[EM] Quick and Clean Burial Resistant Smith

[Forest Simmons]

Let's call it Q&CBRS.

Pre-requisite background:

Candidate X pairwise defeats candidate Y iff candidate X is ranked
above/before/ahead of candidate Y on more ballots than not.

A defeat chain is a sequence of candidates in which each candidate pairwise
defeats the subsequent member of the chain.

Using a "bubble sort" procedure to sort a list of candidates into pairwise
order produces a defeat chain of the listed candidates.

In this way we can easily find a defeat chain that includes all of the
candidates. The first candidate in such a chain is an example of a Smith
candidate. More generally, any candidate who has a defeat chain to any
other candidate is a member of the Smith Set.

Q&CBRS:

First, find the "basic score" for each candidate defined as the number of
ballots on which it is ranked above one or more candidates.

Then let X be the Smith candidate with the smallest basic score.

Finally,  among the candidates not defeated by X, elect the one with the
greatest basic score.

That's it! Quick & Clean!

Note that if there is only one Smith candidate X, then X will be the only
candidate not defeated by X, and therefore the one elected .

In general a candidate not defeated by X will defeat X, and thereby find
itself at the head of a defeat chain to every other candidate, i.e. it will
be a Smith candidate.

When there is only one Smith candidate, that candidate will not be pairwise
defeated by any other candidate.

I repeat the entire method procedure here:

First, find the "basic score" for each candidate defined as the number of
ballots on which it is ranked above one or more candidates.

Then let X be the Smith candidate with the smallest basic score.

Finally,  among the candidates not defeated by X, elect the one with the
greatest basic score.

Where else can you find such a simple, quick, and clean election method?





El lun., 3 de ene. de 2022 4:19 p. m., Forest Simmons <
forest.simmons21 at gmail.com> escribió:

> Good idea!
>
> Although it seems to me that the highest approval candidate would have to
> pairwise beat or tie the approval cutoff candidate X pairwise (which would
> be impossible for a non-Smith candidate to do) ...I could be wrong ... and
> in any case redundancy reinforces communication and understanding.
>
> El lun., 3 de ene. de 2022 3:04 p. m., Ted Stern <dodecatheon at gmail.com>
> escribió:
>
>> Hi Forest,
>>
>> This sounds like an interesting method to me!
>>
>> However, I would change the winning criteria to "Elect the most approved
>> member of the Smith Set".
>>
>> On Mon, Jan 3, 2022 at 11:06 AM Forest Simmons <
>> forest.simmons21 at gmail.com> wrote:
>>
>>> I apologize for the defiant tone at the end of the previous message ...
>>> I must have gotten carried away with the "Dirty Dozen"' theme.
>>>
>>> But isn't it frustrating to you when people use the 2nd law of
>>> thermodynamics (or Arrow and Gibbard-Satterthwaite in the EM context) to
>>> justify their stubborn resistance to any kind of engineering progress?
>>>
>>> In the previous message Q&D Burial Resistant Condorcet was formulate in
>>> the typical "stitched together" form ... "Elect the CW if there is one,
>>> Else ..."
>>>
>>> In this message I would like to formulate a seamless version:
>>>
>>> Let X be the Smith candidate who on the fewest ballots is ranked ahead
>>> of any other Smith candidate. On each ballot approve all candidates down to
>>> X, but include X only when no Smith candidate is ranked ahead of X.
>>>
>>> Elect the candidate approved on the most ballots.
>>>
>>> This method can be described as electing the approval winner when the
>>> approval cutoff is (at the rank of) the weakest of the Smith candidates,
>>> which itself is approved on (and only on) those ballots which do not
>>> approve any other Smith candidate.
>>>
>>> In other words, the approval cutoff is inclusive only when necessary to
>>> ensure approval of at least one member of Smith.
>>>
>>> Since a Smith member is approved on every ballot, the method satisfies
>>> the Condorcet Criterion, i.e. it elects the only Smith member when Smith is
>>> a singleton.
>>>
>>> How does that grab you?
>>>
>>> -FWS
>>>
>>> El dom., 2 de ene. de 2022 7:30 p. m., Forest Simmons <
>>> forest.simmons21 at gmail.com> escribió:
>>>
>>>> Is there any burial resistant Condorcet method simpler than this?
>>>>
>>>> The basic pre-requisite is to understand that whenever there is no
>>>> Condorcet Winner there will be a pairwise cycle, called a "top-cycle" of
>>>> candidates whose members are not defeated by any candidates outside of the
>>>> cycle, just as a Condorcet Winner is a candidate undefeated by any other
>>>> candidate.
>>>>
>>>> Here's the Q&D burial resistant method:
>>>>
>>>> Lacking a Condorcet Winner elect the candidate X having the greatest
>>>> pairwise victory over the top-cycle member Y that has the smallest ratio of
>>>> first to last place votes within the top cycle.
>>>>
>>>> Two examples illustrate the method:
>>>>
>>>> Example 1.
>>>>
>>>> 49 C
>>>> 26 A>B
>>>> 25 B (sincere B>A)
>>>>
>>>> The top cycle is ABCA
>>>>
>>>> Candidate A has the smallest ratio 26/74 of first to last place votes.
>>>>
>>>> Candidate C is the only candidate with a pairwise victory over it, so C
>>>> wins.
>>>>
>>>> Notice how our rule does not reward B for insincerely lowering A to
>>>> (equal) last?
>>>>
>>>> Example 2.
>>>>
>>>> 45 A>B (sincere A>C)
>>>> 35 B>C
>>>> 25 C>A
>>>>
>>>> Candidate C has the smallest ratio  25/45 of first to last.
>>>>
>>>> Candidate B wins as the only candidate with a pairwise victory over C.
>>>> So A's burial of C backfires.
>>>>
>>>> Typically, the faction A that buries or truncates a Condorcet Winner C
>>>> to create a top-cycle cannot by so doing become a pairwise victor over the
>>>> buried Condorcet Winner ... but must (in order to create a cycle) help some
>>>> other candidate B defeat C by insincerely voting B>C.
>>>>
>>>> Our Quick and Dirty method insures that if the sincere CW's rightful
>>>> victory is subverted, it goes to B, not to A.
>>>>
>>>> Is this method quick enough and dirty enough for the FairVote IRV
>>>> promoters?
>>>>
>>>> What objections/criticisms might they have?
>>>>
>>>> Do they have any counter proposal that rivals this one in any way?
>>>>
>>>> If so, let them educate us ... we'll gladly join them if they can show
>>>> us a better way!
>>>>
>>>> If not, then they should join us to educate the politicians, public,
>>>> and last but not least, the academics still stuck in the pre-EM era!
>>>>
>>> ----
>>> Election-Methods mailing list - see https://electorama.com/em for list
>>> info
>>>
>>
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