[EM] Novel Electoral System

[Chris Benham]

Daniel,

In common with the pro-Borda minded, you seem to be making the 
completely unjustified assumption that the multiple candidates are more 
or less equidistant from each other in "issue space" and so it's ok 
infer some maybe-sincere ratings from rankings.

> Imagine an election with 26 candidates, A, B, C... Z, and 1 million 
> voters. Let us suppose that candidate A is unanimously preferred to 
> every other candidate, 1,000,000 to 0, except for candidate Z, to whom 
> she loses by 2 votes, 500,001 to 499,999. Meanwhile, candidate Z beats 
> every other candidate by the same 2 vote margin, and is thus very 
> narrowly a Condorcet winner. Does it really reflect the will of the 
> majority better to declare candidate Z the winner because of his 
> extraordinarily narrow margin over all of the opposition when 
> candidate A is the unanimous favorite versus everyone but Candidate Z, 
> to whom she barely loses?

Yes. The other (related) thing you seem to have in common with them is 
obliviousness to Clones.  Like Borda, I am sure your method fails Clone 
Independence (but maybe less egregiously).

With only ranking information, other things being equal, I don't see any 
justification for rejecting Condorcet (or Smith).  Based on positional 
information we can sometimes guess that some other candidate is higher 
Social Utility, but probably not in a way that is reliable or isn't very 
vulnerable to strategy.

For example say this is an election for a seat in the House of 
Representatives in Australia which uses compulsory full strict ranking Hare:

49 A>C>B
48 B>C>A
03 C>A>B

The A and B supporters very likely only ranked C because they were 
forced to fully rank.  C is the voted Condorcet winner, but it would 
never cross anyone's mind in Australia that C should be the winner.  In 
some places C might have struggled to get on the ballot, and in 
Australia would be in danger of forfeiting his or her cash deposit (for 
not getting a high enough percentage of the "primary vote".)

If truncation was allowed (as I think it should be) then most likely the 
A and B voters would have truncated and A would be the voted CW  (A>B 
51-49,  A>C 49-3).

But aside from that by far the main reason why a method that fails 
Condorcet might be acceptable is that the Condorcet criterion is 
incompatible with other criteria that people like.  Because Condorcet is 
incompatible with Later-no-Help all Condorcet methods are vulnerable (to 
varying degrees) to Burial strategy.

It's a bit like comparing two engines, one performs perfectly with clean 
pure fuel but very badly with dirty impure fuel and another that doesn't 
do quite as well with clean fuel but copes quite a bit better with dirty 
fuel.  I'm thinking of course of the comparison between a Condorcet 
method and Hare.

Chris

On 19/05/2025 9:16 am, Daniel Kirslis via Election-Methods wrote:
> Hi r b-j,
>
> Thank you for this response. I want to address both of your principles.
>
> First, is "one person, one vote". I of course agree completely that 
> each individual's vote should be treated exactly equally, and the 
> K-count does this. You say that "for any ranked ballot, this means 
> that if Candidate A is ranked higher than Candidate B then that is a 
> vote for A... It doesn't matter how many levels A is ranked higher 
> than B, it counts as exactly one vote for A." This is precisely how 
> the K-count works - if A is ranked above B on one ballot, then A 
> advances by one along the 'preferred to B' axis. The number of 
> rankings between them is immaterial to A's position vis a vis the B 
> axis. However, if A is ranked above other candidates on that ballot, A 
> will also advance along those candidates' axes, so it is perhaps not 
> exactly "one vote". But each voter's vote has the same potential power.
>
> To your second principle. You say "I cannot understand why, *if* a 
> Condorcet winner exists, how *any* other method; Hare, Borda, Bucklin, 
> or Kirslis is more democratic than Condorcet." Let me give an example 
> to illustrate, which relates to the principle of majority rule.
>
> Imagine an election with 26 candidates, A, B, C... Z, and 1 million 
> voters. Let us suppose that candidate A is unanimously preferred to 
> every other candidate, 1,000,000 to 0, except for candidate Z, to whom 
> she loses by 2 votes, 500,001 to 499,999. Meanwhile, candidate Z beats 
> every other candidate by the same 2 vote margin, and is thus very 
> narrowly a Condorcet winner. Does it really reflect the will of the 
> majority better to declare candidate Z the winner because of his 
> extraordinarily narrow margin over all of the opposition when 
> candidate A is the unanimous favorite versus everyone but Candidate Z, 
> to whom she barely loses? Many more preferences are violated by 
> choosing the Condorcet winner in this case than choosing candidate A. 
> This is the heart of the issue with the Condorcet winner criteria - if 
> a Condorcet winner exists, a Condorcet method *must* completely ignore 
> the size of the margins of victory, no matter how large. In my view, 
> this curtails the meaning of 'majority rule' in a way that feels 
> undemocratic.
>
> I am not familiar with the Burlington election that you reference, and 
> I will look into it when I have a chance. I don't know what the 
> K-count would decide in that case. But I can try to answer in 
> principle your question "How *possibly* can Candidate B be elected 
> without counting those 3476 voters' individual votes a little more 
> (like 17% more) than how much the votes were counted from the 4064 
> voters preferring Candidate A?" In the K-count, for Candidate B to be 
> elected in this scenario, there would need to be a 3rd candidate (or 
> multiple other candidates) to whom B was widely preferred but A was 
> not. So, B would win because the people who favored A still preferred 
> B to C, while the people who favored B preferred C to A. If you only 
> look at the head-to-head votes of A vs. B, this seems 
> anti-majoritarian, but the point I make in the paper is that you 
> cannot**make valid inferences by decontextualizing the data like that, 
> as doing so can lead you into the logical contradiction of a Condorcet 
> cycle. It is in the very nature of multi-option preference aggregation 
> that the data cannot be decomposed in this way. Another way of 
> thinking about this is - suppose that while A is preferred to B, B is 
> preferred to C, and C is preferred to A, so you have a classic 
> Condorcet cycle. Then, someone must be declared the winner, so in your 
> reasoning, someone's votes will be counted for more than someone 
> else's. And, when resolving this issue, most Condorcet methods will 
> look at the margins of victory, even though they are ignored in the 
> case when a Condorcet winner exists. But if margins matter enough to 
> decide a winner when no Condorcet winner exists, why is it okay to 
> completely ignore them when a Condorcet winner does exist?
>
> The K-count is a way of trying to reconcile Condorcet's conception of 
> majority rule, which looks for majority in terms of each head-to-head 
> matchup, with Borda's conception of majority rule, which seeks to 
> honor the maximum number of individual pairwise preferences.
>
> Thanks again for your response, and thank you for looking over the 
> paper. I appreciate your civil tone and good faith questions, and I 
> hope it is clear that the discussion here is made with full respect 
> and in a spirit of friendly intellectual inquiry. And I welcome your 
> response to these arguments!
>
> I am also considering the questions from other folks and am working on 
> responses to those as well.
>
> On Sun, May 18, 2025 at 3:30 PM robert bristow-johnson via 
> Election-Methods <election-methods at lists.electorama.com> wrote:
>
>
>     Hi Dan,
>
>     I made one pass through your paper, but the interaction with Chris
>     and Andy was helpful.  I understand the definition of your K-count
>     measure, but still don't understand the motivation of it, solely
>     from the POV of democratic principles, which is where I draw my
>     Condorcetist perspective.  Admittedly, I am a hard-core Condorcet
>     advocate, but I am so because of some basic principles.
>
>     I read your section 9 and re-read it, and I still cannot get past
>     how it justifies *any* non-Condorcet method (including your
>     K-count) over Condorcet.  The principles of free and fair
>     elections in a democratic context require, among other things,
>     that our votes are valued equally:
>
>         1. "One person, one vote": Every enfranchised voter has an
>     equal influence on
>         government in elections because of our inherent equality as
>     citizens and this is
>         independent of any utilitarian notion of personal investment
>     in the outcome. If I
>         enthusiastically prefer Candidate A and you prefer Candidate B
>     only tepidly, your
>         vote for Candidate B counts no less (nor more) than my vote
>     for A. The
>         effectiveness of one's vote – how much their vote counts – is
>     not proportional to
>         their degree of preference but is determined only by their
>     franchise. A citizen with
>         franchise has a vote that counts equally as much as any other
>     citizen with
>         franchise. For any ranked ballot, this means that if Candidate
>     A is ranked higher
>         than Candidate B then that is a vote for A, if only candidates
>     A and B are
>         contending (such as in the IRV final round). It doesn't matter
>     how many levels A
>         is ranked higher than B, it counts as exactly one vote for A.
>
>     If our votes are not valued equally, then I want my vote to count
>     more than yours.  If that is unacceptable (understandably) then we
>     must agree to count our votes equally.  In the U.S., too many
>     people have died over that inequality.  So then, in order for our
>     votes to be valued equally, we must have Majority Rule in
>     single-winner elections:
>
>         2. Majority rule: If more voters mark their ballots preferring
>     Candidate A over
>         Candidate B than the number of voters marking their ballots to
>     the contrary,
>         then Candidate B is not elected. If Candidate B were to be
>     elected, that would
>         mean that the fewer voters preferring Candidate B had cast
>     votes that had greater
>         value and counted more than those votes from voters of the
>     larger set preferring
>         Candidate A.
>
>     Those are two ways of, essentially, expressing the same principle
>     in single-winner elections.  For multi-winner elections, the way
>     to value our votes equally would be Proportional Representation,
>     but I don't wanna go there in this discussion.  I would like to
>     stay with single-winner elections.
>
>     Now, of course this doesn't deal with the problem of cycles and we
>     can discuss what the best and most democratic way to deal with
>     cycles is, but I cannot understand why, *if* a Condorcet winner
>     exists, how *any* other method; Hare, Borda, Bucklin, or Kirslis
>     is more democratic than Condorcet.
>
>     If a CW exists and we *know* (from the Cast Vote Record having
>     ranked ballot data) that the CW exists and who that CW is, how is
>     electing the K-count winner, assuming they're different from the
>     CW, more democratic?  Just like with the IRV failures, we will
>     *know* that a smaller set of voters have left that election
>     satisfied than that of a larger set of voters leaving the election
>     dissatisfied.  We will know that the votes coming from that
>     smaller set of voters were more effective in electing their
>     preferred candidate than the votes coming from the larger set of
>     voters that not only preferred someone else, but they preferred a
>     *specific* candidate over the one who Kirslis elected and marked
>     their ballots saying so.  For the very same reason that IRV failed
>     in Burlington Vermont in 2009 or in Alaska in August 2022, the
>     elected candidate will suffer a sense of loss of legitimacy in the
>     election.
>
>     In Burlington in 2009, 4064 voters marked their ballots that
>     Candidate A was a better choice than Candidate B and 3476 voters
>     marked their ballots to the contrary.  (There were 1436 voters
>     that didn't like either A or B and didn't rank either.)  How
>     *possibly* can Candidate B be elected without counting those 3476
>     voters' individual votes a little more (like 17% more) than how
>     much the votes were counted from the 4064 voters preferring
>     Candidate A?
>
>     Now this is a failure of Hare (IRV) but I can construct the very
>     same question for an election decided with Kirslis rules that
>     failed to elect the CW when such exists.  How would you answer
>     that question?  How do you justify satisfying a smaller set of
>     voters at the expense of a larger set of voters that preferred,
>     not just anyone else, but a specific candidate over the Kirslis
>     winner?  I couldn't glean an answer to that from section 9 (or
>     anywhere else) in your paper.
>
>     bestest,
>
>     --
>
>     r b-j . _ . _ . _ . _ rbj at audioimagination.com
>
>     "Imagination is more important than knowledge."
>
>     .
>     .
>     .
>
>     > On 05/18/2025 1:51 PM EDT Daniel Kirslis via Election-Methods
>     <election-methods at lists.electorama.com> wrote:
>     >
>     >
>     > Hi all,
>     >
>     > Thanks so much for the replies. I’ll respond to everyone in this
>     thread.
>     >
>     > Andy - I really appreciate your feedback. Your summary is
>     correct, and your framing of it as one-norm vs. two-norm vs.
>     infinity-norm is a way of thinking about it that I had not
>     considered. It seems like a potentially fruitful lens for
>     understanding it. And, as perhaps you have surmised, I may have
>     been mistaken in the statement about the sincere favorite
>     criteria, but I am working on an analysis of the issue that I will
>     share.
>     >
>     > Toby, making a short summary is a great suggestion. The argument
>     in the paper is admittedly a bit convoluted before it presents the
>     actual method. Here is the simplified way that I would explain it:
>     >
>     > Each voter ranks their preferences, with ties allowed and
>     unranked candidates treated as last-place preferences. Then, for
>     each candidate, you make a plot, where each axis is the total
>     number of times that they were preferred to each of their
>     opponents. So, if the candidates are A, B, and C, candidate A’s
>     plot would have “number of times preferred to B” on one axis and
>     “number of times preferred to C” on the other axis. Candidate B &
>     C could be plotted similarly in terms of their opponents. The
>     winner is simply the candidate who is plotted the farthest up and
>     to the right, or closest to topmost and rightmost point, which is
>     where a candidate who is the unanimous first-place choice would be
>     plotted. The distance from that point is calculated using the
>     Pythagorean theorem, which is where minimizing the sum of squares
>     that Andy referenced comes in.
>     >
>     > The figures in the paper tell the story better than the words,
>     as it is essentially a geometric idea. And, sections 4, 5, and 6
>     can really be skipped - they are more about justifying the
>     approach than explaining it.
>     >
>     > Chris, you asked “Why should we be interested in the "concerns"
>     of Borda (whatever they are)? And so much that we should embrace a
>     method that fails the Condorcet criterion?” Great question. If you
>     look at the Stanford Encyclopedia of Philosophy’s entry on Social
>     Choice Theory, they list Condorcet and Borda as the original
>     pioneers of this thinking
>     (https://plato.stanford.edu/entries/social-choice/). Borda thinks
>     about majoritarianism in terms of votes, while Condorcet thinks
>     about it in terms of voters. Obviously, in FPP elections, these
>     are the same, but the heart of the interest in these questions
>     comes from the tension that arises between them in a ranked-choice
>     setting, where each voter has multiple votes and ‘majoritarianism’
>     is no longer simple to define. Don Saari is a thinker who studies
>     these issues and has argued most persuasively for Borda’s approach
>     over Condorcet methods. In section 9 of my paper, I explain some
>     of my philosophical objections to the Condorcet winner criterion.
>     >
>     > You also asked “Do you propose allowing above-bottom equal
>     ranking or truncation?” Equal ranking is allowed, and unranked
>     candidates are treated as last place.
>     >
>     > And, I am afraid I may have actually been mistaken about the
>     sincere favorite property, so will have to disappoint you there.
>     >
>     > You asked “Who does your method elect in this example?
>     >
>     > 46 A
>     > 44 B>C
>     > 10 C”
>     >
>     > If I am understanding your notation correctly, A would win in
>     this example. The full ranking would be:
>     > A's K-count = 46 = 100-SQRT((100-46)^2+(100-46)^2)/(SQRT(2))
>     > B's K-count = 44 = 100-SQRT((100-44)^2+(100-44)^2)/(SQRT(2))
>     > C's K-count = 28.53 = 100-SQRT((100-54)^2+(100-10)^2)/(SQRT(2))
>     >
>     > As you can see, when a candidate only appears as a first-place
>     or last-place preference, their K-count is simply equal to the
>     number of voters ranking them first.
>     >
>     > Thanks all!
>     >
>     >
>     > On Sun, May 18, 2025 at 12:10 PM Andrew B Jennings (elections)
>     <elections at jenningsstory.com> wrote:
>     > > Hi Dan,
>     > >
>     > > Great paper. Thank you for posting!
>     > >
>     > > It seems like the short version is that the winner is the
>     candidate with the smallest sum of SQUARES of non-victories
>     (defeats plus ties) against their opponents.
>     > >
>     > > Taking the square root and dividing can make it meaningful by
>     scaling it to [0,1] or [0,s] (where s is the number of voters),
>     but doesn't change the finish order.
>     > >
>     > >
>     > > It does seem like an interesting attempt to "square the
>     circle" (great pun) and compromise between Borda and Condorcet. I
>     hadn't realized that Borda and Minimax are minimizing the one-norm
>     and infinity-norm in the same geometric space. The two-norm
>     certainly seems like it should be explored.
>     > >
>     > > I would love to see the proof of non-favorite-betrayal.
>     > >
>     > > Best,
>     > >
>     > > ~ Andy
>     > > On Thursday, May 15th, 2025 at 4:25 PM, Daniel Kirslis via
>     Election-Methods <election-methods at lists.electorama.com> wrote:
>     > >
>     > > > Hello!
>     > > >
>     > > > I am a newcomer to this mailing list, so please forgive me
>     if this message violates any norms or protocols that the members
>     of this list adhere to.
>     > > >
>     > > > I have recently developed a novel method for tabulating
>     ranked-choice elections that attempts to reconcile the concerns of
>     Borda and Condorcet. I believe that it maintains the simplicity
>     and mathematical elegance of the Borda count while incorporating
>     Condorcet's concern with pairwise dominance. Intuitively, it can
>     be understood as ordering candidates by how close they come to
>     being unanimously selected when plotted in Cartesian coordinate
>     space. Here is a link to the paper:
>     > > >
>     https://drive.google.com/file/d/152eNheS2qkLHJbDvG4EwW3jdO4I_NwcX/view?usp=sharing
>     > > >
>     > > > Given its simplicity, I have been very surprised to discover
>     that this method has never been proposed before. I am hoping that
>     some of you all will take a look at the paper and share your
>     comments, questions, and critiques. Ultimately, it is my hope that
>     ranked-choice voting advocates can arrive at a consensus about the
>     best method for RCV and thus strengthen efforts to adopt it and
>     deliver much needed democratic improvements. But even if you don't
>     find the system itself compelling, you may find the method of
>     plotting electoral outcomes elucidated in the paper to be useful
>     for the analysis of other electoral systems.
>     > > >
>     > > > Thank you!
>     > > >
>     > > > -Dan
>     > >
>     > >
>     > ----
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