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 apportioned 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 apportioned score voting to assign candidates a weight in the legislature equal to the number of seats they would win divided by the total number of seats, when there are an infinite amount of seats, the only score on your ballot would effect any candidate’s weight in the legislature 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 definitely 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, to simplify ballots and counting, 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.


Maybe I missed something here in this discussion, but this doesn’t seem to make sense to me at all, outside the possibility of being worthy of watching with a bag of Popcorn just for entertainment purposes.

This kind of power brokering is what makes our current system so broken and I guess the fact that it’s been going on practically forever makes it seem like something we should be discussing, but whatever happened to the voters having this power?

I think we need to LIMIT or DENY our politicians this power with our voting systems - not build it in.

My understanding of the STAR system is that it basically functions to do exactly the same thing as this “asset” system, except that the power is in the hands of the voter and the VOTER gets to decide where his/her second vote goes.

That seems much more appropriate to me.


…and again, I say that if they have to actively deviate from an honest vote in order for the voting method to honor their Exclusive Top Preference, then the method itself is not meaningfully proportional.

  1. The question is not whether they can vote for the compromise candidates, the question is whether they do vote for the compromise candidates.
  2. The scenario I’m talking about under RRV is when the compromise candidate gets nothing because the voters who indicated them an exclusive top preference are fewer than those who cast ballots with the polarizing candidates as the exclusive top preference.

So, does that mean you’re claiming so long as it is possible to do it, it should be meaningfully considered as meeting that criteria?

By that logic, would it be considered “free and open elections” if all voters have to vote in person, having arrived at the polling station no earlier than 3:58 am and no later than 3:59 am, and pass a literacy test before they are given a ballot, which must be turned in no later than 4:06 am? People have the ability to do that, so that means it satisfies the condition, right?

Please allow me to put some constraints on that, using S for Small Party, and A for Aligned Party:

  1. They must include honest ballots, so no significant amounts of bullet voting, and any bullet voting must be roughly uniform across voting blocs.
  2. The small party candidate must be the exclusive top score of at least one Hare Quota (to make it clear that they deserve at least one seat).
  3. The number of voters with A as their exclusive top score must be significantly larger than the number of S voters, larger than the ratio of the scores that S voters gave S vs A (ie, if S voters cast S5 A1 ballots, then A voters must outnumber S at least 5:1)

I maintain that with those constraints, S will not win any seats (despite having a full Hare Quota) unless they bullet vote (hence constraint #1), or happen to win the first seat.

Fine. Prove that RRV/PAV is proportional without bullet voting given the constraints above, because that was the scenario I was describing, I am quite confident that the only way those voters get their actual preference rather than being “represented” by a candidate they think of “not literally Hitler” is through bullet voting.

What’s Appointed Score Voting? Because if that’s just another misspelling of Apportioned Score, your assertion that the only thing that matters is “the highest score you gave” is wrong, something you’ve actively complained about.


If such delegation is publicly announced prior to voting, and is binding, then it wouldn’t be horrible, but that would still have two problems, to my thinking:

First, it basically turns Asset Voting into a bizarre form of RCV, being functionally equivalent except for where the runoff information comes from (being the Candidate preferences, rather than Voter preferences). The theoretical benefit of Asset Voting is that there would be negotiations between candidates after the election.

Additionally, the publicly stated candidate preferences undermines the “determine the will of the people” aspect of voting.

Imagine that Candidate A says their support would transfer to B>D>C, while Candidate B says they would transfer their support to A>C>D, and the A/B party is polling low enough that it’s possible that one of them might win a seat, but it’s possible that neither of them would be seated.
Imagine, also, that a significant percentage of the A/B bloc the preferences A9, B7, C6, D1. I assert that a significant number would rally behind B, because the 2 point loss in utility if B gets seated rather than A would be seen as more acceptable than the 5 point utility loss if neither A nor B gets seated, and their votes transfer to D rather than C.

No, to my thinking, the reason that Asset Voting isn’t just a less democratic version of RCV is that there would be negotiations between the candidates after the votes were all tallied, and having the asset transfer declared prior to the vote would either eliminate that difference (if binding), or be meaningless (if non-binding)


Take the following 5-district and force a Hare quota for all but the final seat.

Liberals ABCDE get 10, 10, 9, 9, 9. (47 total)
Conservatives VWXYZ get 10, 10, 10, 10, 9. (49 total)
Centrist M gets 4.
0/5 seats filled.

If there was pre-election delegation or STV, then the centrist drops out and distributes the votes some way or another. If, on the other hand, we use pure Asset or Asset-STV hybrid, the liberals and conservatives each first claim two seats and then we get this:


Now M can just hold on to the 4% and eventually C must crack and give M the 7%, clinching M’s victory. (C cannot afford to hold on because that might be repulsive to M and X would just win.)

Of course, it may seem like C is a bad centrist for only getting 4% of the vote, but perhaps this was Vote One or Vote As Many As Number Of Winner Slots and most people thought the votes may transfer to the centrist anyway.


Like I said earlier, 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. SNTV is clearly not proportional, but even it passes this bare minimum requirement. This property is supposed to be a proportionality blacklist, not a proportionality whitelist. The reason why STAR-PR cannot be considered proportional, no matter what lens of proportionality you look at it with, is because this property blacklists it from the proportionality club.

The property in which a quota of voters have the ability to force the election of one of their own is just a watered down version of a PR property that all voting methods (from PAV to Elbert’s to STV to Schulze STV) that are considered to be proportional pass: Which is that when a group of voters gives max support to a set of candidates and does not contribute to the election of any other candidates, they are guaranteed to award to that set at least the number of seats that is proportional to the portion of the electorate that that group makes up (±1 seat).

As for the version of RRV and PAV where candidates can win multiple seats (as well as probably a few other methods), they pass an even more restrictive proportionality property: a group of voters with identical ballots that gives at least one candidate a max score will guarantee that the sum of scores one of them gave to the elected candidates will always be at least (max score)(total seats * portion of the electorate in that group - 1).

Because this version of RRV has this property, when voters round down, not up, when converting their honest utilities to ratings*, they will be satisfied with the election results by atleast the amount that proportionality constraints say they should be satisfied by the election results*.

*when their satisfaction of the election results is approximately equivalent to their total satisfaction among all of the elected candidates times how many seats they won divided by the total number they won

That would be a reasonable constant among ranked voting methods, but among rated voting methods, it is stupid. Consider this small 3 winner district that is a part of a much larger legislative body:

33% of the voters rate A 100 stars, B1 99 stars, B2 99 stars, B3 99 stars, C, 0 stars, and everyone else 0 stars
33% of the voters rate A 0 stars, B1 100 stars, B2 99 stars, B3 99 stars, C, 0 stars, and everyone else 0 stars
33% of the voters rate A 0 stars, B1 99 stars, B2 99 stars, B3 99 stars, C, 100 stars, and everyone else 0 stars

Your proportionality constraint says that [A, B1, C] should win. But that would be stupid because most voters who would give A 100 stars, B 99 stars, and C 0 stars, would certainly prefer two B’s over an A and a C. Footnote. The same should be true of the C voters, which would mean that all of the voters would of rather have [B1, B2, B3] over [A, B1, C] which means that your constant would be preventing the unanimously preferred election result from being selected by any voting method that calls itself proportional.

It makes sense that a definition of proportionality may “prefer”* outcome Y over outcome X even when more voters prefer outcome X over outcome Y, because proportionality is often in contrast with the preferences of the majority. However, a good definition of proportionality should not “prefer”* an election outcome Y over election outcome X when voters unanimously prefer an election outcome X over election outcome Y. That is why a good definition of proportionality needs to be able to take into consideration the intensity of a voter’s preference of candidate X over candidate Y.

*By a property “preferring” outcome X to outcome Y, I mean that that if outcome Y has that property, outcome X should also have that property, and if outcome X does not have that property, outcome Y shouldn’t have it either.

Footnote: This would be especially true in a multi-winner district that is a part of a much larger legislative body because in such a scenario, a voter’s total satisfaction among all of the elected candidates times how many seats they won divided by the total number they won of a voter’s utility is a much better approximation of their actual satisfaction with the election results.


…if and only if they do not also provide another, significantly more popular candidate, a score.

Dude. What the fuck. I haven’t been calling your opinions stupid, so where the hell do you get off rejecting my ideas thus?

And what part is stupid in specific?

Is it constraint #1, under which a voting method would only be considered proportional if it doesn’t require strategy to achieve proportionality?

Or is it constraint #2, whereby we only consider scenarios where the smaller faction is large enough to win a seat if they vote strategically?

Perhaps constraint #3, which is the scenario in which I have found RRV/PAV to be majoritarian rather than proportional?

Dismissed as testing for success, rather than testing for failure. I’m not arguing that it can’t get good results, I’m arguing that it can get bad results.

Further, your scenario is bad, and I’ll show you why:

  • 33% A100, B99, B’99, B’'99, C0
  • 33% A0, B100, B’99, B’'99, C0
  • 33% A0, B99 0, B’99 0, B’'99 0, C100

Now, the C bloc, through dishonest voting, can force a result of {B,C,B}, which they would prefer to B,B,B.

This is what I mean about you testing for success. Sure, I’ll concede that it kind of makes sense to elect {B,B,B} in your scenario that you specifically contrived to achieve that goal… But why on earth would they compromise when they know that they are large enough that there is zero downside to Hylland Freeriding?

Further, your entire scenario violates Constraint #1: No significant amount of Bullet Voting; your entire B voting bloc effectively bullet voted, with your results being exactly the same if it were run as Party List with the B bloc voting 99 for all of Party B.


@fsargent Add this to the new multi-winner election theory category?


This is my test of a multi-winner RCV demo–it supports both Droop and Hare quotas: Multi-winner RCV .


Why are you advocating for RCV? It is a really difficult system to count because it cannot be totaled in precincts and then summed centrally without dealing with bribery issues. The problem with any multi-round system is near-ties in an early round clogging the entire process.

(Yes, the website has a very pro-RCV bias to it. I actually checked.)


All of this very complicated deliberation could be totally swept away just by using the tranche-proportional method. It involves absolutely no concerns whatsoever for minorities, “moderates,” “left,” “right,” or political parties. It can be utilized with any kind of score voting, approval voting, single-selection voting, and probably many others. I described it at:

Add-On Multi-Winner Proportional Representation

You start by deciding the number of seats you need to fill:
T = The Total Number Of Tranches (i.e. (one less than) the number of seats to be filled).

The election is held and the votes are counted, and then every one will know the "strongest winner’s total (of votes), which we may call “W”.

Then any high school freshman can calculate the “bases of the tranches” (values of “B” corresponding to tranche numbers “N”) by simply plugging in “W”:

=/ B = W * ( 1 - [(N /( T + 1 )]^2 ) /=

Tranche #1 is the highest (just below the strongest winners position); tranche #2 is just below tranche #1, and so on.

This provides a party/interest-group blind method in which minorities of any sort can vote for their champions and have a reasonable chance of obtaining some degree of representation.


Are there any IRV elections which have failed due to ties? WDS seemed to suggest that it was happening, but swept under the rug.


I just added the link to this website (which I am very new to) when I found out about a serious flaw in RCV. So far, no work as to the problems in multi-winner RCV, just put it out there.