Several articles in this series introduced members of the balanced family of voting systems. Balanced approval voting (BAV) has received the most attention, but there have also been articles on
balanced plurality voting, instant runoff balanced voting, balanced ration voting, instant reverse ranked balanced voting, balanced randomized voting and balanced Condorcet voting. BAV deserved its special attention if only for its simplicity, but importantly because of its membership in another important family, that of evaluative voting systems. Evaluative voting systems ask voters to evaluate all the candidates according to the same rules. In contrast, plurality voting, asks the voter must choose only a single candidate to vote for while saying nothing whatever about any of the other candidates; ranked voting systems fail to be evaluative because, for example, the voter must choose just a single candidate to place first. Voting systems (such as BAV are both evaluative and balanced, and for that reason they avoid the spoiler effect. Moreover, they avoid giving an undue advantage to famous candidates like most other voting systems do.
Ranked voting illustrates a somewhat different sort of family that for convenience we call a nuclear family. What characterizes a nuclear family is that the ballot for one member can work as well for any other member of the nuclear family. For example, the most widely familiar of the ranked voting systems is ranked choice voting, formerly known as instant-runoff voting (IRV). But in another article, we introduced what arguably is a better way for counting the IRV ballots. And Borda voting, along with its infinitely many technical variations, are also members in good standing of this nuclear family. Members of this nuclear family may elect different candidates but at least the ballots used
can be identical. It should be noted, however, that there are deficiencies common to this entire family of ranked voting systems.
A natural question is whether there are other members of BAV's nuclear family. Are there other sensible ways we might tally BAV ballots? Certainly, we could just ignore the opposition votes as a way of using approval voting ballots in a BAV election. But a ballot for approval voting would not generally be suitable for a balanced approval election. Approval voting and BAV are both in the evaluative family and both are in the balanced family, but they are not in the same nuclear family. The question remains whether there are other members of the BAV nuclear family.
An affirmative answer to this question is suggested by a common practice on the internet. If do a search for some sort of product, you will likely find links to the Amazon web site. Noticing the ratings, you might decide to consider only the products getting a four or five-star rating. But you might temper your enthusiasm for even a rating of five by also taking note of the number of customers rating a product. It might be advisable to discount a five-star rating if that rating reflects the opinion of only a single customer.
By analogy, we might consider a modification to BAV that ignores candidates with fewer than a specified number of non-abstentions. The tally would be the same as for BAV, but a step in choosing the winner would be to reject a candidate with fewer than the required count of non-abstentions. The problem being addressed by this is, in essence, the stealth-candidate problem discussed in an article from 2022.
As another approach, we might observe that in a BAV election, the net vote for a candidate will be at most the number of non-abstentions for that candidate. And one might notice that this cap on the net vote is higher for famous candidates than for the less famous. Since the famous candidate can be expected to also have a greater number of opposition votes BAV compensates for this problem, but still, some may argue that the higher cap on net vote still gives the famous candidates an advantage. It is a judgement call, but some people could feel strongly about and this could motivate other variations in BAV's nuclear family.
Notice that while BAV determines the winner based on the net vote, it would be entirely equivalent for BAV to express the net vote as a percent of the number of the total number of voters (this percentage being simply the net vote divided by a positive constant). An alternative system could instead determine the winner of an election according to the candidate's net vote expressed as a percentage of the non-abstentions for the candidate. The effect would be to penalize the most famous candidates (though another effect would be to further aggravate the stealth-candidate problem).
There is no limit to the variations possible with this alternative approach. Instead of choosing the winner based on the percentage of non-abstentions, it could be taken as a percentage of any positive value computed from the number of non-abstentions. The formula for this could easily be designed to penalize candidates with very few non-abstentions (to avoid the stealth candidate problem) but at the same time the formula could be tuned to further depress the prospects of a famous candidate who is favored with so few abstentions.
So, just like ranked-choice voting, BAV enjoys an infinitely large nuclear family, but what is more important is that it (like many members of its nuclear family) enjoys membership in the two very important extended families that together mitigate against duopoly politics (being both balanced and evaluative). It is these smaller extended families that are (given what we know now) most appropriate for use by a democracy. BAV stands out in this family as its simplest member.
