What I want in a single-winner voting system

Given that CES and EVC both now have multi-pronged grading systems for voting systems, I figure I want to share what my own goals for a voting system are, and in particular why I do not back IRV or Distributed Voting.

Most of these are practical in nature. I do not consider failing, say, monotonicity as inherently bad; it is only bad if it causes a legitimate problem, like making it unsafe to support your favorite candidate.

So… here are my goals:

1. Diversity of Opinion

Many people do not like either the Democratic or Republican parties. Personally I have issues with both, but much stronger issues with one of them than the other and so I usually vote lesser of evils.

Approval Voting has the nice advantage that I can support multiple candidates. This means you can start having several candidates run within a party. This is important because you can have, for instance, pro-climate Republicans or pro-gun Democrats, creating more of a meritocracy of ideas.

This also reduces the influence of money, because while it may be strategic to approve one frontrunner and disapprove the other, there are still other candidates who can pull ahead of both.

Score Voting and STAR voting improve on this because one can punish, say, a Republican who wants to end Social Security or a Democrat who wants to let incarcerated felons vote, without having to send them down to zero.

IRV ostensibly would let you rank all the candidates of your party up top. But it can eliminate candidates in the wrong order, and we all know (e.g. from the Yee diagrams) that IRV has an extremist bias. This could even lead to IRV pitting the two extremes against each other, or your side’s extreme with the other side’s moderate (i.e. you lose). Worse yet, IRV ignores your second preferences when deciding which candidate to eliminate.
DV fixes the second problem but only partially solves the first.

Final Grades

  • Approval: B
  • Score: A
  • STAR: A
  • IRV: D
  • DV: C-
  • FPTP: F

2. Resistance to Manipulation

We all wish our voting systems would work as well as we want, but of course voters and candidates will game the system.
Obviously none of these systems can get an A because every system can be manipulated somehow.

  • Score gets a B. I do not actually see that much wrong with strategic voters back-sliding the system to Approval because Approval is a pretty good system and the extra benefits of Score are still there if you want to use them. The system also resists money and strategic nomination rather well.
  • Approval gets a B as well for resisting money and strategic nomination. There is less room for voter strategy, but I think whatever strategy does exist will lead to, at worst, the election of the Condorcet winner.
  • STAR gets a B-. The minus is just because it is very vulnerable to tactical voting (it fails NESD*) and strategic nomination in 3-candidate elections, but the first effect goes down as the number of candidates increases. The worst that runoff-gaming can do in large elections is degenerate the system into Score Voting which is a pretty good system, although if some ideology cannot run 2 candidates then that does put them at a disadvantage. Still, I expect this to be minor because STAR always elects either the Score winner or the Score second-place who should still be a pretty good candidate.
  • DV gets a D just because it is too opaque for me to really figure out how to manipulate it. But this “security through obscurity” is a bad idea and I need to punish it with a low score.
  • IRV and FPTP get an F for well-known reasons.

3. Practical Use

  • IRV: D. IRV has been used before to successfully declare winners, but it still requires a lot more infrastructure to count centrally and has the problem of recounts in early stages dramatically affecting the eventual winner.
  • DV: F, because in addition to IRV’s silliness, DV is very complicated for the average voter to understand.
  • STAR: B. STAR gets knocked down a bit because the runoff step requires a bit of creativity to successfully count without centralizing it. Then again, there are only 2 rounds (not many as in IRV and DV) and it only requires O(n^2) to precinct count.
  • Approval and Score: A+. No real problems here. In fact since they are compatible with each other and Approval is compatible with IRV, I give them a bonus.
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What is this in reference to? I’m guessing you’re talking about Approval-IRV (IRV with equal ranking), but I’m curious if you could give an example of how Approval and Approval-IRV would work together.

Simple. Those who cast IRV ballots have their vote transformed into an Approval ballot that approves all candidates their IRV vote ever counted for. Election stops counting at the final 2.

There are many opinions, and ambiguous definitions; not easy to understand what you really want.
Many of the criticisms made also seem quite subjective.

What I want in a voting system are the following goals:
(1) Votes must all have the same power.
By itself, it’s a trivial goal to satisfy, but it’s fair to say.
(2) Representation of interests.
The vote must best represent the interests of the voter.
From worst to best

  • Single choice
  • Multiple choice
  • Ranking
  • Range / Cumulative

(3) Resistance to tactical votes.
It’s assessed by observing how much the results with honest votes differ from the results with tactical votes. In this regard, I created a criterion called Honesty Criterion.
(4) Resistance to tactical manipulations.
By “tactical manipulations” I refer to all those manipulations that affect the results, even if the voters are completely honest (votes conditioned by the prediction of the results, are considered in the tactical votes).
Only a method that satisfies the IIA can be immune to these manipulations, but in practice no method satisfies the IIA (in theory it’s possible to satisfy the IIA only with unrealistic hypotheses).
In this regard, I created the IWA criterion which, unlike the IIA, can also be satisfied in practice and which guarantees at least the independence of the results from the worst (minority) candidates.

Distributed Voting is the only voting method (in addition to the DMV) that satisfies IWA, Honesty Criterion and offers a good representation of interests.
In DV there are 100 points to distribute which can be complicated on paper (the points to be distributed could be reduced), but in electronic format it is much simpler. You can see for yourself an example of an online survey in which 100 points have to be distributed, to evaluate how complex it really is.

Do you prefer a method with more correct results but more complex to understand, or a method with more incorrect results but easier to understand?

Looking at the definition of NESD*:

A single-winner voting system obeys the “NESD property” if, when every voter (all voters assumed initially honest) changes her ballot to “exaggerate about A & B,” i.e. to now rank A top and B bottom (or B top and A bottom; which depends on the voter and is “honest” about that voter’s preference between A and B alone), leaving it otherwise unaltered – these alterations constitute NES, naive exaggeration strategy – that exaggeration-behavior does not always (ignoring very rare “exact tie” situations) cause A or B to win.

We also define the “NESD*” property (note the star) to be the same as NESD except A and B are to be solely-top-rated or ranked by all voters; we forbid coequal top.

It only really seems relevant for systems that forbid equal ratings/rankings, so not STAR. STAR passes favourite betrayal so you can always rate your favourite with the top score.

Having said that, just looking here:

2-party domination worry (or not). One property-sacrifice that worries me is that STAR fails the NESD* property. That suggests STAR voting might cause a country to fall irreversibly into 2-party domination. To understand this, consider an election where the voters exaggerate their scores for the two major-party contenders A and B (to make them sole-top or sole-bottom), in an effort to increase their impact, e.g.

51% vote “A5, B0, C4,” while 49% vote “A0, B5, C4.”
Here the best-for-society winner probably is C. So it would be bad, with STAR, if only A and B got into the runoff, whereupon A would win. Actually that does not happen, but the ultimate result is the same: The totals in the first round are A255, B245, C400, so the runoff is {A,C}. But then A wins that runoff 51-49.

It’s not just about tactical voting, but that it will generally go for a more majoritarian than utilitarian approach. I think my sequential proportional election with candidates cloned modification would elect C into the run-off twice, guaranteeing their election.

This is wrong actually. STAR does fail favourite betrayal. By max rating your favourite, you can put them into the run-off, which they’d lose. And it could be the candidate your vote displaced from the run-off is preferred by you to the winner, and that the displaced candidate would have won the run-off.

It’s true that STAR fails favorite betrayal. However, it passes a slightly weaker criterion which I think is good enough in practice. That is: if A is at risk of knocking B out of the runoff but then losing to C, it may be strategically correct for an individual A>B voter to strategically vote B over A, but if all A>B voters can pick a common strategy, they can always find one that elects B without any of them voting B>A; that is, they can just all vote B=A.

Note that this “weak favorite betrayal” criterion is NOT passed by IRV.

In practice, I think the two criteria are essentially equivalent. For instance, in a center squeeze scenario where B is the true CW, I believe that B will typically make it to the runoff without any strategy, and that the danger of them getting strategically pushed out of the runoff by C voters strategically voting them lower can reasonably easily be counteracted by A voters strategically rating them higher, without any favorite betrayal.

I think “resistance to manipulation” is a bit of a vague concept. Is it manipulation if you exaggerate your scores in score voting? If you form a party and nominate candidates to defend against vote splitting in plurality? If you follow the polls and use that to inform your approval vote (picking only one of the two front runners)?

I would break this concept down into two things:

  1. “game theoretical stability.” I have also used “median seeking” to express the same concept. It’s very easy to explain with a numerical vote (everyone picks their preferred value and choose the median, which is extremely stable, see https://pianop.ly/voting/median.html ). A system that leads to two parties, and where each tends to win about half the time, leaving half the people angry, is the opposite of stable. You want something that converges toward the center.

  2. lack of incentive to attempt to predict of how others will vote. Plurality and approval don’t do well here – to vote effectively you need to know how others will vote. Condorcet seems to be really good here, you gain very little by following the polls. Such incentives needlessly complicates things for voters (it is extra information they have to keep track of), doesn’t work well for “downballot” elections, and is subject to manipulation by media and other organizations that might have agendas.

I’m not really understanding how this works. Take the election I made for this page, for example. There’s one A>B>C voter who has to commit favorite betrayal in order to elect B on their own. There’s a A>C>B voter and two C>A>B voters who won’t help because they prefer C to B. Finally, there’s 6 B>C>A voters who could indeed help, but obviously don’t qualify as A>B voters. How does your criterion apply to this election?

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Well, yes. Of course a post that starts with “What I want” is likely to be subjective. My goals are based on how I want the voting system to behave in practice.

Then again, that (like all ranked methods) still requires JavaScript while FPTP, Approval, Score, and 3-2-1 do not.

You cut my quote off right before I mentioned that it only fails NESD* for 3-candidate elections! Same with the rangevoting.org webpage you quoted: it later goes on to mention that elections with 2 good consensus candidates can send both to the runoff.

Just so you know, if one of the two Big Factions has 51% of the vote then their factional candidate will be the Condorcet Winner.

I mostly mean the system will not suffer really bad effects (that, say, undermine goal #1) as a result of strategy. So yes, all of those things are “manipulation”, but I do not think that e.g. min-maxing Score is really that big of a deal because the system still elects good winners and still allows candidates to flourish based on their ideas.

Again, a lot of this stuff is intentionally “vague” because they are long-term goals, not mathematical criteria.

If you want to make a serious election, then you always have a computer available. Even if it wasn’t serious, I created a site that could be used by anyone on purpose.
Computational complexity nowadays is the last property to matter.

Right, but I suppose I was questioning whether failing it means it’s very vulnerable to tactical voting at all. Why does failing NESD* mean it must be very vulnerable to tactical voting in 3-candidate elections? And if that is the case, presumably it’s the case when there are exactly 3 strong candidates rather than just 3 candidates. If, for example, there are 4 candidates, but one of them isn’t a strong contender, it might as well be the same as the 3-candidate election without that candidate.

It is a little less vulnerable than IRV, but there are a lot of cases where you are strategically forced to give more stars to the lesser evil if polls show your favorite would lose in a runoff to the greater evil.

It depends. Trading simplicity for accuracy is fine up until the point where there are too many people who would rather retain the old system than go with a new system that they don’t understand for the method to be enacted.

Ok, however now the peper ballot of the DV uses range [0,10] so I have also eliminated the last two big problems left.
Only the failure of the monotony remains, but it’s almost impossible to exploit by tactical votes so in practice it’s not a big problem.

What I want in a single-winner voting system is:
as many voters as possible would feel the proces was as fair as possible and thus the result would be considered legitimate.

A Since voters just like the votingsystem-experts on this forum have opinions, experiences and preferences about votingsystems they should in each election be able to express these preferences.

B the votingsystems should contribute to the singlewinner in a proportional way

Anybody else likes this?

I want a system which does not cause everyone to say, afterwards, “We don’t think that really represented the will of the voters”. Plurality causes this. IRV/RCV causes this. In my practical experience, approval voting does not cause this.


Although those are pretty vaguely expressed, I’d say you both more-or-less described game theoretical stability.

Approval achieves this pretty well if people are able to predict accurately how others will vote. If not, such as in a small election where there is no polling beforehand, less so. Lots of people can end up regretting how they voted because they chose the approval threshold wrong, and will wonder if the outcome would have been different (and “more representative”) if the election happened again now that everyone knew who would be in the lead.

Not trying to pick on approval but I think the importance of accurate polling in producing an accurate result shouldn’t be dismissed.

I quite like plain score voting, partly because of its simplicity but I see why methods like STAR have a head-to-head part as well. But the problem with elimination methods like STAR, 3-2-1 and obviously IRV is that the “wrong” candidates can get eliminated. Or at least, you get weird discontinuities.

But I was thinking you could still combine score with pairwise comparisons without elimination but borrowing from Condorcet methods instead.

When comparing two candidates pairwise, you use their total scores but also give an extra max score to each candidate for each voter that ranks then ahead of the other one.

For example, let’s say there are 100 voters and the score range is 0 to 1. Candidate A gets a total score of 60 and B 55. But A is only ahead of B on 45 ballots, with B ahead on 55. So for A we have 60 + 45 = 105 and for B 55 + 55 = 110. So B is ahead on this pairwise comparison.

I think that if a candidate is rated ahead of another one on more than 2/3 of the ballots where they differ, then they will always win the pairwise comparison, and that if a candidate has an average score of more than max/3 more than the other candidate (on ballots where they differ) they will also always win.

However, these numbers change when the range isn’t continuous. For 0 to 5 scoring (as in STAR) it seems that being ahead on more than 5/8 of the ballots is enough, or having an average score 1.25 higher (equivalent to 0.25 on a 0 to 1 scale). Take the following example:

5 voters: A=5, B=4
3 voters: A=0, B=5

Here we’ve minimised the amount A is rated ahead of B but maximised the amount B is ahead of A.

The total scores are A: 25; B:35

Then we have to add on the max scores for the pairwise wins which makes an extra 5x5=25 for A and 3x5=15 for B. This makes it 50-50 in the pairwise matrix.

A was ahead (minimally) on 5/8 of the ballots and behind (maximally) on 3/8 and it was a tie. So it would be impossible for A to be ahead on more than 5/8 and not win the pairwise comparison.

And if we look at it from a score point of view, B has averaged 1.25 points per ballot more than A with the worst defeat ratio possible and it was a tie. So it would be impossible to average more than 1.25 more than the other candidate and not win pairwise.

We can make these numbers even “better” by using a 0 to 2 scale (so just 3 grades as in 3-2-1 voting).

4 voters: A=2, B=1
3 voters: A=0, B=2

The total scores are A: 8; B: 10. And the pairwise wins give an extra 8 to A and 6 to B making it 16-16.

So it ends up that being ahead on more than 4/7 (approx 57%) of the ballots guarantees you the pairwise win. Also averaging 2/7 (approx 0.29, or about 0.14 equivalent on a 0 to 1 scale) points more per ballot more guarantees the pairwise win.

Then obviously you just pick your favourite Condorcet method to finish off the job and decide which candidate wins overall based on the pairwise comparisons. Of course you can vary it by giving different weights to the scores and head to heads. Instead of giving an extra max score to the winner of the head to head it could be 2 x max score or 0.5 x max score, for example.