One voting system may seem very different from another but when examined closely, the two systems may be quite similar. Though they may use quite different ballots, or count the ballots differently, in elections the outcomes may be identical. Mathematicians use the term isomorphic for these sort of equivalences. Isomorphic voting systems will elect the same winners even though the vote tallies may be different; it may not be particularly obvious but the vote tallies for the two systems might always fall into the same order. Alternative isomorphic representations do provide different points of view, and examining a voting system through such an alternative perspective can sometimes be informative. But often there is one best perspective and that is called the canonical form. Most often the other alternative views add nothing but confusion.
Although it is not often described this way, approval voting can be understood as the score voting with only two scores. Likewise, in the previous article of this series we observed that balanced approval voting (BAV) can be described as a score voting system with three scores. But in each case the score voting paradigm is neither the most natural nor the clearest description. Nonetheless, for approval voting it is an accurate and complete description. For BAV, however, though this description is accurate, it falls short of being a complete description. It happens that there are not just one, but three different (and not isomorphic) score voting systems, even though these three systems use the same three scores.
Unfortunately, with score voting, there has long been some carelessness in specifying how the votes are tallied. What has been ignored is the need to specify what to do when there are abstentions. Surely different ways of handling abstentions could alter election results
The most natural way to understand approval voting as being equivalent to a score voting system is to choose the score system which uses the scores 0 and 1. Likewise, BAV is most naturally understood as equivalent to score voting when score voting adopts the scores -1, 0, and 1; this special choice of scores illustrates what mathematicians call a canonical form. While other choices of scores lead to isomorphic systems which produce the same election outcomes, these specific scores, -1, 0, and 1, make the analogy with BAV particularly easy to understand. For one thing, if we assume there are no abstentions, the net-vote tally for BAV will exactly match the score vote tally. And certainly, if two voting systems produce the same vote tallies then they produce the same election outcomes. Using different scores such as 1, 2, and 3, the vote tallies will be quite different from BAV's net vote even though the election results may be the same.
In the interest of simplicity, mathematical models often ignore some real-world details when they seem insignificant; abstracting only the essential features is useful and usually intentional. But sometimes the importance of some detail is missed. The mathematical model of score voting happens to disregard the fact that, in real elections, voters will sometimes fail to provide any rating. This might happen because the voter's feelings are conflicted about a candidate, liking some things about her but finding her offensive in other respects. Other abstentions might be simply oversights, mistakes of accidentally skipping over a candidate in the list on the ballot. But a quite common reason surely is that the voter does not feel adequately informed about a candidate, perhaps not even recognizing the candidate's name. The BAV voting system deals with abstentions explicitly, but traditionally, score voting does not. This was simply a mistake, and the mistake has led to misunderstanding.
As a practical matter any election must take account of the possibility of no score being specified; the easiest solution with score voting is simply to assign a designated, default score. Score voting theory has long ignored this issue, however, providing no guidance. And that lack of guidance seems responsible for the too common assumption that it must be immaterial which score is designated as the default. That easy assumption is quite simply wrong.
For the canonical system with scores -1, 0 and 1 to be equivalent (isomorphic) to BAV, that default score must be specified as the middle value among the scores. And notice that for the canonical system the scores are chosen so that middle score is 0. Choosing 0 as the default score ensures that the score tally (simply adding up all the scores for a candidate) will be identical to BAV's net-vote count (NV). BAV computes NV by subtracting the number of opposition votes for a candidate from the number of support votes. NV literally ignores abstentions when computing the vote tallies.
But for score voting more generally but using the scores -1, 0 and 1, the default score is not specified; in this cast, the default value might instead be chosen to be -1 or +1. Notice that it follows that there are at least three different score voting systems, each with three scores and it turns out that only one of the three will guarantee that the score tallies will match the net vote for BAV.
Let us see what would happen if we were to choose +1 as the default. It is useful to designate with the letter A, the number of abstentions for a candidate (in the score election). The score tally for that candidate would no longer be NV as in the canonical balanced system, but rather NV+A.
NV seems so appropriate for an election tally that we should question what justification there could be for adding A to a candidate's tally. Should a candidate get an extra vote merely for having elicited an abstention? It seems unfair, especially in considering that the number of abstentions, A, varies greatly from one candidate to another. Candidates with many abstentions get more of a boost in their vote tally than others. And It is likely to be the least famous one will get the largest benefit. The extra advantage might be rationalized as helping the underdog. Though not so democratic this is also not very convincing. But at least this justification does evoke some appeal.
Alternatively, we might choose -1 as the default and then the tally would instead match NV-A, now punishing rather than benefiting the least famous candidates. A few sadistic people might feel that when a man is down, he should be kicked and they might favor this approach, but here we might note that the candidate who is down might be a woman. This approach seems as undemocratic as the other. And there seems to be even less appeal to the idea.
Oddly however, this kick'm-while-they-are-down approach is exactly what most score voting systems adopt. When there is an abstention, the smallest score has habitually been chosen to serve as the default. No doubt following this tradition is why star voting, with its six scores, has selected the smallest score, 0, as the default even though this disadvantages third parties and independents. In at least this respect, star voting perpetuates duopoly politics and the polarization that accompanies it.
Traditionally, negative scores have been avoided with score voting but it has not been uncommon for one of the scores to be zero. Star voting specifies zero as its default score and in fact, there is a good reason for assigning zero as the default score. When you do this, abstention is, in effect, ignored in computing the score tally. And it seems very likely that is what the voters would expect and want.
But we observed in the previous article that, in choosing the winner, it is not the assignment of scores that matters (so long as the scores are equally spaced). What does matters is the position of the default score among the selected scores. Fair and equal treatment of candidates, famous or not requires the default score to occupy the middle position among the scores. And moving 0 to the middle position necessarily requires the lower scores to become negative. Fortunately, though, this arrangement makes the scores appear exactly how we should want them be understood, with zero being neutral, positive scores being votes of support and negative scores being votes of opposition. Using the alternative scores 2, 3 and 4 with 3 being the designated default, 3 would become the neutral vote, 2 the opposition vote and 4 the support vote. Using those scores, some voters might fail even to be aware that there are alternative ways to understand the impact of the scores.
In elections, voters do oppose some candidates, just as they support others. And they want to express how they feel about the candidates. Making it clear on the ballot that a voter can express any of these opinions can only help with voter satisfaction. Conversely, it is hard to imagine any benefit in disguising that voters have the options. BAV does this quite clearly, but the score voting alternative can do it nearly clearly when the canonical form is used.
With score voting, it would be good practice to always include zero as one of the scores and to make that the default. This convention would alert voters (and theorists) how the scoring is to be done and surely that is important.
The canonical form described above using -1, 0 and 1 follows this practice but it is not the only way to represent BAV as a score voting system. It is merely the way that is least likely to confuse. BAV can be represented as the score voting system with scores 1, 2 and 3 (or for that matter, 57, 67 and 77). But the unnecessary mental gymnastics in dealing with these peculiar choices can cause confusion and even lead to misunderstandings. For example, when only positive scores are offered, one might conclude that the scores all represent support, though perhaps to different degrees; and some voters may react to this perception by abstaining. Instead, the canonical form suggests, quite accurately, which scores convey opposition, neutrality or support.
It is important to keep in mind that the choice of scores does not materially alter the score voting system; whatever scores are chosen, election outcomes are unaffected, provided the voting remains equivalent and the default stays in the same position. But how the system is understood can affect understanding and even alter how people vote. And when people vote differently, clearly the election outcomes may change. This chain of effects is difficult to assess and is often dismissed in discussion of voting methods as merely a psychological effect. Psychology does play an important role in elections, however, so such considerations should not be dismissed lightly.
When there are an even number of scores, zero cannot possibly be the middle score simply because there is no middle score. Still, we might try placing the default score, 0, as close as possible to the middle.
Taking six scores as an example, we might choose the scores -2, -1, 0, 1, 2 and 3. The first thing to notice about this choice is that the default value, 0, falls below the center so that abstentions, tallied as zero, act in opposition a candidate; in turn this will disproportionally punish the less-known candidates and favor the famous ones. But another thing to notice is that when a score of 3 is specified in support of a candidate, no other voter can balance this with an equal but opposite score of -3. This unfortunate imbalance gives voters who support a candidate greater voting power than it gives those in opposition. In turn this advantage would likely encourage candidates to play the odds and take a greater risk of angering some voters in the hope of enthusing supporters to vote enthusiastically with a score of 3. Meanwhile, the angered voters lack the power to respond effectively with a balancing opposition vote.
An equally promising choice of scores would be -3, 2, -1, 0, 1, 2. Notice that the position of the default score of 0 has now changed, so this is a truly a distinct and different voting system, quite possibly leading to different election outcomes. A Japanese proverb warns that "the nail which stands out gets hammered down" and with this score voting system, politicians may be advised to take this to heart. In this case, abstentions provide additional support for the least famous candidates, perhaps an encouragement for all candidates to maintain a low profile to boost the number of abstentions.
Also, with this score voting system, it is the opposition vote of -3 that cannot be balanced by any single support vote by another voter. Voting in opposition now has greater power. In turn this makes it especially important for a politician to avoid angering some constituents, even if it would increase support by more. Politicians anticipating an election with this voting system are motivated to avoid standing out. In two different ways, it would encourage candidates to keep a low profile.
It seems difficult to justify ever using a score voting system with an even number of scores. This even includes approval voting, which would be significantly improved by including just one extra
score. And it is not at all clear what one might hope to gain by, for example, adopting six scores rather than five or seven.
Still, it should be said that score voting (with three or more scores) does give voters the opportunity to vote either for or against each of the candidates. It may fail to offer this with an equal opportunity, but at least the voter can, with care, avoid the mistake of abstaining. The election may disguise as weak support the opportunity to vote against a candidate, but it is still possible for voters to see through that disguise and recognize that there is an opportunity to vote in opposition.
It seems hard to imagine why one might use a score voting system that is not balanced, with the default score 0 being in the middle. BAV is the simplest such system but there are others.
Balanced voting systems like BAV provide equal voting power for both support and opposition; they favor neither famous nor unknown candidates. And widespread adoption of BAV will undermine a duopoly. Politicians anticipating BAV elections are apt to avoid taking extreme positions. Even primary elections would tend to compete close to the center rather than at the extremes of public opinion.
