3
  3. A3, B0, C0, D0

Voter #1 clearly contributes the most to A getting a high grade

…but they do the same for B, C, and D, too. As such, I assert that Voter #1 does not actually contribute to A winning in any meaningful way (outside of a “minimum score to be seated” scenario). As such, it is down to Voter #2 or Voter #3.

To ease the decision, let us add in a 4th voter:

  1. A3, B4, C5, D4

Any subset of those voters including Voter #3 results in a guaranteed victory for A. In order for A to win without Voter #3, however, you need both to include Voter #2 and to exclude Voter #4.

So what happens if we run a 2 Seat party list election using “Highest Score for Winner” as our metric for “Contributes most”?

Seat 1: A:3.75 > C:3.25, remove Voter 1 (5) and 2 (4)
Seat 2: A:3 > C:2.5

Compare that to what we get with “Difference from Average”:

Seat 1: A:3.75 > C:3.25. Remove Voter 3 (3-0.75 = 2.25) and 2 (4 - 3.25 = 0.75), the totals for voters 1 and 4 being 0 and -1, respectively.
Seat 2: C:5 > B=D:4.5

While it feels right to declare that the ballots with the highest score for a given candidate contribute most to them winning, because (again, outside of victory threshold scenarios) the only real question is the relative contributions the ballot made to the various candidates.

If you have a superior answer as to how to determine that question, I would be very glad to hear it, but the gut reaction that everyone (including myself, mind!) seems to have of “highest score obviously contributes most” seems pretty clearly wrong when you start approaching it critically.

After all, isn’t that the reason that people assume that Min/Max, Approval Style Score voting is “obviously” the most effective method? Because it maximizes the relative contribution between candidates you like and dislike?


Well, at the very least, there would need to be some sort of tiered membership. It’s not feasible to give everyone who votes for themselves an office, staff, a salary, or even a physical presence in chamber debates. (Another issue would be deciding who gets access to classified documents.) So there will have to be a minimum level of support for someone to become a full member. But if membership is tiered, there might be pressure on voters to support likely winners, as someone with access to the resources of the legislature will probably be able to provide more effective representation than someone who missed the threshold. That would be bad because it would entrench the high-tier incumbents.


It is one thing for a voting method to fall in a proportionality grey area and it is another for it to just break spontaneously (and by breaking, I mean a voting method that is marketed as proportional resulting in an outcome that is not proportional by every conceivable measure). In any truely proportional voting method, a quota of the electorate should be able to force the election of one of their candidates regardless of how anybody else votes. PAV, RRV, Harmonic, PSI, Least Squares, Elbert’s, Monroe’s, STV, Schulze STV, CPO-STV, etc. (and even some non-proportional voting methods such as SNTV and Cumulative) all have this property. Not checking for this property prior to popularizing STAR-PR is the mistake the equal vote coalition made when they released this. Now many STAR voting supporters are advocating for a non-proportional voting method. I don’t want you to make the same mistake the equal vote coalition made which is why I am asking if you can prove that your voting method can not break in the way I described.

I have a lot of homework and a big test that is worth 20% of my grade, so I was hopping that you could run code that checks for monotonicity, proportionality, and resolvability in the way that I described. It is your method, so the burden of proof is kind of on you to show that it is actually proportional, monotonic, and resolvable. But I suppose I could run those checks in Java this weekend when I have more time on my hands.


You cite RRV as having this property, except as I showed in that thread (or one of them), smaller party voters (a fraction of the size of the major ones, but with a full Hare quota) can only get proportional results if they force them (via bullet voting). Once that started, I expect would degenerate into SNTV with Score Ballots.

…but the other thing I showed was that Apportioned Score doesn’t require anyone vote strategically to get reasonably proportional results.

…isn’t that just a STAR version of RRV?

Oh, goodness… because they reweight the entire ballot, that still means that your 5/(1+X) is going to be greater than your 4/(1+X), and therefore the entire weight of your ballot is still considered for your preference in the Runoff Step… and since RRV already trends majoritarian in Party List scenarios, and the Runoff Step is inherently majoritarian… I bet that completely kills proportionality for anything other than maybe groups/parties of comparable size, huh?

And thank you for it. This is why I originally put this to the google group: I don’t trust myself to make a perfect method, and want as many people as possible to try breaking it.

No, no, that’s fine. I’m just trying to figure it out, because I can’t imagine how it would come about, and can’t test things I can’t fathom.

That said, given that the “Find winner, check to confirm, find new winner” is analogous to the
“Find centroid, check to confirm that the elements that ‘belong’ to that centroid have that as their centroid, recalculate with the new centroid” procedure of K Means Clustering (which does resolve, even in multidimensional spaces), I’m pretty confident that it will, but I need to get around to writing such tests.


I said that a quota of the electorate should be able to force the election of one of their candidates regardless of how anybody else votes. They can instead use their voting power to elect compromise candidates as well. Just because a quota of voters may not always chose to use their voting power to elect one of their candidates doesn’t mean that their right to do so should be stripped away from them. When a potentially proportional voting method crashes in full blue screen fashion is when that voting method strips the ability for a quota of voters to get a proportional result and I don’t believe any voting method that crashes in that way is proportional. What I am going to test this weekend is if appointed score voting gets a proportionality blue screen where it is impossible for a quota of voters to elect a candidate. Passing that blue screen test is the bare minimum a voting method needs to go through to be proportional which is why it is an impotent property to check for among potentially proportional voting methods.

Not necessarily, since while RRV and PAV fail the favorite betrayal criterion*, they only fail it because of free riding so if a voter’s favorite candidate isn’t popular enough to get elected, unlike with SNTV, there wouldn’t be a good reason to not vote for their favorite when their favorite is less popular. The problem with proportional voting methods is that as you increase the number of seats, the number of max scores it strategically makes sense to give decreases (i.e. in a 1 winner race, you might want to approve ≈1/2 of the candidates, but in a 2 winner race, ≈1/3, and in a 3 winner race, ≈1/4). The reason why this happens is because they are more proportional and in a purely proportional election (with fractional vote weights) there is no reason to vote for more then a single candidate. The way some PR methods get around this is by just not using as much information (ex: if you used appointed score voting to distribute a million seats among 3 candidates, the only score on your ballot would effectively matter would be the highest score you gave which means as the number of seats gets large, the voting method is just bullet-a-fying your vote so you don’t have to). There are definitly pros and cons to having the voting algorithm do your strategic voting for you (ex: bullet-a-fying your vote for you when there are a million seats), but one of the cons is that there are impossibility theorems about the amount of strategic voting the voting the voting method can handel for the voter so the more strain you put on the voting method to make voter’s votes have more strategic weight, the more opportunities you open up for voters to vote strategically in other ways i.e. more favorite betrayal and the loss of other nice properties. Making voting methods is like playing wack-a-mole. Each time you hit a mole representing a form of strategic voting, another mole pops up someplace else. I believe that the most honest forms of strategic voting are the forms that still result in a semi honest ballot (where the voter just has a threshold between their favorite and least favorite candidate) even if that threshold leans closer to their favorite candidate as you increase the number of seats because of it’s proportionality. I am still working on creating such a voting method, but am a long ways away from anything concrete.

*under the assumption that voters have monotonic preferences, i.e. If they prefer A to B, they will always prefer [X, Y, A] to [X, Y, B], and [X, A, A] to [X, A, B], and [X, A, B] to [X, B, B], and [A, A, A] to [A, A, B]. I know that this assumption may not always be true for every voter, but it’s a good approximation of voter preferences and it is impossible to have any form of FBC among this class of voting methods without this assumption.


“It’s not feasible to give everyone who votes for themselves an office, staff, a salary, or even a physical presence in chamber debates.”

Fine. Offices, staves, salaries, and physical presences could be limited to the 100 with the most count of proxies. But the others still get to listen to the debates and vote their proxies on legislation. Drawbacks?


I like the idea of losers delegating to winners, but it should happen publicly, before the election, not after.


As I noted earlier, the legislators who win a physical presence might be able to provide more effective representation than those who can only participate remotely. Voters who care whether their rep is physically present should be able to rank alternate reps in case their first choice doesn’t get in. The ballots should have 1) A ranking of possible reps and 2) A default option for the rep they want the most if none of their choices get in. So if a voter’s preference of reps was:
A with a physical presence > B with > A without > Any other configuration,
then they would vote a ranking of A>B; Default: A.