Topics created by cfrank

@rob said in Are Equal-ranking Condorcet Systems susceptible to Duverger’s law?:

So what is that algorithm? I mean, it could be Condorcet, and I have no problem with that, but I can't see how the term "proportional representation" applies. It just sounds like multi-winner.... there isn't anything proportional about it.
The only way the term "proportional representation" would apply (by my understanding of the term) is if we assume that there are some number of parties, and each candidate and each voter is in one and only one party. If a 3rd of the voters are in the Bull Moose party, then a third of those elected should be in the Bull Moose party. The further you get away from that, the less "proportional representation" seems to be a meaningful descriptor.
All of my complaints regarding PR (and with so many people's insistence that it is so much better than single winner methods such as Condorcet methods) are based on the assumption that voter X below is considered to have "better representation" if d, f and e are elected (because candidate d is very close to voter X), than if a, b and c are elected.

For the algorithm, you will be aware of Single Transferable Vote, which gives PR without any mention of parties. If voters happen to vote along party lines, it gives party PR, of course. That's just one example, without having to mention obscure methods invented by people on this forum.

And as for a, b, c versus d, e, f, I discussed that in the other thread here. I'm not sure it's worth quoting though because it's quite long.

@toby-pereira Right.... but of course "the popular vote" isn't so meaningful with ranked ballots either.. I assume he means "the most people who picked a candidate as their first choice," and admittedly this has become associated with "popular vote" and even "democracy" simply because that's been how it's been done for so long.

Really, though, the term "popular vote" makes sense when contrasted with "electoral college." But not so much as it is being used here.

Steve Forbes did similar things with the thing he is most known for, the flat tax. Similarly, it sounds so nice and simple and logical as long as you don't actually think too hard about it.

@Andy-Dienes A thought occurred to me that makes me believe the sensitivity to Condorcet components is actually not odd at all.

The thought can be best illustrated with three candidates {x,y,z}. In this case, there are two kinds of balanced Condorcet cycles, namely (xyz+yzx+zxy), and (zyx+yxz+xzy). Individually, and even combined, these Condorcet components should indicate a tie by any reasonable deliberation, since the candidates are interrelated by perfect interchangeability.

For the same reason, a ballot set that is a scalar multiple of

(xyz+yzx+zxy)+A(zyx+yxz+xzy),

where A is a non-negative real number, should equally indicate a tie.

However, consider what would occur if one of the candidates, by new information that has a negligible effect on the relationship between the other two candidates, were discovered to be an invalid winner. Without loss of generality, we can say that candidate is z. It seems reasonable to simply remove z from every ranking in the ballot set, which would transform the ballot above into

(2xy+yx)+A(2yx+xy)=(A+2)xy+(1+2A)yx

At this point, it is clear that the preferable choice between x and y, conditional on the invalidation of z, can be decided from the information available in the combination of the two oppositely oriented Condorcet components. If 1>A we should prefer x, and if A>1 we should prefer y.

In other words, combinations of Condorcet components at least provide conditional information about the preferences between candidates. If this information is considered relevant, which I think it ought to be, then ignoring Condorcet components is improper.

Informally, while Condorcet components should indicate a tie, this can be considered due to a normative balance of potential marginal dissatisfaction among the voters. If some of this dissatisfaction is actualized as a sunk cost, then the margins are changed. In a sense, this is what is being analyzed by considering the pairwise elections—the normative marginal dissatisfaction of the electorate as a whole is being minimized when the Condorcet winner exists and is chosen.

This I think upholds the concept of the Condorcet winner as a canonical choice, given that one exists.

@cfrank Not long ago I posted this, I assume we are talking about the same thing? I usually call it BTR. Tldr: I like it.

https://www.votingtheory.org/forum/topic/230/bottom-two-runoff-condorcet-irv-hybrid

@robla yes definitely, I agree. The separation here is not explicitly by ballot type, but the space of voting systems according to the way the graphic is constructed seems to be roughly split into two camps along an axis, where systems on one end of the axis seem to have cardinal/score-like ballots while those on the other end tend to have rank-order ballots. Somewhere in the middle of those for example is STAR, which has characteristics of both possibly due to the way the ballot is operated upon. Then there seem to be roughly two other dimensions (according to my rough embedding), and those dimensions in theory might somehow characterize the algorithms.

This specific map though is independent of how the systems are actually used, so the similarities being depicted will probably have a lot to do with the mathematical structure of the systems compared with the degree to which the similarities relate to how the systems operate in practice. As @rob is addressing, it would be more interesting to create a map using practical criteria based on empirical data rather than (and/or in addition to) absolute criteria.

@andy-dienes in this example there are more scores than there are candidates. The same problems remains though when there are only three scores for three candidates, and I can see why this is a significant issue. With the method I described, no matter what the majority does with strict rankings, the middle candidate will win, even if the ballots are

A>B>C [99%]
C>B>A [1%]

Which is absurd. And without strict rankings, the majority can guarantee their top candidate’s victory by bullet voting anyway. I don’t like it but it is what it is. I think you’ve convinced me.

@cfrank Geez I hate being boring and just agreeing with @Andy-Dienes again, but.... yeah. Cardinal ballots (i.e. 0-5) I think are the easiest for voters. Higher resolution cardinal ballots are even better, where it is practical. It makes sense for the votes I am proposing we do in this forum, since it gets more information out of each voter. (0-10 with as many decimal places as you want)

But really, the best methods will just ignore the information that goes beyond ranking, or if they do use it, it is as a last resort.

I'm not overly concerned with bullet voting, as those voters are choosing to not use all the power granted to them. I wouldn't characterize it as a vulnerability, any more than people choosing not to vote, or choosing to equally rank candidates (if that is allowed).

As for what I think is most sellable to the public: probably ranked not cardinal ballots, simply because RCV is a thing. I think you could sell a Condorcet method while just calling it RCV. Bottom-2-runoff is good in that it is the smallest change to RCV. A simple version of minimax is good because it can be based on a pairwise matrix alone, which makes it most easily precinct summable.

Recursive IRV is.... well I'm kinda obsessing over it at the moment. But I won't say more at the moment until I understand it better.

@cfrank said in Condorcet with Borda Runoff:

Can you explain why it doesn’t make sense to try to elect a candidate that is not highly divisive whenever possible?

I personally think a method should indeed do that, but trying to label that a "supermajority" might be misleading or at best, vaguely defined. Supermajority what? Supermajority might indicate something like 70%, but the other variable is "degree of approval." So does it mean something like, over 70% thinks they are pretty good, as opposed to a simple majority where over 50% thinks they are awesome? If so, I would describe that as the system "favoring moderate candidates."

Regardless, I still suspect that a Condorcet compliant method will accomplish such a goal (favoring moderate candidates) better than any given method that isn't Condorcet compliant -- at least if you also want to avoid strategic manipulation and all that ugliness.

Notice that FairVote's primary complaint about Condorcet systems seems to be almost the exact opposite of yours. They think that Condorcet systems favor moderate candidates too much. (Then again, maybe you consider "moderate" to be significantly different from "supermajority support". I am considering them roughly the same, if I properly understand what you mean by "supermajority.")

@cfrank Then whichever party has more candidates than the other is advantaged. This makes the clone failures even worse

@andy-dienes I think in principle alternative dynamical rules could vary as much as different voting systems vary from each other already, and the benefits of utilizing adaptive procedures would need to be balanced against the detriments to complexity.

I defined one method before that uses past election data or predefined distributions to measure a formal construct of consensuality among the candidates, and then chooses the highest scoring candidate according to that metric. The distributions could be fixed, or could be updated to include or adapt to newer election data in any prescribed way that seems reasonable.

I think the specific kind of procedure would depend on the kind of information that is being utilized.

As an explicit example that I considered, for a score system I defined a candidate to be "(S,P)-consensual" if that candidate was scored at least an S by at least a P-fraction of the electorate. Then I defined the (S,P)-set of candidate C as the collection of all (S,P) such that C is (S,P)-consensual, and essentially measured the “size” of their (S,P)-sets.

The way of measuring the size used probability distributions for the fraction of the electorate that scored a random candidate at least any fixed score.

@rob I'm just going off of the definitions, no assumptions needed 🙂

Agreed that voters' ballots will likely change depending on the candidates in the race in Approval more with a higher probability than in most other voting rules.

@Toby-Pereira to answer your question I'm looking into it. Arrow's Theorem actually has quite a few (slightly different) formalizations, and it looks like what I said is technically not true for the version defined on Wikipedia since that one only allows (strict) linear orders, but I feel quite sure I saw a formalization where the domain was all (weak) linear orders. I will try to find it.

@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.

@cfrank yes, well I guess the problem can exist at a lot of different levels.

Individuals can certainly vote strategically, and effectively so, under FPTP. Under approval, they almost have to. (unless they truly do think in simplistic black and white "like" and "dislike" terms)

I am also concerned about forming parties and eliminating candidates through primaries (etc), which FPTP strongly incentivizes. To be honest, that is the biggest problem because it causes so much polarization.

I am less clear on how organizations could game it through computation, although I don't doubt it is possible. They certainly do that with gerrymandering, with a lot of sophistication.

@brozai I don’t think what I’m suggesting is the same as the Kemeny-Young method, though it may be related. I added another example, and changed some of my wording to be more clear. I’ll definitely look into those systems you mentioned, I know I like ranked pairs.