Among the articles in this series on balanced voting, the one that has received the most attention has been the recent one, What Could be Wrong with Ranked-Choice Voting?. That article is a survey of the earlier articles about ranked voting. No doubt this outsized attention is due to the fact that this system of voting, more properly called IRV (Instant Runoff Voting) has had such widespread promotion and appeal.
A topic that has had less, but nonetheless significant recent attention relates to AV (Approval Voting) and its relationship to BAV (Balanced Approval Voting). There are six articles that address this relationship in some detail, so it seemed useful to me to write a guide for those who want to explore this other topic; this is the purpose of the current article you are reading.
Starting at the beginning would be the article, What Might be the Best Voting System?, that defined both AV and BAV. AV is the voting system that asks the voter to simply mark which of the candidates the voter supports. The candidate with the highest support count wins election. BAV is the balanced version of AV which instead asks the voter to mark not only which candidates the voter supports, but also which candidates the voter opposes; the net approval (support count minus opposition count) is computed for each candidate and the candidate with the greatest net approval wins election. Important to BAV is that, as with AV, the voter is free to skip over (mark neither approval nor disapproval) for any number of the candidates.
In another article, Neither Positive nor Negative but Balanced Voting, I defined a balanced voting system as any voting system that gives voters an equal opportunity to vote for or against any particular candidate. That definition raised some confusion that oddly seems to stem from a misunderstanding of what an equal opportunity might be. In any event, a body of opinion seemed to grow that AV was balanced because with AV, the voter can express disapproval of a candidate by expressing approval for all of the other candidates. To be fair this overly simplifies their argument, but it the essence of it. This confusion about definitions was addressed in the article, Approval Voting and Confusing Language and again in the later article, What's Wrong with Approval Voting.
The mistaken view that AV is in fact balanced persisted, however, resting on a somewhat mathematical argument that is detailed in a more recent article, Opposites. This article lays bare the error in that argument, which shows how the argument in question resembles a magic trick. Specifically, the attention of the audience is distracted (in this instance with talk of mappings and preservation under those mappings), the real action (of altering the meaning of ballot check-boxes) is easily missed. While in performance of a magic trick, we give the magician credit for being fully aware of the deception, in this respect the analogy breaks down; it would seem that it was likely the magician who was fooled by his (or her) magic trick.
The next article, A Voter's Quandry, provides an example to show, from a voter's perspective, how very different AV (along with that change in the labels of check-boxes) actually is from BAV. Making Choices, makes this same point, but less explicitly and from a different perspective.