Are The Balanced Single-winner Voting Systems Equivalent To One Another?

By “balanced” voting system, I mean one that conforms to https://www.equal.vote/theequalvote .

By “equivalent”, I mean they identify the same winner, given voter sentiment, once the voters have grasped one of their best strategies for the given prescription of the system, for a large electorate.

Do you know of any case where anyone has applied two or more differently-described balanced systems to an hypothetical electorate and found that the systems settled on different winners?

In regard to thinking about the range of different balanced prescriptions, I don’t know that I have any elegant and comprehensive system of subclassification to describe; however, I can mention at least a couple of ways that a system could (appear to) differ from Range Voting, taken as the paradigm of the category.

One variation is to restrict the inputs while keeping the tally the same. Two examples of this kind of deviation are “vote for and against” and “vote for or against”.

A second dimension of variation from Range as starting-point is to specify multiple rounds of tallying whereas standard Range has just the single round of tallying. Various rules can be proposed for deciding how many rounds to use and how many candidates to eliminate in each round.

A respondent has already alerted me (maybe in this very discussion forum) to the possibility that under systems that tally in more than one round (instant-runoff systems), a faction could gain unfair advantage by trying to promote from an earlier round to a later round a candidate who is not popular enough to win the election but who can help displace from the later round one or more candidates that that faction does not like.

A variation on multiple rounds is multiple pollings, or true runoff. This may suffer from the same risk as mentioned just above for instant runoff.

Is the following argument valid?

Suppose two prescriptions for constraints on the ballot and procedures for
tallying both conform to the balance constraint. Since they conform, it follows
that they provide equal power to the voters, one voter to another voter.
Suppose there is an hypothetical electorate (description of voter sentiment)
for which the two prescriptions settle on distinct winners. Let’s call the two
system prescriptions A and B. Since A provided equal power to the voters,
one to another, and B produced a different outcome form A, it follows that B
advantaged some voter and disadvantaged some other voter unfairly.
Therefor B did not provide equal power to the voters, one to another. However
this contradicts our assumption above that B would provide equal power to
the voters, one to another. Therefore, one of the assumptions must be wrong.
Either all balanced systems are equivalent, or the balance constraint does not
assure equal power.

Hmm, I don’t know but given this bit:

And that score and approval both become Condorcet methods if voters have the right knowledge and are fully strategic, your question could become “Are all balanced systems equivalent and specifically Condorcet systems?”