[EM] Why All the Fuss?

[Forest Simmons]

Correction below ...

On Sun, Feb 26, 2023, 10:58 AM Forest Simmons <forest.simmons21 at gmail.com>
wrote:

> 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.
>

Whoops... A defeats B.

So the burial can succeed if it is enough to make B beat C, but not enough
to make C nominally "worse" than B ... a delicate, hence risky balance.

Which nominal standards of "worst" make this balance most precarious if not
impossible?


> 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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