Multi Member Voting Systems


Hi everyone,

I’ve been wondering what the best, simplest voting method might be for proportional representation for an organization like the US House. Let’s say you’re a state, you have the number of seats you can send to the house based on your population – how do you divide them up evenly?

I’d heard good things about Proportional Approval Voting, but never really got a chance to dig into it properly. Is this the right thing?

What would you recommend?


What about Asset Voting?


@NoIRV can you link to a description of that?


It involves politicians negotiating with each other for power, which could be pretty entertaining.


Well, that’s mindblowing. That’s a super compelling system. I would absolutely watch that on c-span, popcorn in hand. This is one of the better-written articles from Warren too. I wish it were easier to discover things on the site.
There are conversations on how we can better organize election system knowledge to lower the barrier to entry for the subject. Warren has said that everything’s basically open source. It should be easier to categorize and identify the details of voting science.
I swear, 40 years from now someone’s going to win a Nobel, but only after we get these methods inacted in policy.


Yeah, Asset Voting is what I was going to suggest as well.

Warren made an interesting “improved MMP” that uses Score Voting and Asset Voting instead of Plurality Voting and party lists.

Love it!


Well, I seriously think there’s a 10-40% chance that STAR and Approval Voting both pass this fall in their respective cities. Odds that at least one passes could be 75% or more, given the political climate we’re in.


Choice of Representation. The citizen chooses his representative, and she has his proxy for voting on legislation.


The election will happen in 6 days. I really hope that we get real world experience out of both systems.


(S)PAV (Thiele’s Method, by any other name) is a good first step, but it fails Consistency (see 15.4 in Svante Janson’s analysis).

Further, as I demonstrated back in the google group, with groups of sufficiently different size, majority groups tend to obscure the existence of smaller aligned-but-distinct groups, which at best will lead to Hylland Freeriding (ie, not voting for someone you like so as to ensure someone else gets a seat) and examples of bullet voting, or at worst, major party domination to the exclusion of smaller parties that make up more than a Hare Quota of voters.

I’m partial to my own Apportioned Score Voting, which has Approval, STAR, 3-2-1, etc, variants. Basic idea, as you can see from the images, is that it takes the good stuff about STV, and removes the broken parts (IRV, ranked ballots), replacing them with better versions (cardinal method of choice, cardinal ballots).

I am less keen on Asset Voting, because anything where politicians decide who among them gets seated makes me nervous; I mistrust career politicians enough already, and to give them power to broker among themselves?


From @Ciaran’s link:

In the case of Score/Range voting, I recommend “Descending order, by score on ballot for W minus average score on that ballot.”

That is why apportioned score voting is non-monotonic and not clone-proof. I strongly recommend just descending order by score on ballot for W, full stop. That way, you still pass the monotonicity criterion, the clone-proof criterion and the independence of irrelevant alternatives criterion. There is no other definition of “strongest support to candidate W” that apportioned score voting can use while still passing the independence of irrelevant alternatives criterion. And it just looks nicer because what you get when you calculate the average score a voter gives to each candidate is a meaningless value that can’t be used without introducing a lot of problems.

If you were going to use that archaic meaningless average, I would recommend weighting each candidate’s weight in their contribution to that average based on their “likelihood” of winning or else voters would be able to manipulate their ‘average score given to each candidate’ score by writing in as many candidates as possible because each time they give five stars to a write in candidate, they increase their ‘average score given to each candidate’ score by the same amount that giving five stars to the other front runner would even though giving five stars to the other front runner would effectively cancel out the weight of their ballot. However, the problem of defining a good scalar measure of how close each candidate is to winning (or if winning, how far they are from losing) such that all of the scalars add up to 100% introduces a whole new can of worms. And even if you were able to figure out how to go about doing that, your method would still fail IIA, so I strongly recommend just calculating each voter’s contribution to a candidate independently of their contribution to any other candidate as I described above.


Also, another (and arguably much bigger) problem with your version of apportioned range voting, is it’s not even a proportional voting method.

Scenario A: Suppose that 40% of Democrats give the Democratic candidate 5 stars, the two Depublican candidates 0 stars and like most voters they don’t bother to use their available write-ins. The 60% of Republicans like-wise give the two Republican candidates 5 stars, the Democratic candidate 0 stars and don’t bother with the write-ins either. The Republican wins the first round and (50% to 33% depending on which quota you use) of their votes are exhausted, allowing the Democrat to easily win the second seat.

Democrats (40%): Democrat=5stars, Republican1=0stars, Republican2=0stars
Republicans (60%): Democrat=0stars, Republican1=5stars, Republican2=5stars

1st winner: one of the two Republicans

Democrat’s contribution to winner: 0 - “0” = 0
Republican’s contribution to winner: 5 - “0” = 5

After quota allocation:

Democrats (40%): Democrat=5stars, Republican1=0stars, Republican2=0stars
Republicans (10% to 27%): Democrat=0stars, Republican1=5stars, Republican2=5stars

2nd winner: the Democrat

Scenario B: The Republicans max out the number of write-in scores they are allowed to give (suppose each voter is allowed to rate 5 write-in’s) by giving every one of their family members 5 stars. This did lower their “contribution to the winner” score by a stupidly large amount, but it wasn’t enough because in the first round, the Democratic voters “contribution to the winner” scores are still 0 so the republicans are going to need to do a bit better if they want to steal the 2nd seat from the Democrats.

Democrats (40%): Democrat=5stars, Republican1=0stars, Republican2=0stars
Republicans (60%): Democrat=0stars, Republican1=5stars, Republican2=5stars, Mom=5stars, Dad=5stars, Grandma=5stars, Grandpa=5stars, Other Grandma=5stars

1st winner: one of the two Republicans

Democrats contribution to winner: 0 - “0” = 0
Republicans contribution to winner: 5 - “4.125” = 0.875

After quota allocation:

Democrats (40%): Democrat=5stars, Republican1=0stars, Republican2=0stars
Republicans (10% to 27%): Democrat=0stars, Republican1=5stars, Republican2=5stars, Mom=5stars, Dad=5stars, Grandma=5stars, Grandpa=5stars, Other Grandma=5stars

2nd winner: the Democrat

Scenario C: Same as scenario B but the instead of giving the two Republican candidates 5 stars, the Republicans give them 4 stars.

Democrats (40%): Democrat=5stars, Republican1=0stars, Republican2=0stars
Republicans (60%): Democrat=0stars, Republican1=4stars, Republican2=4stars, Mom=5stars, Dad=5stars, Grandma=5stars, Grandpa=5stars, Other Grandma=5stars

1st winner: one of the two Republicans

Democrats contribution to winner: 0 - “0” = 0
Republicans contribution to winner: 4 - “4.125” = -0.125

After quota allocation:

Democrats (0% to 7%): Democrat=5stars, Republican1=0stars, Republican2=0stars
Republicans (60%): Democrat=0stars, Republican1=4stars, Republican2=4stars, Mom=5stars, Dad=5stars, Grandma=5stars, Grandpa=5stars, Other Grandma=5stars

2nd winner: the other Republican

Thus, a violation of proportionality. The Democrats gave the Democrat max support without contributing anything to the first republican’s victory in the first round.

Because the Republicans gave the Republican candidates less support in scenario C then in scenario B, this is also an example of apportioned score voting’s violation of monotonicity.

Using my more simplistic definition of “contributes most” would fix both of these errors, and then some.


Um… Parker? Your math is wrong, and therefore your conclusions don’t follow.


The Democrats’ contribution to the winners is 0 - (5+0+0)/3 == -1.667

In all three scenarios, the Republicans contribution is considered greater than that, even Scenario C, where their contribution was negative (due to a crazy number of write-in slots), the contribution from the Republicans was counted as more than one full point greater than that of the Democrats.

Indeed, even if you allowed for an infinite number of write-ins, the difference from average for the Republican ballots would never drop below the -1.667 that the Democrats gave them, without dropping below the point of beating the Democrat (it would have to be an average below 3.333, at which point the Democrat would win the first seat anyways, because .6* 3.333 == .4*5).

Further, you’re glossing over the Confirmation Step. Consider the flow within the purple box; if by some bizarre arithmetic twist, a larger number of Democrats were selected for the Quota, the confirmation step would find that, no, the Democrat wins that quota, and that first Republican would not be seated.

Under your alternate math version of Scenario C, the “Who wins Quota Q?” section would proceed as follows:

Quota Q:
80% D5, R0, R0
20% D0, R4, R’4
Total: D4, R0.8, R’0.8

D wins Quota Q, so replace R’ with D, and find a new Quota that most strongly supports D (that, in this scenario, would be the same quota as before). Then, having confirmed the seating of D, and removed all of the Democrats and 10% of the Republicans (I prefer Hare Quotas for this), the second seat would go to one of the R’s, and you’ve got a D,R scenario instead of an R,D scenario, maintaining proportionality.

As such, your scenario does not show a violation of Monotonicity, because even if we used alternate math, lowering the scores for both Republicans does not increase the number of Republicans seated. Indeed, by lowering the score of the Republicans, they would go from being the first seat to the second.

…and break others. You cited violation of IIA as a problem, but I actually consider the way it does so a feature in most cases, and it was made thus intentionally. Imagine a scenario with more than two factions.

Say there were two ballots that you had to choose to set aside as being “Satisfied” by A being seated:

  1. A4, B0, C0, D0
  2. A5, B4, C0, D2

Your simplistic definition of “contributes most” would select Ballot 2, as having the higher score for A than Ballot 1 did. That would leave the decision between B, C, and D to a voter that expressed no opinion.

Further, it would be problematic in a Party List (or, if you prefer, Presidential Elector) scenario; by artificially lowering their score for A from 5 to 4, Voter 1 could keep themselves in the voter pool longer, thereby effectively lowering the scores for B, and D by 4 and 2 points, respectively, while only lowering A’s score by 1 point.

Thus, your (and @Jameson-Quinn’s) recommendation to go with “Decreasing Scores for Candidate” rather than “Difference From Ballot Average” would introduce the opportunity for an analog of Hylland Free-riding. It wouldn’t be foregoing a vote for the candidate, but it would be a non-monotonic result if, in aggregate, that changes the result for the second seat from B to A.


33% A4, B0, C0, D0
33% A5, B4, C0, D2
33% A2, B0, C5, D3

Clearly, this should work out to some set {A,B,C}, right?

With my metric:

  • A is seated (average of 3.667), ballot group 1 is removed (4-1 > 5-2.75 > 2-2.5)
  • D is provisionally seated (average of 2.5) by the quota comprising ballot group 3,
    • C replaces D in the Confirmation Step, having a higher score than D in that quota
  • B is seated

final result of [A,C,B]

With your method:

  • A is seated (Average of 3.667), and group 2 is removed (5 > 4 > 2)
  • A wins a second seat (Average of 3), and group 1 is removed (4 > 2)
  • C wins the final seat

Final Result: [A,A,C]


Ciaran, with Asset the politicians can only negotiate to distribute the votes they receive from the electorate. And if they elect a person whom the electorate almost universally hates, then it was our fault for giving those people votes. Besides, Asset should break 2PD especially if the Senate is elected with Score or STAR.


Again, why is that the option of the candidates rather than the voters?

I get that we’re talking about representative democracy, and the advantages thereof using representatives as an approximation of the will of the people, but I simply don’t see how inserting another approximative layer is helpful. Isn’t that drifting further from the will of the voters, towards the will of politicians?


Because the other ways of getting proportional representation are either too complicated or have anti-PR pathologies or are not precinctable or lead to 2 party domination. For example, STV (which automates the transfers) can eliminate centrist candidates prematurely because it only counts first place votes. RRV is non monotonic and is not precinct countable.

Politicians ALREADY are corrupt and make bribes, and besides, with Asset there will eventually be enough members of each party to prevent exceptionally bad people from winning. So (especially if the senate gets Score or STAR voting) Asset will reduce that problem.

With Score Voting you can have a corrupt Democlican who is accused of 35 sexual assaults and is part of some bizarre cult (but not everyone knows that), and an honest Democlican that has no scandals at all, and there is no way for the Repubocrat to win off the vote splitting.
With Asset Voting, as long as there are enough honest Democlicans running, they can at the very least prevent the corrupt ones from winning, and maybe even convince the corrupt ones (who hopefully have fewer assets) to donate their assets to the honest Democlicans because otherwise too many Repubocrats will win.


To make it practical, you’d have to have some additional rules to bound the size of the legislature, such as by requiring candidates to win a minimum share of the vote to actually be seated (say, 0.2%). You’d need ranked ballots so that voters can be matched to a later choice of rep if their earlier choices lose (otherwise, voters might be pressured to support incumbents so their votes count somewhat.)
An interesting concept (granted, maybe not a good one) might be to replace ‘elections’ with having voters ‘register’ their vote in a manner similar to how people register with parties (although the votes would have to not be publicly available.) So you could change it at any time.


Thanks for pointing that error out. I would try to fix my example because I still believe that under your old version of allocated score voting that didn’t include the conformation step, there are still proportionality and monotonicity errors out there. But that’s a lot of work, and as you pointed out, no longer relevant because of the conformation step.

So what do I think of this step? Well, …

  1. It seems a bit sporatic, and If I had to guess, it probably isn’t monotonic. I recommend running your election method under code that generates lots of random elections and checks for monotonicity violations when comparing different random elections.

  2. Have you proved that this step can’t result in a loop where the voters who “contribute the most to A winning” the most prefer B, the voters who “contribute the most to B winning” prefer C, and the voters who “contribute the most to D winning” prefer A? Because it would be really embarrassing if your voting method broke in a public election in this way.

  3. Have you proved that your method is proportional? When a quota of voters give the maximum support to candidate A and do not contribute to electing any other candidates, that it is impossible for candidate A to lose? If you can’t prove whether this is true about your method, at least write code to generate a lot of random elections and test your method for this property.

  4. Score voting doesn’t have a conformation step which means that the method you are using each round to chose the winner before you exhaust ballots isn’t score voting. If you allow the quota used to be an adjustable parameter, then allocated score voting isn’t a true multi-winner adaption of score voting since it only reduces to score voting in the single winner case when a specific quota is used.

Perhaps those are indeed the candidates that should be elected if we are talking about electing candidates in the traditional way were each candidate can only win once.

And that’s what the results should be when electing the candidates in the non traditional way. Every single voter gave A a higher score then B. A was unanimously preferred over B. If all the voters who prefer A over B also prefer [A, A, C] over [A, B, C], then the voting population as a whole would unanimously prefer outcome [A, A, C] over [A, B, C]. So please explain, why is [A, B, C] better?


“To make [Choice of Representation] practical, you’d have to have some additional rules to bound the size of the legislature…” Why? Doesn’t the real work happen in committee? In the full congress or the committee of the whole, less-popular legislators could participate remotely.


What makes you say?

I’m not skilled in proving negatives, but it doesn’t seem like that would be likely; the scenario in which this has come up, that caused me to add that step in the first place is when you have a scenario where a compromise candidate is initially selected, but then the quota that had the greatest impact on them drew (almost) exclusively from a minority faction that approached a quota unto themselves. The example that first triggered this concern was a slightly skewed normal, with numerous candidates, and one of the centrists got a second seat, but drew almost exclusively from their side of the curve.

I am having a hard time imagining a scenario where you would get this sort of Condorcet Loop going, but I suppose that is a risk that needs a Catch condition, at the very least. Can you come up with a hypothetical scenario that would cause that?

When I ran it with an extrapolated 2016 California Presidential data, it was closer to the actual Quotas (Hare or Droop) than RRV was, as you may recall from the google group.

That doesn’t follow; that’s like claiming that because IRV doesn’t have a “reallocate excess” step, STV doesn’t use IRV to determine each winner.

…which is why I don’t believe that the Quota should be adjustable. I strongly believe that Hare is the only appropriate quota for Apportioned Score Voting, because Droop, etc, are largely attempts to account for the fact that you must disregard the preferences of some portion of voters under Ordinal methods. Because I’m pretty sure that no such failing exists under Cardinal voting methods (pick your poison), there is no need to make concessions to that failing.

As such, with Hare Quotas, the Confirmation Step, the entire purple box, is superfluous in Single Seat Apportioned Score, just as it is in Single Seat STV. After all, the Hare Quota for a single seat would be 100% of the ballots.

Indeed, that’s further argument for insisting on Hare Quota: if you keep apportioning Hare quota after Hare quota, after the penultimate quota, all that is left is the single quota that will elect the last seat in a standard Score election.

Responding to the rest in another post.