Here is the definition of Frohnmayer balance according to Equal.Vote:
A voting system has counter-balances if "for each possible vote expression that one voter may cast in an election, there exists another expression of the vote that another voter can cast that is in balance, such that the outcome of the election is the same whether both or neither votes are counted.
A voting method passes the Equality Criterion if every possible vote expression has a counter-balancing vote expression and if the counting system produces the same election outcome when any pairing of a vote expression and its counter-balancing vote expression are added to the tally."
This may seem reasonable and important, but actually it is almost trivial due to the implicitly restrictive scope addressed by the definition, and In fact, we can produce a modified version of any voting system that almost trivially passes the Equality Criterion by suitably defining what it means to “count” a vote and for a vote to be “added to the tally.”
Here is how. Take your favorite voting system. Now modify it by allowing each voter to submit two ballots in an ordered pair instead of one, along with an indication of which ballot is to be “positive” and which is to be “negative.”
Allow as many negative ballots to cancel out with matching positive ballots as possible. Then, ignore the remaining negative ballots and proceed with the election considering only the remaining positive ballots. Or perhaps more simply, check to find “balanced pairs,” of the form (+A,-B) and (+B,-A), and cancel them. Proceed with the election considering only the positive parts of all unignored votes.
This method has counter-balances, because if one voter submits +A and -B, then another voter can submit +B and -A. The ballot includes the positive and negative parts, so these are both just “vote expressions” that cancel each other out. The two will cancel out as per the election process, so indeed the result will be the same whether both or neither votes had been “counted,” because of the way the modified algorithm functions.
However, this modification is trivial and does not truly grant much favorability for the new voting system over the old, especially for systems with complicated ballots, since it becomes less likely for a negative ballot to match a positive one, unless voters have nearly perfect information. Such a modification might significantly affect plurality voting, but can’t possibly have much of an affect on many other superior systems.
I imagine that working toward the spirit of the definition might involve imposing a metric on ballots and determining degrees of cancelation, categorizing ballots into “types” and allowing voters to cancel out one ballot of a particular type rather than a specific ballot, or allowing each voter to submit many “negative” ballots depending on the complexity of the ballots of the original method. That all seems arbitrary and complicated. Perhaps one could require the ballot space to be a topological Abelian group, but again, what is the topology? What is the addition? How do we evaluate a ballot sum into an election result? Maybe you like the idea of voting systems that conform to a canonical topological Abelian group, like Range voting does, but that doesn’t mean they are superior in practice to systems that don’t. In fact, more likely they have their own inherent restrictions, pros, and cons.
Anyway, Frohnmayer balance as it is currently defined is clearly not restrictive enough to achieve in itself what is purported, i.e. “Equality.” Instead it seems to be a low-bar, ad hoc abstraction of a specific class of systems that turns out not to be very useful for evaluating voting systems in general, or maybe I’m missing something. Your thoughts, rebuttals, and/or qualifiers, if I might ask. Thank you!