Since the very start of this series on Balanced Voting, the idea of "balance" has come under misunderstanding, criticism and from me, just a little regret. "Balance" is just a name but it does evoke some expectations. I chose to use that name because it seemed to fit the concept I had in mind, which was to let voter feelings of opposition have equal but opposite strength as compared to feelings of support. The name still seems appropriate, but from a marketing sense others suggest a better name to have chosen would be negative voting. Sure, we are so accustomed to only voting for a candidate, from a marketing perspective that may caught peoples attention better.
But while not opposing the term, others have embraced it but give it a significantly different meaning. Probably this is an unintentional confusion due to ambiguity about the meaning of the noun "vote". Be that as it may, I continue to find people perpetuating the opinion that Approval Voting (AV) is balanced. That is their right of course. It all depends on what is meant by "vote" and of course by "balance". Perhaps it could even depend on what the meaning of "is" is.
AV invites voters to indicate any and all of the candidates the voter wishes to support and the winner is the candidate who receives the most support. This is the simplest example of a score voting system. More generally, a score voting system asks the voter to choose, for each candidate, a numeric score to indicate a level of support. Typically, the permitted scores are consecutive numbers, often 1,",5 or perhaps 1,...,10.
With AV, possible scores are 0 and 1 though significantly, with AV the voter is not explicitly asked to choose one of those two numbers. This is a subtle difference in how the ballot is worded but that could have a psychological difference that affecting the vote. A likely way this might happen, for voters who are asked to choose one of the scores 0 or 1, is that a voter may fail to specify either of those values. This can be a a critically important possibility.
From the standpoint of who wins election, there is considerable freedom to choose the values to use as score. AV, when described as a score voting system, typically (and most intuitively) uses the scores 0 and 1 but it could just as easily use -13 and 810 and the winner would be the same. More generally, for any selection of scores, we can multiply all of the scores by any positive number and add any number at all to the available scores without changing the winner.
Balanced Approval Voting (BAV), when described as a score voting system is typically described using the scores -1, 0 and 1, but the same winner would be chosen using the scores 12, 14 and 16. Such a departure from convention seems pointless but probably confusing. But this freedom suggests a way to clarify what makes for balance by exploring voting systems that are just a little out of balance.
Score(12, 14, 16) is in balanced with the center of balance being the middle value, 14. On the other hand, Score(12, 14, 17) is just a tad out of balance because it gives an extra weight to voters who support a candidate. Likewise, Score(11, 14, 16) also puts its thumb on the scale but in the direction of opposition.
Now what about Score(-1, 0, 99999999)? That really puts votes of support in command, with opposition essentially ignored. For all practical purposes there is no operational difference between AV and Score(-1, 0, 99999999), the winner will be the same candidate. But notice that it will take 100 million votes of opposition to overcome just a single vote of support. From that perspective it is hard to understand why AV should be considered balanced.