[EM] Why All the Fuss?

[Forest Simmons]

In the context of elimination methods (like IRV, Coombs, Baldwin, rtc, as
well as all of our "worst-elimination" methods) the temptation for a
faction to bury (insincerely lower on their ballots relative to one or more
other candidates) a candidate C in order to help some candidate A win
instead of C ... this temptation arises when C defeats A pairwise, but the
A supporters, by lowering C, get C eliminated at some earlier elimination
step so A and C are not competing head to head.

Note that this burial ploy will not work  wih IRV elimination, because
lowering C on a ballot where A is already preferred over C will not
decrease C's first place support ... so it cannot get C eliminated earlier
... since IRV elimination prioritizes low first place support.

Coombs elimination, on the other hand prioritizes high last place counts
for early elimination, so the burial ploy has a good chance of succeeding
under Coombs.

Note that the feature that gives IRV immunity to burial is the same feature
that makes it vulnerable to the Squeeze Effect.

So is it possible to have immunity to burial and squeeze in the same method?

Yes, our "worst-elimination" methods have immunity to both... immunity to
squeeze because of Condorcet efficiency and immunity to burial because in
the above ploy, to eliminate C earlier (whether by burial or some other
means) must backfire as long as the ballot change preserves C's pairwise
win over A.

It does preserve C's pairwise win over A in the case of burial ... because
A was already ranked ahead of C by the buriers before the burial.

So how does this fact make C's elimination before A backfire?

Because according to our method...when C reaches "worst" status .... it is
eliminated only "after any and every candidate defeated by it [including
A]  is eliminated"

In other words, if and when C reaches "worst" status (with or without the
push downward from A supporters), it takes down A with it. So it doesn't
matter if our nominal standard of worst is "fewest first" or "most last" or
anything else ...  if it speeds up C's demise, it also speeds up the demise
of any candidate that C defeats pairwise.

In the three candidate case ... C is the sincere CW, and wins if C is
eliminated, sothe other candidate B is the sincere Condorcet Loser.

The A faction buries C under B, which creates a beat cycle ABCA.

A thinks this cycle gives it a chance at winning ... which it would under
most elimination methods.

But not under ours, because, on the one hand A cannot win unless B or C is
"worst" ... and ...

If B is worst it takes A down with it because B defeats A in the cycle ...
then B defeats C.

On the other hand, if C is worst, it takes A down with it, leaving B as
winner.

So burial of the A faction's second choice results in the election of their
anti-favorite B ... a complete backfire of the burial gambit!

I hope that.explanation clarifies the main reason for the clearing out of
the candidates defeated by the pivot candidate, i.e. the nominally "worst"
candidate, at each elimination stage ... see there really is a "method to
our madness".

You may remember I once proposed a Quick & Dirty method that simply said
elect the "best" candidate that pairwise defeats the "worst" Smith
candidate.

That's a shortcut rule of thumb that will elect the same candidate as our
"worst-elimination" methods do whenever there are no more than three Smith
members ... but the short cut is not Landau efficient ... so I don't
recommend it.

The main defect of the shortcut is that it requires some knowledge of Smith
... which our "worst-elimination" methods do not require.

So even though Q&D is shorter ... it is neither quite as good nor quite as
simple.

If you have any question about any other method that you would like to
compare with its nearest "worst-elimination" method ... it could interest
other readers of the EM list, too.

Remember "worst" is a nominal, tentative judgment that can hardly go wrong
... since the direct pairwise comparisons trump the tentative judgments if
there is any disagreement.

Good sources for "worst" candidates are losers of other methods.

Also losing candidates in strong pairwise defeats ... for any decent gauge
of defeat strength.

Enjoy!

Forest



On Sun, Feb 26, 2023, 8:37 AM Forest Simmons <forest.simmons21 at gmail.com>
wrote:

> The ElectoScope aka Yee Diagram makes clear both the problem with and the
> solution to the Center Squeeze phenomenon ... elimination methods that
> judge "worst" by size of the Voronoi regions ten to suffer from the defect.
>
> But the cure is easy and sure ... no eliminations of undefeated candidates.
>
> All Condorcet Efficient methods have the same Yee Diagram ... the win
> region for a candidate is its entire Voronoi polygon, no matter how small.
>
> Next ... burial ...
>
>
>
>
>
>
> On Fri, Feb 24, 2023, 3:49 PM Forest Simmons <forest.simmons21 at gmail.com>
> wrote:
>
>>
>> Why can't we just have majority rule? Why all the fiss?
>>
>> Many a student of my "Math for Liberal Arts" class asked me that question
>> during the decades I taught the Community College course by that name.
>>
>>  That's the reason Joe Malkovich's contribution to the textbook was so
>> important ... his examples of ballot profiles for which no two of several
>> different majority rule methods agreed on who should be elected.
>>
>> Most if not all of these methods start out with the phrase..."Elect the
>> majority winner if there is one, otherwise cull out the weakest (meaning
>> democratically weakest) candidates one by one until there is a majority
>> winner among the remaining."
>>
>> But there is no agreement on what constitutes "democratically weak' ...
>> and it makes a big difference!
>>
>> So what can we do?
>>
>> One thing we have tried without much success is to suggest that the next
>> best thing, lacking a first preference majority winner ... is to elect the
>> candidate unbeaten by any majority comparison with another candidate.
>>
>> But just as there is no guaranteed outright majority winner ... neither
>> is there any guarantee of the existence of a pairwise unbeaten candidate.
>>
>> It turns out that the best we can guarantee along these lines is the
>> existence of at least one candidate that can pairwise beat in two steps
>> every candidate that he cannot defeat in one step (by a majority of the
>> participating voters).
>>
>> Such a candidate is said to be "uncovered."  We're going to need a better
>> word than that if we want to get anybody on board with this minimum
>> guaranteeable standard of "majority rule."
>>
>> Let's say a candidate is "democratically strong" if it has a beatpath to
>> every other candidate ... and is "very strong majority pairwise" if it has
>> a beatpath of one or two steps to each of the other candidates ... each
>> step being a pairwise victory by a majority of the participating voters ...
>> meaning voters expressing a preference.
>>
>> Then the "Strong Majority Pairwise Criterion" (SMPC) is satisfied only by
>> methods that always elect uncovered candidates.
>>
>> Contrast that with the weaker, relatively impotent Condorcet Criterion
>> which is satisfied by any method that elects an unbeaten candidate "when
>> such a candidate exists" ... the copout escape clause in quotes letting the
>> method off the hook whenever things start to get interesting.
>>
>> Another way to express compliance with this SMPC criterion is "Landau
>> Efficient."
>>
>> Every method under the "Worst-Elimination" umbrella is seamlessly Landau
>> Efficient ... it effortlessly (and without fanfare) satisfies the SMPC ...
>> no matter what nominal standard of worst is instantiated into the umbrella
>> template.
>>
>> Who can name even one commonly known election method that is Landau
>> efficient?
>>
>> What's more ... no matter the nominal "worst" criterion, the method will
>> be more or less burial resistant ... as I will explain presently.
>>
>> I suggest that proposals for any method under this umbrella, include
>> verbiage to the effect ...
>>
>> "When there is no majority winner or any candidate that a majority of the
>> participating voters rank ahead of each of the other candidates ... cull
>> out one-by-one the nominally "worst" candidates as well as any
>> democratically weaker candidates (as determined by majority ballot
>> preferences) until there is a majority winner among the remaining
>> candidates."
>>
>> This umbrella is so robust that the choice of nominal "worst" is not
>> overly critical.  The main thing is to keep it simple enough that (1)
>> voters can easily understand and relate to it, and (2) it can be
>> efficiently and transparently tallied by precinct without multiple passes
>> through the ballots.
>>
>> Complicated "worst" criteria are the ones that tend to introduce crowding
>> and teaming distortions ... smallest Borda score is a example of this kind
>> of "worst" criterion ... pun intended.
>>
>> Anti-vote splitting can be easily ensured (in general) by allowing
>> equal-top whole counting, and multiple truncations in large elections.
>>
>> In the continuation I will explain why this method tends to backfire on
>> buriers.
>>
>> At some point those who have power to advocate for one method over
>> another need to understand them beyond the surface heuristics that appeal
>> to the impatient public.
>>
>> Among other things enlightened defenders of electoral democracy need to
>> understand the "squeeze effect" and "burial ploys" ...
>>
>> To be continued ...
>>
>> -Forest
>>
>>
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