Score Voting ideal quorum


For our WA ballot initiatives for Score Voting, we need to decide on a quorum. See this for reference:

I had proposed we just add 100 (or 1000?) zeroes to the denominator. A fixed number.

You could also do something dynamic: add a number of zeroes that’s a function of the number of voters who cast a vote in that race, or the number of total voters, or the number of registered voters, or something like that.

Bear in mind that we’re really only addressing write-ins here. With any candidate whose name appears on the ballot, surely enough zeroes will be cast to prevent the election of any crazy dark horse candidates. But I don’t think the weighting formula should only apply to write-ins, if for the sake of simplicity more than anything else.


11.1111…% of ballots cast? i.e. make 10% of the votes zero votes that give every candidate a zero. This number is probably too high but I feel like it’s better to play it safe in the first ever use of score voting in governmental elections (if you don’t count 3rd party primaries)


Isn’t it simpler to just use sums rather than averages?


Are all of the initiatives going to be in places where run-offs are required? Are there any municipalities in Washington state that are not required to use run-offs? I don’t mind the run-offs too much but they do prevent us from using score voting’s favorite betrayal criterion as a talking point, so…


Of course. But there are substantial benefits from using average too. Discussed here:


Top two is a WA state requirement.


I like the simplicity of sums.

But I also liked an old proposal by Eric Sanders:

Individual abstention value = (raw average) * (% non-abstentions)

Total value of abstentions = individual abstention value * number of abstentions

Final score = raw total score + total value of abstentions

After some math, I found this that for a lesser known candidate with a higher average to beat a well-known candidate with a lower average, the % of non-abstentions they need is:

u = average of candidate with %non-abstentions
v = average of candidate with 0 abstentions

%non-abstentions = 1-√(1-(u/v))

If it’s exactly equal, they tie. If %non-abstentions is larger, the lesser known candidate wins. Note that this assumes the range of scores starts at 0. If it doesn’t, then subtract the minimum score to u and v first.

For example, if a well-known but devisive candidate gets an average of 5, while a candidate who is lesser known, but universally loved by those who know him, gets an average of 9, then the lesser known candidate needs to be known by more than 1/3 of the electorate to win.


The alternative that I was discussing with Clay was that of having a minimum denominator.

S = Sum of votes cast for that candidate
C = Valid votes cast for that candidate
O = Valid Votes cast for that office

Then for any candidate, their average would be S divided by the greater of C or floor(O/2)+1

That way, regardless of how many people weighed in on a given candidate, their actual sentiment among a majority of voters would be at least as high as that average.

  • It solves the problem of a small “sample size” by ensuring that the candidate’s average opinion among a majority of voters would be at least that high.
  • As the number of voters expresses opinions approaches a majority, the average asymptotically approaches the average of those voters who expressed an opinion.
  • As such, there is no artificial deflation of the average scores for candidates who are scored by every voter.
  • There is no minimum “cutoff”
  • The math is relatively simple
  • It (technically) allows the possibility of a well-loved write-in winning


Simpler, yes, but it accentuates Ballot Access questions. For example, imagine if a candidate that the electorate at large considered a worthy option were compelled to run a write-in campaign by exclusionary ballot access requirements. Further imagine that, for whatever reason, they were only able to get their message out to 75% of the electorate.

At that point, a candidate whose name is printed on 100% of the ballots, but is universally considered to be 20% less worthy than our hypothetical write in, simply being printed on the ballot, in the voting literature, would offer that person an advantage; 100% * 2 > 75% * 2.5.

This distortion would be exacerbated by the fact that a significant percentage of the voters are likely to scale their opinions to fit the scale offered them. That means that a voter may give a Printed candidate a maximum score even if they would have reserved that maximum/minimum score for the Write-In (scaling the other scores as appropriate) had they known they were running.


Ehhh, I guess. I think “The addition of a quorum and arguments about what number to use complicates score voting enough that it isn’t adopted” is a much more realistic scenario than “Someone can’t get on the ballot but can somehow win the election anyway and everyone just accepts the result even though that candidate got only a tenth of the votes of the people on the ballot”.


I think either 4096 or 2*sqrt(V) is fine.


My issue with this is that it creates a discontinuity. As the number of valid votes increases for you, your numerator increases, but your denominator doesn’t. But then all of a sudden when half the voters have voted for you, your denominator starts increasing too.

So you can have a situation where you add n identical scores to candidate Bob and Alice, and that causes Bob to overtake Alice. Weird.


100 voters
Bob has 50 points from 20 voters = avg of 1.0
Alice has 60 points from 50 voters = avg of 1.2

We add 10 5’s.

Bob has 100 points from 20 voters = avg of 2.0
Alice has 110 points from 60 voters = avg of 1.83

Other key points:

  • Simplicity: a constant adder to the denominator is dead simple, and any more complexity might just make sum-based preferable to average-based

  • Clarity: If you only mildly distort the denominator, then the “weighted average” is still pretty accurate and meaningful. If you use a large number of zeroes, then it becomes a pretty convoluted number. At that point I’d want to publish the total points, and total number of voters who cast a vote for each candidate, so people could calculate the actual averages if they wanted to have insight into actual support.


Um Clay, that is simply a result of adding “no opinion” scores to score voting, and it’s also the reason why score voting no longer passes the participation criterion when the “no opinion” option is added.

Consider that:
1 voter gives Bob 0 stars and Alice 5 stars.
9 voters gives Bob “no opinion” and Alice 0 stars.

Alice has a higher average then Bob, but even without any quorum, adding enough Bob and Alice 5 stars votes will change the election result from Alice>Bob to Bob>Alice. This is certainly weird, and one of the reasons why I’m not a huge fan of a “no opinion” option. I believe that voters should be forced to decide on a score for a particular candidate, except for when there are two many candidates to rate them all (aka write-in candidates). No quorum rule will save you from this issue. The only way to fix it would be to get rid of the “no opinion” option all together or count each “no opinion” as a predetermined score (like zero) but that would kind of ruin the point of having such a “no opinion” option.

I do agree with clay that good quorum rules should be somewhat continuous. It should also be smooth.

By smooth I mean that if x voters rate candidate A Y stars and if (constant - x) voters rate candidate A Z stars, if we plot A’s resulting rating f as a function of x, then the function f(x) should be infinity differentiable when x is between 0 and the constant. The add baseline zero ratings to a candidate’s average quorum is an infinity differentiable quorum, thus it is smooth. So is Eric Sander’s method, so it is also smooth.


The issue I’m pointing out is even more severe, and I apparently need a better example to demonstrate that. If you have a mark from fewer than 50% of the voters, each additional rating increases your numerator but not your denominator, but then after that it increases your denominator too. There’s a discontinuity.

There is clearly a benefit to average-based Score Voting, as the most qualified candidate may be some policy wonk who’s relatively unknown. The Lady Gaga vs. Craig Venter example could not be a better demonstration of this. Whether it’s worth the complexity is up for debate. I think it is, if you can have an extremely simple quorum rule, like just adding n zeroes.


I think that would be covered under whether a quorum rule is “smooth” (infinity differential, as described in my previous post) or not. The discontinuous quorum rule that you are talking about would not pass the “smoothness” test. If you have some voters rating candidate A 5 stars, and some other voters rating candidate A 0 stars, if you were to plot a graph of A’s score as you slowly change the 5 stars to 0 stars, the point where the denominator starts/stops increasing would not have a derivative.


You mean Ciaran’s proposal? The one I’m proposing is smooth—you just add a constant to the denominator. You can even make that “constant” a function of the number of voters (it’s constant for all voters, within a given election).


And as I said in a previous post, that one does pass the smoothness test. It’s also the quorum that you should probably use. For every 9 ballots cast, add a zero ballot of which rates every candidate 0 stars.

Both the candidate’s average scores should be reported (so candidates have a good idea of how much support they got since the zero ballots make it look like they were less popular then they actually were) and the candidate’s quorum adjusted average scores should also be reported (so we know how close each candidate was to winning).


In my personal experience, the quorum rule is by far the toughest aspect of score voting to explain to people, though I understand it’s necessary political cover for enacting an average based score voting scheme. Unfortunately, this catch-22 is going to make it difficult for any average-based score voting scheme to pass.
If you must have a quorum rule, it should be a proportion of total votes cast, such as 10%, since a raw number of automatic zeros will have a much different impact on a state legislative race than a gubernatorial one. Technically this that would lower the maximum score from 9 to 8.1818, but that could be adjusted by multiplying averages by 11/10.


Yeah, I’m thinking the initial implementation should just be sum-based. KISS. If we get over that hurdle, then we can get fancy. I think I’m done here.


Here is a link to a post in the old forum where I evaluated several different quorum rules on some criteria:

“Adding zero votes equal to some fraction of the number of voters” meets the same criteria as the “soft quorum” rule, but it might sound a little more sensible to the average person. Hard to say.

In the end, though, I agree with Clay. I think the basic sum-based rule is the best. Because I think the participation criterion is very valuable. And nobody has ever shown me an example election where a quorum rule elected a candidate that was obviously better than the sum-based winner.