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Why is Balanced Approval Voting so Hard to Understand?

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BAV seems so easy to explain. You allow the voter to check one of two boxes for each candidate, one box to show support and the other to show opposition. And you allow the voter to check neither or in fact both. The net vote count for each candidate is the number of voters checking support minus the number who check oppose. The candidate who gets the largest net votes wins election.

This seems so easy and natural. So I was surprised that many people just didn't understand it; even more surprising the people who had the most trouble understanding were the very people who were experts at voting theory. Unfortunately, though understandably, these people insisted on re-casting BAV as something familiar, as an example of score voting. Relevant to this is a warning I included in a recent article:

What is the score?
What is the score?
(Image by Brian O'Donovan from flickr)
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... I feel compelled by experience to warn against interpreting BAV as an example of score voting. Arguably it may be, but that line of thinking, while in no way helpful, tends to be misleading (so if you don't know what score voting is, don't worry about it; that is actually a good thing).BAV seems so easy to explain. You allow the voter to check one of two boxes for each candidate, one to show support and the other to show opposition. And you allow the voter to check neither or in fact both. The net vote count for each candidate is the number of voters checking support minus the number who check oppose. The candidate who gets the largest net votes wins election.

Now, reluctantly and in spite of that advice, I will now address Score voting. These are voting systems that adopt a ballot on which each voter assigns candidates a score. Scores are typically, but not necessarily in a range, such as 1 to 3. The scores are tallied (added) for each candidate and the winner is the candidate with the greatest tally. Among people familiar with the topic, a well known fact is that you can multiply the scores by any positive integer and get an equivalent voting system. Likewise, you can add any integer to the specified scores. Here, by equivalent it is generally meant that the relative positions of the candidates is unchanged; the winner (or winners if there are several) will remain unchanged, but importantly this is true only if the voting remains equivalent; in other words, the assignment of scores for the smallest, middle and largest scores (along with any other positions) must remain the same from one of the two equivalent systems to the other.

Often unnoticed though was that score voting was not a real voting system but an abstraction. In a real election there will be voters who, regardless of the instructions, skip over candidates. In the abstract theoretical model this is ignored through the device of simply declaring it not to happen, but in a real election it is necessary to decide how to handle such ballots; they surely will occur. One might decide to just skip over those ballots when performing the tally, but that has the same effect as adding zero to the tally. In effect, the system with scores 1, 2 and 3 becomes the system with possible scores 0, 1, 2 and 3. Note, however that the system will be unchanged precisely when 0 is included as one of the scores. This should motivate us to insist that 0 must be one of the scores; there is no harm in insisting this since there necessarily are equivalent systems that include zero. The advantage of insisting 0 be included as a score in fact makes the model conform to a real voting system. In these real-world score voting systems though, adding a constant to the scores does not always result in an equivalent system (multiplication of the scores by a positive constant will still result in an equivalent system, however).

With this convention, the score voting system using scores -2, -1, 0 will no longer be equivalent to the system with scores -1, 0 and 1 or with the system using scores 0, 1 and 2 because the votes and the tallies will change as the abstain votes are added at different positions among the scores. This convention of using the value 0 as a default score when no score is specified makes critical the position of 0 among the scores. A detailed numerical example of this can be found in an earlier article .

Let me remind you that BAV is defined without resort to scores at all. But with the added requirement that zero must be one of the scores and that in tallying ballots you simply skip over instances where no score is specified, then the score system with scores -1, 0, 1 equivalent BAV (however with the boxes labeled -1 and 1 instead of oppose and support).

The phenomenon of an abstract model leading to mistakes is not at all limited to voting systems. It could be seen as what Einstein noticed about the Newtonian model of mechanical motion. A recent podcast at Intelligence Squared discussed this kind of mistake in general and with respect to economic theories. The entire podcast is well worth hearing, but this specific topic came up about 17 minutes into the podcast.

It may help clarify this issue to recount a particular objection that was made to the notion of a balanced voting system. In particular, Approval Voting (score voting with scores 0 and 1) was claimed to be balanced. Part of the issue was the linguistic issue of whether not supporting a candidate means the same as opposing that candidate. People do say such things, but I would contend there is a neutral ground lying between support and opposition.

But that same confusion can be expressed in numbers. The observation was presented that score voting with -1 and 1 as the two allowed scores is equivalent to approval voting (score voting with the scores 0 and 1). As noted before, this is the case for the traditional abstract model that assumes all candidates are scored as either -1 or 1. But assuming ballots that specify neither are ignored, then 0 surreptitiously becomes another possible score. Approval voting, when moved into a real-world election naturally transforms in the real-world model to BAV. There are other possibilities of course. For example, entire ballots could be rejected when any candidate fails to be rated. But that seems a much less attractive alternative, particularly if write-in ballots are anticipated.

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Attended college thanks to the generous state support of education in 1960's America. Earned a Ph.D. in mathematics at the University of Illinois followed by post doctoral research positions at the Institute for Advanced Study in Princeton. (more...)

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