[EM] Coombs method and typical RCV hybrid

[Forest Simmons]

El lun., 24 de ene. de 2022 3:53 p. m., <culitif at tuta.io> escribió:

> Hey Forest,
>
> Thanks so much for the feedback and the kind welcome! That certainly is a
> lot to digest, but I did find your definition of it really helpful. I've
> been kinda busy recently, but I'd definitely love to take a crack at
> coding/implementing and visualizing the system at some point in the near
> future!
>
> Would a system that uses the implicit fractional approval score really be
> equal to a system that takes the first and last place votes for each
> candidate?
>

Let F(X), L(X), and MR(X), respectively, be the number of ballots on which
X is voted First, Last, or Middle Ranks, me respectively.

Then the total number of ballots is the number T = F(X)+L(X)+MR(X), so your
score CULI(X)=F(X)-L(X).

 Now substitute the RHS of these two equations  into the expression

CULI(X)+T,
and simplify the result:

(F(X)-L(X)) +(F(X)+L(X)+MR(X))

= 2F(X) +MR(X)

Now divide both sides by two:

(CULI(X)+T)/2=(2F(X)+MR(X))/2

.5*CULI(X) +.5*T = F(X)+.5*MR(X)

So multiplying your score CULI(X) by a positive constant (.5) and adding
another constant (.5*T) yields the fractional approval score that most
people prefer, because it eliminates negative scores that bother some
people.

All of this is based on two facts

1. x>y if and only if x+c> y+c

and

2. p>q if and only if r*p>r*q,
for any positive r.

Put these together and you have


rt+k>rv+I  if and only if t>v
(povided r is positive)

"Positive affine transformations preserve order."


> -Culi
>
>
> Jan 23, 2022, 11:29 AM by forest.simmons21 at gmail.com:
>
> Culi,
>
> Great site!
>
> And great first EM post.
>
> Your "first minus last" score for each candidate is equivalent to what
> some of us call the implicit fractional approval score, which is your score
> plus the total number of ballots, all divided by two.
>
> Yours is simpler for people that are comfortable with integer arithmetic.
> The advantage of ours is that it is just the number of Top votes plus half
> of the number of non-Bottom votes ... no possibility of getting negative
> scores.
>
> Now here is the best (IMHO) use of your suggested scores:
>
> While more than one candidate remains, among these, eliminate the highest
> score candidate that does not pairwise defeat the one with the lowest score.
>
> Unlike the elimination method you suggested, this method, "Fractional
> Implicit Approval Chain Climbing," FIACC, is monotonic, clone free, and
> Banks efficient.
>
> As long as equal rankings and truncations are allowed, we can confidently
> (but humbly) affirm that FIACC is the best known "Universal Domain" method
> with these three properties.
>
> Universal Domain means RCV style ballots only.
>
> A Banks candidate is one that stands at the head of a maximal chain of
> candidates ordered by pairwise victory.
>
> Every Banks candidate is also a Landau candidate, which means it has a
> beatpath of two or fewer steps to any other candidate.
>
> If a method is not Landau efficient, then (embarrassingly) sometimes it
> will elect a candidate that is covered by some other candidate. This other
> candidate has a valid complaint or Pareto dominance over the winner with
> respect to pairwise wins, ties, and losses:
>
> Candidate X covers Y iff it not only defeats Y, but also defeats every
> candidate defeated by Y.
>
> Of the well known methods, only Kemeny-Young and Copeland are Landau
> efficient. But they both lack monotonicity, among other failings in
> comparison with our simple FIACC method.
>
> In particular, unlike Copeland, FIACC is highly resistant to Chicken and
> Burial attacks.
>
> And unlike K-Y, FIACC is computationally tractable, while K-Y is
> non-polynomially hard.
>
> I know that's a lot to digest ... but you can do it one bite at a time
> with patience.
>
> The method is simple ... and the three basic properties (monotonicity,
> clone independence, and Condorcet efficiency) are easy to understamd.
>
> It's only the Banks and Landau efficiency (that distiguish it from Ranked
> Pairs, Schulze, River, etc) that are a challenge to fully appreciate.
>
> To emphasize the simplicity of the method itself, I repeat the complete
> definition here for reference:
>
> While more than one candidate remains, among these, eliminate the highest
> score candidate that does not pairwise defeat the one with the lowest score.
>
> Assuming you know what "scores" we're talking about and what "pairwise
> defeat" means, that procedure completely and unambiguously defines the
> method.
>
> Do you know anybody who can do the same for their favorite complete method
> with the same unambiguous precision in fewer than twenty-five words?
>
> [Copeland is not by itself a complete method, just like Condorcet is not a
> complete method ... they require "completions" to resolve ubiquitous ties
> and cycles.]
>
> Once again, welcome, and thanks for your great first post to the EM list,
> and invitation to your site.
>
> -Forest
>
> Forest
>
>
>
> El sáb., 22 de ene. de 2022 12:03 p. m., <culitif at tuta.io> escribió:
>
> Hello all,
>
> I'm Culi, I'm a recent subscriber. Took a social choice theory in college
> and have wanted to make visualizations for electoral methods ever since. I
> recently finally got some time to create something like that!
>
> It's basically a tool that compares the outcome of an election in RCV,
> Coomb's RCV, and a third method which I have yet to find out the name of
> (I'd appreciate help with it). It's all explained more on the site, but
> basically it tries to take into account both first-choice and last-choice
> picks into deciding which candidate to drop every round.
>
> I'd love to someday expand the tool to show how a number of other
> single-winner electoral methods would result in the same election. I built
> a similar tool a while ago in Python but never got to deploy it. I only got
> so far as to simulate the election in FPTP, RCV, Borda Count, Coombs,
> Copeland, Quadratic Voting, and Contingent Vote.
>
> Now that I have web development skills I'd love to rebuild it and make it
> into an educational tool to let people compare different voting systems.
> I'd also love some day to code out some of the electoral methods discussed
> here on this mailing list!
>
> Anyways, here's what the site currently looks like (I'll have a better url
> later I promise). I'd love any feedback and suggestions for the name of the
> third voting method:
>
> https://elegant-shaw-2cb49a.netlify.app/votevote
>
> Best,
> Culi.
> ----
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> info
>
>
>
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