In conversation after conversation about the single-winner context, my interlocutors express doubt in the importance of balance.

In defending my position that balance matters, I invite you to join me in analyzing how Choose-one Plurality Voting empowers the ruling class. In talking about such an analysis, I will highlight a role played by lack of balance. I contend that this role is central to the mechanism.

Background readings:

The "Cash3" document by Warren D. Smith, Ph. D. "The Equal Vote" by probably substantially Mark Frohnmayer and his father. The "Prisoner's Dilemma" thought experiment and paradigm for real experiments, from Game Theory.

[More later.]

@Marylander said in Approval vs. IRV:

[IRV] will strike out everyone but the top two, then pick the most preferred candidate between the two.

The current system does that just fine.

@BlainCellars

An accurate system yields high voter satisfaction efficiency and/or Bayesian regret. It is not so much about getting the right winner given the ballots but about getting this right winner in the case of strategy. @SaraWolk can point you to the VSE research showing how different systems do.

Equal and honest are also about strategy resilience.

Some of the other criteria are more straightforward like being expressive and simple. Those have an inverse relationship.

@Jack-Waugh It would depend on how incomplete rankings were dealt with, but the major parties would probably try to run many candidates to make it more difficult to eliminate them, and try to coordinate the bottom of their supporters' ballots to try to control the order in which their opponents are eliminated. Major parties would still have the advantage of just having more resources to communicate strategies to their supporters. On the other hand, they would face a disadvantage of being more prominent targets for opponents to try to eliminate early, and if these opponents are coordinated, even running many candidates won't help. However, if the major parties agree to coordinate with each other to keep minor parties from being competitive (or even just specific minor parties), they could make it very difficult for those parties.

Major parties could also use the fear that they will be eliminated first if they do not reach a majority to discourage defections to minor parties.

@Ted-Stern said in Preference Approval Sorted Margins:

I would have thought that anyone interested in ranked methods would know the basics of Condorcet.

I have no interest in Ranked methods. Sorry

@Casimir said in ICA/ICT/SICT - So what about "improved Condorcet" methods?:

Why would that be preferred? And does this only work when explicitly ranking/approving both, or also for bottom ranks? In that case one could state the rule as just: "A candidate is unbeaten if there is no other candidate that is ranked higher by more than half of all voters." Which also would be easier to calculate and visualize.

If ties count in favor of both candidates when they are compared, then at the bottom ranks, voters would want to make up a ranking for those candidates just because it would hurt whoever is put lower in the ranking.

Generally, this system reminds me of a system I thought about a while ago but I don't know if I wrote about it anywhere. (The thinking behind them is basically the same.) It works like Minimax Condorcet with co-equal ranks, but there is an approval-like cutoff in the ranking. For tied candidates above a ballot's cutoff, the ballot counts for both candidates in that pairwise matchup. For tied candidates below a ballot's cutoff, the ballot counts against both candidates in that pairwise matchup.

Ultimately the thing that I really don't like about that system is that the notion of a coequal ranking is deceptive: coequal rankings sound like they are intended to indicate that you support both candidates equally, but the system is designed so that they really serve a strategic purpose.

It's easy to show that MJ fails the opposite cancellation criterion.

2: A/Excellent, B/Very Good, C/Reject
1: A/Good, B/Very Good, C/Reject

Medians: A/Excellent, B/Very Good, C/Reject
A is elected

We can add a pair of opposite ballots to change this result.

2: A/Excellent, B/Very Good, C/Reject
1: A/Good, B/Very Good, C/Reject
1: A/Good, B/Excellent, C/Poor
1: A/Acceptable, B/Reject, C/Very Good

Medians: A/Good, B/Very Good, C/Reject
B is elected

Demonstrating that MJ fails the cancellation criterion is a bit more difficult since we must consider all possible cancelling ballots for A/Good, B/Excellent, C/Poor.

2: A/Excellent, B/Very Good, C/Reject
1: A/Good, B/Very Good, C/Reject
1: A/Good, B/Excellent, C/Poor
1: A/?, B/?, C/?

Medians: A/?, B/Very Good, C/Reject

Luckily it's possible to construct an election where B and C's medians are independent of the final ballot (as I did here), so we only need to consider the possible ratings for A. An Excellent rating will lead to A keeping their median rating of Excellent and winning, but no other rating will. A Very Good rating creates a tie between A and B which is broken in B's favor, and anything lower leads to B winning as well.

Now all that remains is to find another election in which A/Good, B/Excellent, C/Poor cannot possibly be cancelled out by a ballot with an Excellent rating for A.

2: A/Poor, B/Acceptable, C/Reject
1: A/Good, B/Acceptable, C/Reject

Medians: A/Poor, B/Acceptable, C/Reject
B is elected

2: A/Poor, B/Acceptable, C/Reject
1: A/Good, B/Acceptable, C/Reject
1: A/Good, B/Excellent, C/Poor
1: A/Excellent, B/?, C/?

Medians: A/Good, B/Acceptable, C/Reject
A is elected

So MJ fails both of these formalizations of Frohnmayer balance.

@masiarek

Here is my link https://docs.google.com/document/d/1ASC5BS10rCfAYZWGeCyS7dKdKc4p5wwI6DHs4F7ScGc/edit

@Jack-Waugh My apologies. I did not notice that your question about a voter favoring both front runners was directed to @Rob. I mistakenly thought it was directed to me based on a strategy WDS proposed in his paper for IRV simulations.

The WDS strategy assumes the two front runners are significantly distinct and any voter tempted to vote strategically would have strong preferences. If I recall correctly, the WDS strategy for IRV (Strategic Hare STV) was to give the maximal rank to the better of the two front runners and the worst rank to the lesser front runner, then use a moving average to fill out the ballot by giving the highest remaining rank to the next most likely alternative that was better than average and the lowest remaining rank to the next most likely alternative that was worse than average.

That strategy is certainly codable for a simulation, but it also seemed like a direct lead-in to your question about a why voter that liked both front-runners would vote strategically, and framed the question in the real world. Hence my response. Hopefully, it is at least an interesting scenario.

@Marylander Yes agree completely that it makes more sense to make it more difficult to trigger them in the first place. The whole thing needs a rethinking, it doesn't seem to be designed for our highly polarized world.

Of course, if any of the methods advocated in this forum were used to elect the governor in the first place, there would be far less likelihood of there being a motivation to do a recall.

@Jack-Waugh you’re right about For-and-Against, the modification I suggested is much less significant because it requires the pairs to match exactly, whereas For-and-Against counts all the positives and all the negatives. Just another example of how arbitrary the concept of “balance” in this formalism is. I liked your strengthening of the argument, you’re absolutely right about that as well, because it virtually eliminates any level of information voters might have about other ballots.

Maybe there is something more tangible that F-balance is trying to point to, but it isn’t clear what it might be. In my opinion, with respect to the concept of balance, the strengthened construction should by all reason send equal.vote back to the drawing board, especially if they can’t even formalize a valid counter-argument.

@cfrank said in If there are only two candidates, could FPTP be improved upon?:

Do you really think it’s a better idea to have peanut butter sandwiches than to have fruit cups?

Food allergies are a special case where we "override democracy" for some greater ideal. That's very different and I don't think can be accounted for by voting.

Should I be able to claim an allergy (real or metaphorical) to a certain candidate? Claim that if that candidate gets elected, it will cause me to get physically sick or otherwise is simply intolerable? I think we'd have complete chaos, and it would no longer be democracy if such absolutes can be allowed. If someone is on the ballot, they should be considered a viable candidate, full stop.

I think it’s clear that empathy is necessary here to come to a sociable solution

I wish everyone was fully empathetic and cared for everyone else, rather than being selfish. Meanwhile my daughter wishes that unicorns were real and that cotton candy rained from the sky. 🙂 Seriously, though, that just sounds like wishful thinking, and the realization that that sort of thing doesn't work for large numbers of people is why voting was invented in the first place.

I'm all for systems that bring people together rather than driving them apart. There are ways to align people's interests with the goal being social harmony. But assuming they are that way to begin with? I don't see how that is going to work at all.

I have suggested one before involving allowing voters to send in ballots, making the candidates anonymous except for their score distributions, and then having the electorate vote on which candidate to select according to those score distributions only

Ok I missed that one but I don't understand it from your short description. Is this two votes? I don't understand why people would vote on score distributions when they don't know who the candidates are. Seems more efficient, and just as good, to just agree upon a formula for it ahead of time, so you don't spend millions of dollars bringing people in just to vote on how to resolve votes, for every single election. It sounds like all that would do would be make it more complex, and obscure any potential bad effects from being directly analyzed. But mostly, who is going to want to go in and vote on anonymous score distributions? I also don't see it having a positive effect with regard to the food allergy example, it's not like people would see someone giving a particularly low score to PB&J (not knowing that the choice is PB&J) and conclude that they must have an allergy, rather than just being picky.

Also, are you suggesting it would be better than majority vote for a simple binary choice, i.e. two candidates? It doesn't seem like you could vote on score distributions for two candidates, so if not, this really doesn't apply.

@jack-waugh said in STAR vs. Score:

But systems that lead people to think (correctly or not) that strategy in nomination is no longer helpful to their cause, will likely lead to races having more than three candidates.

You are right that STAR isn't as good with more than three candidates.

What it comes down to, from my perspective, is that to reduce the incentive to strategically exaggerate (*), you need to minimize any reliance on "strength of preference."

STAR does this by doing a pairwise comparison as the last step. A pairwise comparison by its nature doesn't consider strength of preference (as you can see when it is a 2-candidate race in simple FPtP).

Condorcet methods try to do it all with pairwise comparisons. This reduces incentive to exaggerate even further. But since we can't guarantee there will be a Condorcet winner, we'll never get it to zero.

However, my position all along has been that getting it all the way to zero would be nice, but isn't necessary. If you get it close enough to zero, attempts to be strategic will have just as much chance of backfiring as they have of helping. A Condorcet method, including one with a very simple "tie breaking" formula, is good enough. STAR may or may not be good enough. Score is not good enough. Again, this is my opinion, but I it does come from a pretty solid game theoretical foundation.

* technically, incentive to exaggerate isn't the only thing we are trying to reduce. We also want to reduce vote splitting, which creates the incentive to strategically nominate, which in turn causes partisanship and polarization. Finally, we also want to aspire to "one person one vote", so each person has equal voting power. All of these things are accomplished by reducing the consideration of "strength of preference" in the tabulation.

@cfrank regarding Condorcet methods, some professors I've been in contact with did a series of three works on the methods "Split Cycle" and "Stable Voting" you may like to read.

https://arxiv.org/abs/2004.02350
https://arxiv.org/abs/2008.08451
https://arxiv.org/abs/2108.00542

Regarding multiwinner rules with ranked ballots there is a nice analysis here

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7089675/

One of the main takeaways of that one is that choose-one is actually not too bad when there are multiple winners (although as we know, it is quite bad for a single-winner election).

@brozai that’s perfectly OK. I was not trying to rationalize the product scoring method, although it is not difficult to do. I was just trying to investigate this line of reasoning and was pleasantly surprised that the product scoring scheme came out naturally. Being in the 27th position is not even well-defined in every election, nor even if so is it necessarily a positive event.

I wasn’t trying to select a winner based on whether they best explain the ballots, the positional score value log(K) literally is just the Shannon information content of the event that a candidate is in the Kth slot given already that the candidate is not in any higher slot.

This means that the score given to a candidate under the product score scheme is essentially the amount of information that is lost in trying to determine the position of the candidate in a sequence of ballots, assuming that the candidate is uniformly distributed across the positions rather than assuming that the candidate is always found in the highest available position. More to the point, if we divide by the size of the electorate, then as the number of voters increases, it also converges to the average number of slots in a random ballot that will not need to be checked in order to determine the position of the candidate if we always make that assumption that they are in the highest unchecked slot, starting from the highest position and moving down by one slot if the candidate is not found.

I definitely think that a concave scoring scheme is important so that compromises are emphasized. The logarithm accomplishes this and has this information theoretic translation so that at least the scores are theoretically meaningful in one sense.

Just as an illustration of this effect, suppose we had just two ballots and three candidates a,b,c, and let the ballots be

c<b<a
a<b<c

Under a standard scoring scheme, all of the candidates are tied with a total of 4. Under the product scoring scheme, in contrast, b beats a and c with a score of 4 compared with scores of 3, arriving at the only reasonable compromise available from these two ballots.

@brozai said in Approval is Sincere:

I think it might be a little stronger than just Monotonicity?

Yes, I think you are right

@culi If you are ranking them, I don't see what added benefit separating them into two lists makes.

Approval voting does indeed force you to make this binary choice. For simplicity's sake, you can word it "those you want and those you don't want", but a sophisticated voter probably won't (or shouldn't) see it that way, since the real world is neither so binary nor so absolute.

For me, "approve" might mean "who do I think is better than who I think is most likely to win?" or something like that. Which unfortunately burdens me with trying to keep track of polls or otherwise estimate who is likely to win.

That is still a binary, of course -- if the election is approval, you have no choice but to see it as a binary. But you don't need to see it as an absolute measure of "want" vs "not want", but instead can see it as relative to other candidates (typically seeing candidates that seem to have little chance of winning as being less relevant to your decision).

The point is, once you've ranked the candidates, you have given all the information needed. Deciding where the "approval threshold" should be provides no meaningful additional information, and simply adds a burden to voters -- guessing who will be front runners -- when they really shouldn't have to worry about that.

In case the above isn't clear enough, here is an example. Say there are ten candidates, A-J. I might rank them as follows:

C>H>F>B>E>J>I>D>G>A

Now you ask me to split that into two lists, for instance:

Approve:
C>H>F>B>E

Disapprove:
J>I>D>G>A

Assuming I am a sophisticated voter that is thinking about strategy, there is one and only one reason for deciding to split the lists between E and J: my estimate at the chances of various candidates to win. For instance if I think E and J will likely be the front runners, that would be a natural place to split the lists.

But that has nothing to do with my preferences, only my predictive skills. And I can't see why anyone would want that to be a factor.

@cfrank Ok, I guess I can't disagree with that, especially since I consider this all hypothetical anyway.

I've just been looking through old threads and found this one.

@nevinbr said in Least-bad Single-winner Ranking Method?:

Objectively, the answer is Game Theory Voting. However, it is basically impossible to explain to, well, anyone.

This method was also discussed at greater length on the old CES Google Group here. But the point is that while it is optimal in the sense that you might want to optimise this one particular thing, you might not want to optimise that one particular thing, so it would be wrong to say that it's objectively the best method.

In fact, when you look at what it is trying to achieve, it becomes quite clear that it's really just a bit of game theory fun rather than a method seriously trying to optimise something that a human would want to optimise. And you can see that by an example that you gave in the CES thread.

To expand on that last point, consider the following election with 3 candidates and 100 voters:

49 ABC

48 CAB

3 BCA

This has a Condorcet cycle:

A beats B by 94

B beats C by 4

C beats A by 2

At first glance, we notice that A has the largest victory and the smallest defeat, as well as the highest Borda total. However, the GT method elects A just 4% of the time.

Can that really be optimal?

Let’s think about it.

Okay, B gets crushed by A and barely squeaks by C. We don’t want to get crushed because that is bad for our long-term average, so B probably should not win. And if B loses (or didn’t run at all) then C defeats A.

It seems that A’s victory over B is mostly ephemeral. As much as we would like to score that +94 to improve our long-term average, the only way it happens is if we pick A and the system we’re up against picks B. But since B shouldn’t win, we expect the other system won’t pick B either.

In particular, if we usually pick A and the other system usually picks C, then we are going to lose frequently. We would rather be that other system and pick C most often, which is exactly what GT does. The optimal distribution is:

A wins 4%

B wins 2%

C wins 94%

Moreover, in any 3-way Condorcet cycle, the probability of each candidate winning is always proportional to the margin of victory between the other two candidates. And this is provably optimal.

In the example ballots, it seems clear that A is the best winner. However, under this method, because B definitely isn't a good winner, other methods won't select B so the A>B pairwise win might as well be ignored. Better concentrate on A v C instead and since C wins that, C is the overall best pick.

So what this method does is not pick the candidate that is somehow judged to be best for society, but the candidate that has the best average margin of victory against a candidate picked by another method playing this same game. And this method would never lose on average against another method (though obviously might tie - e.g. against itself). I don't see how this optimality relates to real life at all and why it would be good for us to adopt it.

To make this clear, the ranked pairs method is a Condorcet method that a lot of people like. But I could devise a method that elects a candidate that pairwise beats the ranked pairs winner whenever one exists. Otherwise elect the ranked pairs winner. According the metric used here, this method is better than ranked pairs - when these two methods are viewed as the only choices at least. But is this method better tban ranked pairs by any reasonable measure? Of course not.

An interesting academic exercise, but nothing more. Certainly not objectively the best single-winner voting method.

@jack-waugh

That "generalized RCV" should just be called "RCV".

I do not consider it to be terribly forthright of FairVote to quietly relabel the method they were promoting from "IRV" to "RCV". There are two disingenuous motives displayed.

First, around 2010, it wasn't just Burlington repealing IRV, there were some other places that did, even though they did not have the failure that Burlington had in 2009. "IRV" was a label losing cachet and, without changing a thing about the product they were selling, they just changed the label, implying that it might be "new and improved" in some manner or just not the IRV they were selling before. Of course that is not true.

Then the other disingenuous motive is that FairVote is implying that Hare RCV (or IRV) is the only way to tally the ranked ballots. Like Borda or Bucklin or Condorcet are not RCV even though they use virtually the same ballots (Condorcet allows equal rankings on a ballot).