Back to Equal Weighting

@cfrank, you convincingly (in a mathematical sense) pointed out that a horrible voting system could still meet the constraint described at equal.vote/theequalvote . However, I suggest that the constraint can be used validly to compare voting systems that both belong to a class I will call "additive."
First Past the Post, Score (including Approval), Vote For and Against, Vote For or Against, and cumulative voting can all be tallied the same way. The differences among them lie solely in what restrictions are placed on the ballots. In each of these systems, a ballot that is valid in the system can be reduced to a vector of numbers where each index is associated to a candidate. The tally sums the vectors and selects the candidate having the highest total. I call these systems additive because the key operation is just a sum.
It is also possible to describe systems that use multiple rounds of tallying as additive or not in each round. Cardinal Baldwin (and its reduction STAR) and IRV fit this category.
As for comparative systems like Llull, they are not additive, however, I feel, without articulating why and how, that they belong to some class that makes the Frohnmayer balance constraint somewhat valid.

@jackwaugh yes I agree that once sufficient additional constraints are imposed on a voting system then it may be that Frohnmeyer balance attains a more substantial status. I think (based on my limited understanding) that it seems difficult to formulate precise constraints on voting systems that are not either very restrictive or else very unrestrictive.
If I understand your meaning, an additive system is one where each ballot is a fixed, realvalued vector, and the result is determined only by the vector sum of all of the ballot vectors. I think that's definitely a reasonable class of voting systems to consider. My issue with Frohnmeyer balance is that the construct was being used to dismiss systems that did not fall within such a restrictive framework.
I actually don't know anything about Llull, could you link me to that?


@andydienes said in Election security under IRV:
@jackwaugh said in Election security under IRV:
you cannot do so based on your preferences alone, but have to take into account your estimate of the stances of the other voters [...] For a system to be ethical, it must accord equal power to the voters.
And thus, for a system to be ethical, it must only allow two candidates ¯\_(ツ)_/¯
Incorrect inference.
I suggest that a system provides equality if it meets two conditions:
 Frohnmayer balance
 Additivity (broken down in one way or another, e. g. by candidate or by pair of candidates).
The system is additive iff all the decisions it makes depend only on sums over some interpretation of the ballots.
Score Voting (no matter what the range) is a simple system that meets both constraints.
Here is a more complex example:
Ranking ballots are collected, with equal ranking permitted.
A voter is permitted to write the special phrase "all others" in one rank. In this case, any candidates not named on the ballot will be in that rank, in effect.
If there is a candidate that the ballot omits to mention, and it does not employ the "all others" key phrase in any rank, there is assumed to be an additional rank tacked onto the bottom, which contains all candidates not explicitly mentioned.
Tallying proceeds in rounds. Each round eliminates a candidacy. When one candidacy remains not eliminated, that candidate wins the office.
Each round scores the candidates whose candidacies are not yet eliminated by a prior round (i. e. the candidates remaining in the running) by consideration of the top and bottom rank on each ballot from among the ranks that identify (implicitly or explicitly) any candidates who are still in the running. The score of a given candidate is the count of ballots that include that candidate on top minus the count of ballots that include that candidate on the bottom. Again, the "top" and "bottom" are read from the ballots in a way that ignores candidates not in the running.
Based on that scoring, the tallying procedure pinpoints the bottom two candidates.
Between those candidates, the one preferred to the other by more ballots continues on in the running to the next round, and the other candidacy is eliminated. 
Analysis wrt Additivity
There are three decision points in the procedure. One is where it is decided whether a winner has been found. Second is the decision between the bottom two, for which candidacy to eliminate, and third is the lessthan, equalto, or greaterthan decision used in sorting the candidates by score in order to find the bottom two. The choice of the winner just depends on the other decisions, so it is additive if the others are. The decision of which to eliminate from the bottom two is pairwise additive, because it can be decided based on a preference matrix, which is a sum over the ballots. The remaining decision is more complex, because it depends on conditions set up by the prior rounds.
Analysis wrt Balance
Given a vote, its antivote can be constructed by inverting the ranking, including any rank that is implied by the rules rather than expressed.
Returning to the Last Decision
It's the last decision I named, but the first to be executed.
Consider the first round of tallying.
Each ballot's contribution to the score for a given candidate is 1 if the candidate is in the top rank, 1 if in the bottom rank, and 0 if in one of the middle ranks (for analysis, we can ignore singlerank ballots on the grounds that these are abstentions). The score for each candidate is the sum over the scores given to the candidates by the ballots.
Consider a subsequent round of tallying.
The voters have had equal power in determining which candidates remain in the running, by induction.
The determination of the scores comes from the ballot ranks that still have candidates in the running in them. If they are looked at as modified in that way, the decision is additive over them. The voters have had equal power in determining this modification, so I say the subsequent rounds do not break the equality of power among the voters.

@jackwaugh said in Back to Equal Weighting:
I suggest that a system provides equality if it meets two conditions:
 Frohnmayer balance
 Additivity (broken down in one way or another, e. g. by candidate or by pair of candidates).
The system is additive iff all the decisions it makes depend only on sums over some interpretation of the ballots.
I don't think either of these criteria are useful, nor do I see why they should be required for "equality."
Any anonymous voting method can be trivially modified to satisfy Frohnmayer balance.
Likewise any voting method is "additive" by summing up # of ballots with ranking x, # of ballots with ranking y, # of ballots with ranking z, etc.
What's wrong with just using anonymity as a test for equality?

Compare FPtP to Approval, for example. Gandhi and King run to split the antiHitler vote. Under FPtP, it takes two million antiHitler voters to balance the power of one million proHitler voters. Therefore, the power of a single antiHitler voter is only half that of a single proHitler voter. FPtP violates the Wesberry v. Sanders decision, which specifies that the "weight and worth" of the votes must be "as nearly as practicable" equal.

@jackwaugh This example is evocative but not particularly useful for coming up with any kind of mathematical definition of equality.
I posit that anonymity is about as close as you can get to a formalization, unless you want to start measuring something like influence

mathematical definition of equality
I don't know whether balance implies equality, but for sure, imbalance implies inequality.
Vote splitting implies inequality. I don't know whether the converse is true, but suspicion and the precautionary principle would say to assume so until it is proven otherwise, and so prefer balanced additive systems.
The one reason to deviate temporarily from following strictly the signal of balance and additivity is as Rob B. says, a lot of mindshare has been taken up with Hare and so maybe we could get some quicker progress by marketing something that looks to the Hare enthusiasts like almost what they have been humping for, but where the seemingly minor modification we ask for would in our opinion ("we" being those not taken with Hare) work much better against vote splitting than Hare would.

@andydienes said in Back to Equal Weighting:
Likewise any voting method is "additive" by summing up # of ballots with ranking x, # of ballots with ranking y, # of ballots with ranking z, etc.
Rats.
The following construction is basically due to @cfrank; this tool NodeBB seems to make it hard for me to find the original post from him in which he gave the construction, but here is an equivalent one:
Ballots have two parts  a chooseone part and a matching part. The matching part is an integer from 1000 through 1, but not zero, or an integer from 1 through 1000.
Before the tally, the ballots are searched for pairs of matching ballots. They match if the sum of their matching parts is zero. Each pair of matching ballots is thrown out. If one ballot matches more than one other ballot, only the one and one of the matches is thrown out.
The tally then proceeds as for Chooseone Plurality FPtP.
The system conforms to Frohnmayer balance (@SaraWolk @Sass).
For practical purposes, the system clearly does not provide equality. There will be practically no matches and so the system reduces to ChooseOne, which clearly does not provide equality, because it splits votes.
Is the system additive? As you point out, it can be treated as additive by summing up the ballots having any given vote type.
So I don't know the mathematical definition that will restore equality to prominence.

@jackwaugh The existence of this kind of construction is precisely why I am so obnoxiously pedantic to everybody about how they are defining the terms they are using; great example.

@andydienes and rightfully so. Now how can we mathematically embody the intuition that throwing out pairs of votes, when compared to keeping all the votes and tallying them to arrive at scores (which may have pairwise indexing) that are sums over the votes, violates something?
[edit]
Suppose we have the system I described above, with the rule of matching ballots and throwing them out in pairs.
Let's say the candidates are Gore, Bush, and Nader and I vote Nader, 511. Let's account for the votes in terms of their effects on the pairs of candidates. In fact, let's ignore NaderBush and concentrate on NaderGore. What is the effect of my vote on the NaderGore accumulator? It has to be canceled by a vote of Nader, 511.