@cfrank

I have no kind words about quadratic voting. It has no apparent rationale for the way it's designed, and it has very high front-end complexity which is an absolute dealbreaker. Voters should not have to do math in the voting booth. It has the hallmarks of something invented by grifters to bamboozle people who don't know any better, like NFTs.

I want to introduce a technique of combining two multiwinner approval methods using what I call vertical composition. The basic idea is that there are two "phases", both of which use approval ballots as their inputs, and the first phase filters out some candidates from entering the second phase. Voters, however, do not cast simple approval ballots, but ballots with trivalent values for each candidate - it is important to note that this is distinct from 0-1-2 score voting. The values are as follows:

• Approve in both phases (AA)
• Approve in first phase, disapprove in second phase (AD)
• Disapprove in both phases (DD)

The hypothetical fourth value, Disapprove in the first phase and Approve in the second phase, is excluded because it doesn't make sense.

My basic use-case for this technique is when the first phase is simple bloc approval voting that filters out anyone with more than, say, 75% disapproval (this threshold is a parameter), and the second phase is some proportional approval voting method.

This specific method would be useful for an organization that wants proportionality in its internal democracy but also wants to set up a barrier to prevent bad actors ("entryists") from joining the organization and winning a few seats proportionally while antagonizing everybody else in the organization. Voters would use DD votes to filter out the bad actors, and AD votes for candidates from different factions that are participating in good faith. A relatively high first-phase disapproval threshold, such as 75%, would make the method resistant to people overusing DD votes.

The quota rule sounds reasonable at first glance, but the more you examine it, the more unreasonable it appears. First, the quota rule is known to be incompatible with two criteria:

• Population monotonicity
• Independence of irrelevant ballots

The latter criterion is most revealing, because the quota rule requires voting methods to try to represent "unrepresentable" voters, and punishes the largest factions when unrepresentable voters are present. By unrepresentable voters, I mean voters who vote for very small parties that aren't big enough to win even 1 seat. The quota rule counts these voters as part of the denominator, meaning that the larger parties have lower quotas, get fewer seats, and some of the smaller parties need to be bumped up from 0 to 1 seat to compensate. Note that the unrepresentable voters aren't voting for these smaller parties that are getting seats, they're voting for even smaller parties and just warping the entire election by changing the denominator of all the quotas. If there are a whole lot of unrepresentable voters, the quota rule tends towards Chamberlin-Courant-like behavior, where the top n factions all get 1 seat, because their quotas are so low that 1 is the upper bound for all of them.

I don't think the quota rule should be placed on a pedestal, and I don't think methods should be bashed for violating the quota rule if they pass independence of irrelevant ballots and population monotonicity. PAV and Seq-PAV do this, as well as D'Hondt party lists - they basically ignore the unrepresentable voters because of how the math works.

I believe that the later-no-harm criterion is so confusingly worded that people think it means something it doesn't. It's defined in terms of not harming a candidate, but people think that it means that adding a second, third, ranking etc. will never harm the voter. So they think that LNH means that every voter has an incentive to give their full ranking.

Because IRV fails the participation criterion, it's possible that the voter can be harmed even by adding their first choice. Sure, this will never change the winner from their first choice to someone else, but it could change the winner from their second choice to their 15th choice.

LNH is meaningless without the participation criterion and it's trivial to show that failing the participation criterion means that a voter can be harmed by adding their second choice as well (e.g. their first choice gets eliminated and their second choice causes a participation criterion failure).

The only method that satisfies both LNH and the participation criterion is... plurality voting. Or some kind of plurality-equivalent ranked voting which only looks at the first choices. And here it's clear that it only satisfies LNH by completely ignoring everything else after the first choice.

@wolftune
I noticed something similar with reweighted range voting when I tested it years ago. I had simulations with a "gray party" that got the same medium score from every voter, alongside colored parties that got the maximum score from their party voters and the minimum score from everybody else. I noticed that as I varied the score for the gray party, it would suddenly "explode" from having no seats to having all the seats.

I devised a fix, which I later learned was the Kotze-Pereira Transform.

The main thing to watch out for in any modification of score voting is "does this create an incentive for turkey-raising." Turkey-raising is when a tactical voter raises bad candidates ("turkeys") above last place / zero score. It's extremely bad and it's infamous for ruining the Borda Count.

Score voting itself does not have turkey-raising (there's never a reason to give bad candidates anything other than 0). STAR doesn't have it either, because it's a combination of two methods that don't (score voting and some kind of automatic two-round system.)

This method looks like a combination of score voting and BTR. BTR has turkey-raising. There's an incentive to put turkeys ahead of a competitive rival in order to increase the rival's chance of being eliminated. So this method might have turkey-raising as well.

@masiarek

Condorcet method, so the main thing (in my view) to look out for is whether or not its cycle-resolving rule makes it immune to turkey-raising or not.
I've heard that a few Condorcet methods are immune to turkey-raising but I don't have more info about this.
Perhaps, in the event of a false Condorcet cycle generated by tactical voting, the smallest-margin victory will always belong to the "turkey".

@cfrank

It's in this paper from 2006
https://rangevoting.org/stratapproval4.pdf

@cfrank

I have no kind words about quadratic voting. It has no apparent rationale for the way it's designed, and it has very high front-end complexity which is an absolute dealbreaker. Voters should not have to do math in the voting booth. It has the hallmarks of something invented by grifters to bamboozle people who don't know any better, like NFTs.

@cfrank

That could end up removing candidates that many voters regarded as unrepresentative of themselves but not antagonistic to them.

The idea is that voters would use AD votes for factions that are potential coalition partners for their faction. Some voters can be expected to overuse DD votes, which is why the first-phase disapproval threshold is high.

@cfrank The first phase would eliminate some candidates who might otherwise win in the second phase.

These would be candidates who've actively antagonized everybody outside their faction. In order for the first phase to matter, such a faction would have to be smaller than 1 - the first phase disapproval threshold, but large enough to win at least 1 seat proportionally.

Believe it or not, I was in an organization that was faced with such a scenario. They were using STV and the troll faction's candidate came in 1st because they all bullet-voted for him.

@cfrank That's what I said, the "phases" are part of the algorithm and voters only cast a ballot once.

I want to introduce a technique of combining two multiwinner approval methods using what I call vertical composition. The basic idea is that there are two "phases", both of which use approval ballots as their inputs, and the first phase filters out some candidates from entering the second phase. Voters, however, do not cast simple approval ballots, but ballots with trivalent values for each candidate - it is important to note that this is distinct from 0-1-2 score voting. The values are as follows:

• Approve in both phases (AA)
• Approve in first phase, disapprove in second phase (AD)
• Disapprove in both phases (DD)

The hypothetical fourth value, Disapprove in the first phase and Approve in the second phase, is excluded because it doesn't make sense.

My basic use-case for this technique is when the first phase is simple bloc approval voting that filters out anyone with more than, say, 75% disapproval (this threshold is a parameter), and the second phase is some proportional approval voting method.

This specific method would be useful for an organization that wants proportionality in its internal democracy but also wants to set up a barrier to prevent bad actors ("entryists") from joining the organization and winning a few seats proportionally while antagonizing everybody else in the organization. Voters would use DD votes to filter out the bad actors, and AD votes for candidates from different factions that are participating in good faith. A relatively high first-phase disapproval threshold, such as 75%, would make the method resistant to people overusing DD votes.

@masiarek

Condorcet method, so the main thing (in my view) to look out for is whether or not its cycle-resolving rule makes it immune to turkey-raising or not.
I've heard that a few Condorcet methods are immune to turkey-raising but I don't have more info about this.
Perhaps, in the event of a false Condorcet cycle generated by tactical voting, the smallest-margin victory will always belong to the "turkey".

@cfrank

It's in this paper from 2006
https://rangevoting.org/stratapproval4.pdf

@cfrank

Bullet voting is not the dominant strategy in score voting or approval voting for that matter. The dominant strategy is to identify the top 2 frontrunners, max-score the one you prefer, and min-score the other one. Then you max-score anyone else you like more than the frontrunner you just max scored. For some voters, that's a bullet vote, but not all.

If everyone uses this strategy, it elects the actual Condorcet winner. That's a theorem that's been independently discovered by several mathematicians.

@cfrank

I've heard that it's possible to make some Condorcet methods immune to turkey-raising by designing for a property called "dominant mutual third burial resistance." But I'm not familiar with examples of methods that do this, and it's been hard to find information about it. And in any case approval and score voting solve this problem in a much less complicated way.

@masiarek

I'm fairly sure it's straightforward. Look at the final two candidates, and any candidate ranked below either of them on a voter's ballot is red.

Edit: Also I think that example image is slightly wrong, the ballot that has D > F > B > ... should have B in black.