[EM] Bucklin

[Forest Simmons]

It seems to me that we need a version of Bucklin that simulates the
following Nanson like manual approval runoff:

After each approval round eliminate every candidate whose total approval
count in the current round has come out to be less than halfway between the
lowest and the highest approval count of the current round.

If you had 1000 candidates, you could expect about ten rounds to get down
to the Lone Ranger.

So a range of zero to ten should be more than adequate for the following
"instant" version of "Midrange Approval Runoff":

Lacking a ballot  CW ... then implement the following "tie breaker" ....

For R from zero to nine ...
... for each candidate X, let  X's Tie Breaking Approval TBA(X) be the
number of ballots on which X is rated strictly above R.
Then eliminate every candidate X such that TBA(X) is strictly less than
halfway between the min and max values of TBA for the current round R.
Next R.

This instant version of approval runoff has now become my favorite use of
score/range/grade/judgment ballots.

For comparison, Traditional Nanson eliminates below mean Borda Count
candidates.  Use of median instead of mean, and level of approval instead
of Borda, improves on Nanson in two ways ...
1. It makes the method much easier to hand count.
2. It confers clone independence on the method.

Nanson is known to be Condorcet Efficient ... which leads me to suspect
that this method's Condorcet Completion to be a seamless patch ... as
seamless as possible while preserving the Favorite Betrayal Criterion (if
it is indeed preserved in the stand alone version ... in other words,
without the fiat compliance with the Condorcet Criterion).

In comparison with other median based methods like Majority Judgment and
Bucklin ... well judge for yourselves ... but I think its simpler to
understand and to execute.

Everybody loves "Usual Judgment" until they try to wrap their mind around
its exhaustive (and exhausting) tie breaking procedure.

This method is as conclusive as any extant Condorcet method.

fws





On Thu, Nov 2, 2023, 8:49 AM C.Benham <cbenham at adam.com.au> wrote:

> There has been more than one version of Bucklin that has been used or
> proposed.
>
> I think it is now generally agreed that least bad is to use some sort of
> limited-slot grading/rating ballot,
> with voters free to give as many or as few candidates as they like the
> same grade and to skip grades if
> they want.  (So it is just a version of Average Ratings)
>
> If ratings ballots are used then above-bottom equal ranking should get
> the whole (not fractional) votes
> interpretation, but if say a ballot equal top-ranks 3 candidates that
> ballot gives a whole vote to each in
> the first "round" but then "sits out" rounds 2 and 3.
>
> (Not doing that was shown to make the method fail mono-raise).
>
> If we are talking about one of these versions then we are talking about
> a method that meets Favorite Betrayal
> and Majority for Solid Coalitions and Later-no-Help.
>
> But there is a very strong incentive for voters to just submit approval
> ballots, and in a competitive election with
> informed voters the extra complexity versus simple Approval doesn't seem
> to buy much.
>
> 40 A>B
> 30 B
> 09 C
> 02 X
>
> 81 ballots.
>
> This example highlights the method's disadvantages compared with IRV/RCV.
>
> I don't consider meeting Majority for Solid Coalitions to be an adequate
> standard of majoritarian representative goodness.
>
> I propose the "Dominant Coalition" criterion:
>
> *If a the number of ballots on which a set S of candidates is
> ranked/voted all below no outside-S candidate is greater than the maximum
> pairwise opposition that any inside-S candidate gets from any outside-S
> candidate, then the winner must come from set S.*
>
> The single-candidate version (that could be relevant for a method that
> fails Clone-Winner) is "Dominant Candidate".
>
> *If the number of ballots on which candidate X is ranked/voted below no
> other candidate is greater than X's maximum pairwise opposition,
> then X must win.*
>
> Another criterion I like (and I think I coined) is Irrelevant Ballots
> Independence: adding or removing ballots that contain no information
> relevant to any of the remotely competitive candidates should not change
> the result.
>
> Another criterion met by IRV/RCV but not Bucklin is Mutual Dominant
> Third : "if a set S of candidates that pairwise beat all the outside-S
> candidates are voted above all the outside-S candidates on at least one
> third of the ballots then the winner must come from S."
>
> In the example A is the Dominant Candidate and the Mutual Dominant Third
> candidate (and so of course the CW) but the Bucklin winner
> is B.
> (A lot of people like IRV's compliance with Later-no-Harm. Of course
> here if the A>B voters had truncated then A would have won.)
>
> But if we remove the 2 X ballots the winner changes from B to A,
> (because the majority threshold lowers from 41 to 40, so now there is no
> second round) a failure of Irrelevant Ballots independence.
>
> Chris B.
>
>
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