This is a quick note about an alternate voting system which occurred to me while talking to Scott Garrabrant about 321 voting.

Above everything else, you should know that 321 voting is very good. It rates very highly in VSE, and is quite difficult to warp with strategic voting.

The basic idea of 321 voting is: a ballot has three options for each candidate, "approve", "disapprove", and remain silent. The three candidates with the most approval votes are the semifinalists; the semifinalist with the most disapproval votes is knocked out to get the two finalists; then the winner is the finalist who is above the other in the most ballots.

This basic idea is very vulnerable to clone candidates, but we'll get back to that problem later.

Scott's Babysitting Story

Scott explained why 321 voting is a good idea with the following story (apologies for any errors): he had to babysit three children. He told them that one of them would have to go to bed an hour early, as decided by a vote near the end of the evening. The children spend the whole evening being really nice to each other, in an attempt to not be the one with the early bedtime.

The analogy with 321 voting is that you get 3 parties, but (unlike in a politically polarized system) each party is trying to please the others, in order to avoid being the one eliminated in the disapproval step. If the three parties were each quite unpopular with each other, a fourth, more innocuous party could come in and displace one of the three.

One might argue that this isn't so different from a two-party system, where the loser competes for the popular vote by moving closer to its opponent's position when it has too little support, resulting in two nearly identical parties with little differences. But I would disagree. In a two-party system, each party only needs the support of about half of the people. This can support a high degree of political polarization. In a 3-party system under 321 voting, you have several forces working against this:

• The approval-voting thing where merely getting the support of any fraction of the people isn't enough, because some other candidate could get more support. Getting the support of 1/3rd of the people isn't enough to become a semifinalist; someone else might still out-compete you by getting even more support. So you have to fight for every voter; you can't alienate any.
• The disapproval mechanism takes this even further, actively punishing you for really alienating anyone. No matter how large your base is, you'll never win an election in 321 voting if you anger more people than some other candidate.

Elimination Disapproval (ED)

321 voting is simple enough to explain quickly, and the disapproval mechanism is quite cool. Looking at it with a mathematician's eye, however, it looks quite inelegant: a different rule is applied at each stage.

Here's a rule which generalizes the final 2 steps of 321 voting:

At each stage, eliminate the candidate who is ranked as the lowest candidate on the most ballots. Remove that candidate from the rankings in all ballots (revealing a new "lowest") and repeat. Last candidate left standing wins.

This requires ranked ballots. It's kind of like opposite-land IRV: like IRV, it eliminates candidates one at a time; but rather than eliminating the candidate who is least often ranked at the top, we eliminate the candidate who is most often ranked bottom.

When there are three candidates remaining, each ballot ranks the 3 candidates, so the ranking ends up like approve/neutral/disapprove. Most disapproval gets eliminated.

When there are two candidates left, of course, this acts like 321 and elects the candidate who is most often ranked higher.

I call the proposal ED for Elimination Disapproval. ("Disapproval Elimination" would be more natural, but DE is just less fun to say than ED.)

Clone Problems

If Pity Party (a relatively unpopular party who should not be elected) manages to get lots of similar candidates on the ballot, this has the effect of spreading the disapproval out. Let's suppose voters don't know how to order the Pity Party candidates (they only know how much they dislike Pity Party relative to other parties), so they rank them randomly within a pity-party block. Then each candidate individually has 1/N the disapproval it should, at each stage, because disapproval is absorbed by other pity party candidates.

If everyone ranks Pity Party below all other candidates, this won't matter, because a Pity Party candidate is eliminated until there are none left. However, if Pity Party actually has some support, this could be a major problem; by spreading around the disapproval, the Pity Party clones could survive elimination until there are no other parties left.

A simple fix for this problem is to allow people to rank candidates equally. However, voters are then left trying to figure out when to do so -- there's a complex strategic question of expressing finer-grained preferences vs exercising more disapproval power. So I don't like that idea very much.

321 voting can also have a clone problem, to some extent; if we get 3 clone candidates as semifinalists, then the remaining steps don't matter. (This makes 321 voting devolve into approval voting, which is not bad, but is worse than 321 should be.)

At the beginning of this essay, I described "the basic idea" of 321 voting. Real 321 voting has a tweak to reduce the clone problem. Actually, there are two proposals:

• The 3rd semifinalist can't be from the same political party as the first 2 selected.
• Or, if you don't want to bring political parties into it: the 3rd semifinalist is chosen only using ballots which didn't approve of the other 2.

In either case, you're creating more variety in the semifinalists, which helps ensure that the disapproval step is meaningful.

However, you have to admit that those are hacks. They don't look like they belong in an elegant voting system. The non-partisan solution problematically encourages voters to not approve of candidates who are likely to be one of the first two semifinalists, in order to prop up their preferred candidate's chance of being the 3rd semifinalist. And both solutions could still let the first two semifinalists be clones, which means having a clone can still help you bypass the disapproval step.

So how can we do better?

Representativeness

We can select the semifinalists via STV or any other multi-winner election technique that promotes representativeness. The point of these methods is to elect representative bodies (such as parliaments etc) in such a way as to be as representative as possible. This means they can help us get rid of clones -- bringing clones in to a "representative" set generally won't be a very good way to represent the distribution of opinions of the people, unless the clones really are quite popular with a sizable segment of the population (in which case they may deserve to be semifinalists after all).

So, here is my proposal: use STV to select N candidates, and then use ED to eliminate candidates down from N to 1.

Getting a bunch of clone candidates on the ballot shouldn't make your chances any better, now, because your clones don't end up in the representative set of N candidates.

I call this RED, for Representative ED.

The parameter N is quite interesting, because it gives us a way to trade off between center squeeze and the clone problem:

• N=1: we use STV to select one candidate, and don't use ED at all. This is just IRV, and has the same severe center squeeze problem as IRV: once we're down to 3 semifinalists, the single most representative candidate is quite likely to be eliminated, because that gives a more representative set of 2 candidates. For example, you'd eliminate a centrist to keep a democrat and a republican. But then you're forced to choose between the democrat and the republican at the last step, when the centrist would have been the better choice. This is going to be a problem with any iterative-elimination technique which seeks high representativeness at each step individually.
• N=2: use STV to select two candidates who jointly best represent the population, then use ED to narrow it down to one. This is almost exactly the same thing again, and still has a severe center squeeze problem.
• N=3: Use STV to select three candidates, then use ED to get it down to one. This is pretty similar to 321. However, unlike 321, there is a center squeeze: to get 3 candidates, STV might reject a 4th candidate who is more centrist than any of the three.
• N=4: use STV to find a representative set of 4 candidates, then use ED to get down to one. Again, this can potentially result in a center squeeze if the 4 candidates are 4 extremes around a center.

As N gets larger, there's less of a chance of center squeeze, but more of a chance that clones come in and mess up the voting method. The chance of a clone problem really depends on the number of non-clone candidates. Ideally you'd set N to be exactly the number of "legitimate alternatives", if you could define such a thing, to just weed out the clones without resulting in any center squeeze.

The geometry of the center squeeze problems is pretty interesting. If you think of possible candidates as a multidimensional space, then a more ball-like space of voters makes center squeeze more probable, while a more elongated (hotdog-like) distribution of voters means you'll get centrists in the representative set with smaller N (such as N=3).

Overall, I don't think RED is a necessarily a great method (although I'm curious how it would rank in VSE). It has more of a center squeeze problem than 321 does. We can set N to minimize that, but it's sort of like there's no good setting of N -- we never completely eliminate the center squeeze, and we never completely eliminate the clone problem, either. 

Still, it's an interesting technique. The trade-off here between the clone problem and center squeeze makes me wonder if, in general, there's some kind of trade-off between clone immunity (IE, inability to change election outcomes via clones) and center-squeeze immunity.

Approval and score voting are both 100% clone immune and 100% center-squeeze immune. Score voting is literally as good as you can get (at least in VSE terms!) if there's no strategic voting, and devolves into approval voting under strategic voting. So, if there is some kind of trade-off between center squeeze and clone problems, it must be in the territory of methods that are better than approval voting (even given strategic voters). That's consistent with my observations, since 321 has clone problems.