Why Frohnmayer Balance may be Unimportant without High Levels of Voter Information

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!

I think the claim is that Frohnmayer balance is necessary for a method to be “good”, not that it is sufficient. So coming up with an alternative version of a method to make it pass and saying that the method is now worse doen’t really demonstrate anything.

I don’t think it’s important that, within a given election, there will be ballots that cancel each other out. Just that casting the opposite ballot should cancel out the original.

Anyway, that all said, I don’t think this criterion is the be-all and end-all. There are lots of criteria that would seemingly be good for a method to pass, but you can’t get them all. For example, the participation criterion. On the surface, this seems like one of the most important of all.

  • In a deterministic framework, the participation criterion says that the addition of a ballot, where candidate A is strictly preferred to candidate B, to an existing tally of votes should not change the winner from candidate A to candidate B.
  • In a probabilistic framework, the participation criterion says that the addition of a ballot, where each candidate of the set X is strictly preferred to each other candidate, to an existing tally of votes should not reduce the probability that the winner is chosen from the set X.

And yet almost all methods fail this. The ones where you just add stuff up (plurality, score, approval, Borda) all pass, as do apparently Descending Solid Coalitions and Descending Acquiescing Coalitions. And in terms of methods that are any good, it’s score and approval.

On the surface, this seems much more important than Fronmayer Balance, but most people are willing to sacrifice it because they don’t want to be committed to score or approval. So when it comes down to it, Fronmayer Balance can’t be set in stone either.


@Toby_Pereira I’m not claiming that the method is now worse, I’m mostly saying that it’s basically the same as before. I can agree that Frohnmayer balance is perhaps a “necessary” condition, but what I’m saying is basically it’s an extremely weak condition. The reason I’m saying that is because I see Frohnmayer Balance being touted a bit as a standard for equality in voting, and in particular used to promote Approval or Range voting systems over other systems, whereas almost all other systems can be modified to satisfy the criterion with minimal deviations in behavior. Of course, the modification is somewhat complex, whereas Range and Approval systems satisfy the Equality Criterion without a complex modification, but if the modification doesn’t really change much about the system, then why should we even make the modification in the first place?

Basically, Frohnmayer Balance itself doesn’t actually alter the important characteristics of most voting systems, and is something that can reasonably be cast aside in favor of characteristics of those systems that are positive in order to reduce the complexity of the ballot and of the algorithm. Imposing Frohnmayer Balance is like using 3.1415926 for PI rather than 3.14159. Sure, it’s a better approximation, but who cares?

Definitely not the be all or the end all, or in my current opinion even the be much or end anything. I think participation is important as well and I understand that properties can’t all be satisfied at one time. All the more reason to try to discover which properties matter and which properties don’t.

Is there any (good) system that benefits from sacrificing Frohnmayer Balance?

Ranked pairs, at least to my knowledge, is a very good system that does not satisfy Frohnmayer Balance. I could be wrong, but I don’t know if its modification would gain much at the cost of added complexity, and if that’s true then in reverse demodification would lose little for the boon of reduced complexity. In general, who knows. There might be a fantastic system that doesn’t satisfy Frohnmayer Balance, and I don’t think the fact that it doesn’t satisfy that property is necessarily rational grounds for dismissal of the system.

In fact, it seems that Frohnmayer Balance can in some cases produce incentives for dishonesty and strategizing. For example, consider a ranked choice system. If voter 1 ranks four candidates as (A>B>C>D, D>C>B>A), and voter 2 would rank (D>C>B>A, A>B>C>D), but doesn’t want their vote to be canceled out, they might swap C and B and rank (D>B>C>A, A>B>C>D).

What, using margin of victory instead of winning votes? That’s not really any more complex.

Even by starting with Plurality (not my favorite by a long shot) and simply adding the ability to provide a negative vote in addition to the positive vote, balancing it massively changes things.

What it does (as I mentioned earlier) is eliminate the main detrimental effect of vote splitting. No longer is there an incentive to do strategic nominations to avoid vote splitting, because now both positive and negative votes are split. Compare here how the candidates on the right split the vote under Plurality (FPTP), while under Approval and For and Against, they don’t.

For and Against, because it is less expressive than Approval (especially with lots of candidates), takes a bit longer to find that optimum, game-theoretically stable equilibrium. But both of them, due to being balanced, find the right equilibrium. Plurality stupidly picks “a” (on the left) simply because it there aren’t other candidates near it drawing away the “for” votes.

I think there are better choices than either of these – ones that don’t have to go through this “equilibrium seeking” process (due to incentivizing paying attention to how others are likely to vote) – but still.

Even without “perfect information” or anything close to it (note that in the simulator, some voters have better information that others), balanced systems do way better. (you can notice this as the state at the beginning, before it has shuffled it all around as the savvy voters adjust their votes)

Everything else being equal, balancing is a huge improvement because it splits the positive votes the same as the negative votes.

I conceded that Frohnmayer balancing can have significant effects in systems where voters have high information. The plurality system lets voters achieve high levels of information, since typically there are few candidates and the ballots are extremely simple. A voter has no real trouble predicting what the ballot of their “opposite” will look like with high probability.

But it has significant effects even without voters having high information.

This is how it looks at the beginning, where everyone is just voting completely naively. In F&A, since they don’t know how others are going to vote, they vote for their favorite, and vote against their least favorite. Under Approval they simply approve the half of the candidates that they like best.

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Both of them do significantly better than FPTP. Approval does better than F&A simply because each voter supplies more information about their preferences.

But both of them would make strategic nomination (i.e. eliminating candidates due to concerns of vote splitting) extremely difficult if not impossible in the real world.

I don’t think it is possible for voters in a vote-for-one system and an understanding of the ideological tendencies of the candidates not to have intrinsically high information. Their “against” votes are very likely to serve their interests. I don’t think that’s always the case. I may be wrong.

I think the only way to be sure is to test the modification I described above on non-Frohnmayer balanced systems that require more information from their voters.

If you think voters will always have high information, I’m not sure your point then.

I think there are big differences between, say, a presidential election, and an election for city council. With the latter, there is a good chance voters have no clue who is likely to be a front runner. Likewise, in our recent vote for a domain name for the forum, no one really knew how others would vote. A good voting method shouldn’t expect that they do, in my opinion.

Regardless, my simulator (which I plan on putting on Codepen and making available for hacking and tweaking and adding new voting methods when I get a chance) tries to show both scenarios and everything in between. That’s why it animates the search for a Nash equilibrium. (it also has various variables you can adjust to specify how savvy voters are)

Even so, “high information” is a tricky concept, because it is a big feedback loop.

They won’t always necessarily have high information. For example, a ranked choice system with 5 candidates would require a voter to select one in 120 of the possible orderings to cancel out. Of course they can narrow it down, but as you mentioned there is a feedback loop. I agree that high information should not be expected.

Ok, so what is your point regarding Frohnmayer balance, then? I’m saying that everything else being equal, it is a significant positive.

It does not address the other issue, which is whether or not a system gives an advantage to voters who are best at guessing how other people will vote. That is simply a different issue. If you haven’t noticed from other threads, I have concluded that Cardinal Baldwin is probably about as good as you can get on that issue.

It is probably true that Cardinal Baldwin isn’t perfectly balanced, but I think it is very, very close. A ballot that is the exact opposite of another ballot should come extremely close to cancelling that other ballot out. But maybe not exactly.

My point is that the significance of the positivity is I think quite smaller in magnitude than purported for most alternative voting systems, and that the effect of this property can actually be measured to a degree if one accepts the modification I suggest as an indication of what a voting system “would be like” if it were to also be Frohnmayer balanced.

Alternatively, I am pointing out that there is a fairly trivial method to modify any system to make a “Frohnmayer balanced version.” My hypothesis is that the effect of this modification is relatively insignificant regarding election outcomes for voting systems with complex ballots. Maybe I’m wrong. But if my hypothesis is accurate, it shows to a degree that a lack of Frohnmayer balance is in many cases not sufficient grounds for the dismissal of a voting system.

Ok, well I think I can get behind that to a degree. I don’t think it is a black and white thing. As I said, my current favorite method doesn’t appear to be exactly balanced, but it seems to be very very close. That method also seems to always elect the Condorcet winner, if there is one, but I can’t prove it and suspect there are can be cases where it won’t, but they’d need to have razor thin margins.

I don’t agree with the approach of comparing voting systems by counting how many desirable properties they have, without regard for how significantly they violate them. Often, close enough is good enough, and the best method is the one that strikes the right balance between a lot of priorities.

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I totally agree with that. But in the instance of Frohnmayer balance, it may possibly be shown that the property itself is not very relavant in certain cases, regardless of the degree to which a system satisfies it. In other words, under certain circumstances, perhaps we have one fewer property to worry about at all.

If this is a reasonable model, in general I think universal comparative analysis of properties through modified versions might be a good way to determine the circumstances under which particular properties are relevant. That’s basically what For-and-Against is for Plurality, but it’s important to look at the other systems too to get a good indication of the general relevance of a particular property.

The trick to that kind of analysis would be to find some universal modifier that takes in a voting system and returns a “version” of that system that is guaranteed to satisfy the property we want to analyze. My intuition is that we got lucky with Frohnmayer balance, but maybe there are other properties that are also amenable to this kind of treatment.

For example, perhaps for the participation criterion, one could analyze a voting system where individuals are allowed to vote versus the same system where individuals are compelled to vote. In this instance my intuition is that high expressiveness in a ballot would cause the property to become less relevant.


When I was doing that simulator a year or so ago, I got kind of excited about the For and Against method, not so much as a real alternative, but as a teaching tool, to explain why FPTP is so broken. If you simply balance the “for” votes with “against” votes, suddenly there is no incentive for parties to eliminate candidates, and the number one issue disappears.

It’s still far from ideal. Then again, if the whole country switched over to For and Against tomorrow for every election, I’d probably get a new hobby because that is good enough for me.

So in that sense it is interesting. I don’t see the concept of “equal vote,” as pushed by that organization, to be the end all and be all, though. I prefer wording it differently, for instance: “let’s use a voting system that doesn’t force us into two opposing parties that do nothing but fight with one another, so we can end this nightmare and solve real problems.”

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I 100% agree. Although I do find the subject fascinating in its own right, and might still think about it.

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I see your modification as a little bit like a modification to make a method pass independence of clones by considering groups of candidates that are ranked adjacently on every single ballot. Yes, this method might now pass in some technical sense, but not in spirit.

This could also apply to your modification. By finding pairs of ballots that cancel out exactly, it would be ignoring ballots that almost cancel out. And if it’s quite an imbalanced method to begin with, it could still be left with large asymmetries.

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Some would argue that there are advantages in using winning votes over margins - I think it helps with the plurality criterion and makes it less vulnerable to favourite betrayal. So could Fronmayer Balance be worth sacrificing?