[EM] Worst Loser Elimination 2.0

[Richard Lung]

Elimination methods may have a policy mandate but they do not have a scientific or knowledge mandate. They violate conservation of (preference) information. Elimination independent of scale of preference is precisely the fault, found over two centuries ago, by Pierre-Simon Laplace, according to JFS Ross, with Condorcet pairing elimination of candidates. It does not take into account the relative importance of higher to lower preferences. Why Laplace sided with Borda amounted to his opening the way to rational counts of preference ranges. Of which Gregory supplied the definitive statistical method to next preferences, of weighting in arithmetic proportion.

It is doubtful whether the Condorcet Winner (or conversely Loser) applies to more than single winner elections, the least democratic, lacking multi-member range of representation. This perhaps may be demonstrated by the consideration that individual candidates hold an over-lapping group or "party" of opinions. Voters support for parties themselves become less and less partisan with more particular choices of parties in proportionally elected multi-member constituencies. So, it may be the CW becomes less and less an absolute individual choice, within the wider context of personal proportional representation, in multi-member constituencies.

Regards,
Richard Lung.




On 1 Mar 2023, at 2:47 am, Forest Simmons <forest.simmons21 at gmail.com> wrote:

After many thought experiments I'm leaning towards the following Worst Loser Elimination version:

Elect the pairwise undefeated candidate if there is one ... otherwise ...
until there is an undefeated candidate ... among the uneliminated candidates, eliminate the candidate whose maximum pairwise support is the smallest ... after first eliminating any candidates it defeats pairwise.

One thing I like about this version is that it only needs as input the pairwise support information ... unlike elimination methods that depend on Top or Bottom ballot counts, it is applicable to tournaments.

To me, that is the true spirit of  Universal Domain methods ...  binary choices based on ordinal information .. no inference of strength that is based on how high or low in the rankings ... a subtle influence that subverts the purity of binary comparisons as a basis for voting.

I'm not against methods that do make use of expressions of intensity of preference ... only methods that pretend not to, but actuually do. 

The pairwise information is efficiently precinct summable with one pass through the ballots. The practical importance of this feature is hard to over estimate.

Example1

x: A>B>C (Sincere is A>C>B)
y: B>C>A
z: C>A>B

Assume (y+z)>x>max(y,z), so the A faction is the largest, but not a majority of the electorate.

The maximum pairwise support for C is its support against A by the B and C factions ... the winning votes of the defeat of A by C: y+z= n-x, where n=x+y+z.

The max pairwise supports of the other two candidates A and B are n-y and n-z, respectively, which are both larger than n-x ...  the smallest value of max pairwise support ... making C the candidate to eliminate (after eliminating the candidate defeated by it).

So in the context of this kind of three faction cycle with complete ballots, the candidate that defeats the favorite of the largest faction is the first one eliminated ... after first eliminating the candidate it defeats.

How does this discourage burial? 

The candidate on the losing end of the weakest wv defeat is the one with the greatest hope of successfully burying its victor... and when it is the favorite of the largest faction, it has the greatest hope of coming out on top of the cycle resulting from that burial.

Example 2

48 C
28 A>B
24 B (sincere B>A)

Evidently a Chicken defection by the B faction ... putting the Sincere CW into a cycle in which its max pairwise support is 28 votes against C. The respective max supports of B and C in this cycle are 52 and 48, both larger than 28.

So A is eliminated after the candidate it defeats ... the defector B.

In example 1 the MinMaxPS candidate took down with it the candidate responsible for its burial.

In example 2 the MinMaxPS candidate took down with it the defecting candidate that put it into a cycle.

In both cases the MinMaxPS candidate was the Sincere CW.

In both cases the manipulating candidate had high hopes of winning the cycle it created by its manipulation ... because almost all Condorcet efficient methods break cycles at weakest wv defeat in the beatcycle chain.

Our method also eliminates that defeat ... but both winner and loser of that defeat ... making the manipulator's gambit fail, instead of rewarding the loser of the weakest wv defeat like naive Condorcet methods do ... which thereby reward the manipulator responsible for creating the cycle.

So this method takes very precise aim at taking down Chicken Defectors and Buriers that hope to defeat the Sincere CW by creating a cycle in which it (the Sincere CW) has become the weakest link.

Our method counters that strategy by taking out the entire weakest link ... both winner and loser, unlike RP, CSSD, River, MinMax(wv), etc.

Can the general public appreciate this? 

How do we help them see?

Through a wide variety of examples where ordinary Condorcet fails to elect the Sincere CW while rewarding the cycle creator?

But also we need to make clear that burial of the Sincere CW is no problem for IRV .... only because IRV is not a Condorct method ... it's not even expected to elect the CW in the first place .... instead of burying the CW it eliminates it by the Squeeze effect ... a much more common problem than Burial ... one that is only overcome by serious insincere compromise ... which belies the IRV promise of sincere voting safety.

The squeeze effect happens with sincere votes. Burial cannot happen with sincere ballots. The Squeeze distortion of the democratic will, will be more common than Burial ... as long as voters vote sincerely ... which ordinary voters tend to do when not urged to vote lesser evil above their favorite.

So don't throw out the Condorcet baby with the burial bath water. Keep Condorcet ... but make sure it is Burial and Chicken resistant Condorcet ... not vulnerable to the most tempting insincere manipulations.

If that required some elaborate multiple pass version of Condorcet ... like Smith//IRV, BTR-IRV, RCIPE, etc ... or even anything as elaborate as Ranked Pairs ... then it might not be worth it.

But voters like (and practically demand) elimination methods ... like ours ... with the same simplicity as MinMax the simplest well known Condorcet method (but not Burial resistant or Landau efficient).

Let's formulate MinMax(wv) as an elimination method to facilitate the comparison:

Until there is a candidate that is undefeated among the uneliminated candidates, eliminate the candidate whose Max Pairwise Support is minimal.

Here's our manipulation resistant version:

Until there is an candidate that is undefeated among the uneliminated candidates, eliminate the candidate whose Max Pairwise Support is minimal, after first eliminating any candidates defeated by it.

-Forest










> On Sun, Feb 26, 2023, 9:47 PM Forest Simmons <forest.simmons21 at gmail.com> wrote:
> The Symmetric Gross Loser notion of nominal "worst" seems to work well for burial resistance ...  but it is not very chicken resistant.
> 
> Here's a better one ... one I mentioned before .... where the nominally worst candidate is the loser of the strongest defeat... where defeat strength is gauged by the number of ballots on which the victor outranks the loser plus the bottom count of the loser... we abbreviate this gauge as wv+lbc... winning votes plus losing bottom count.
> 
> Remember that a candidate's bottom count is the number of ballots on which it outranks nobody.
> 
> So the method is to elect the CW if there is one ... otherwise ...
> Until there is an unbeaten candidate among the remaining candidates, eliminate the nominally worst candidate after first eliminating the candidates it beats pairwise (if any).
> 
> The nominally worst candidate is 
> the loser of the single strongest defeat, gauged by av+lbc ... winning votes plus losing bottom count.
> 
> This method seems very promising for both burial and chicken resistance according to initial hand counts of some standard test cases.
> 
> Simulations will tell.
> 
> Here's my question: do simulations carry any weight with the public? Or do they just care about choice of buzz words and phrases like democracy; majority rule, etc?
> 
> -Forest
> 
>> On Sun, Feb 26, 2023, 3:57 PM Forest Simmons <forest.simmons21 at gmail.com> wrote:
>> Here's the cleanest notion of "worst" in this contex:
>> 
>> The "nominally worst" candidate is the Symmetric Gross Loser (SGL) defined as the pairwise loser between the candidate with the greatest pairwise opposition and the least pairwise support.
>> 
>> I am suggesting specializing our "worst-loser" elimination method to the following:
>> 
>> While there is more than one remaining candidate, from among them eliminate the current Symmetric Gross Loser SGL after first eliminating every candidate (if any) pairwise defeated by this SGL.
>> 
>> Elect the last candidate to be left standing (or eliminated).
>> 
>> This is the version I would like to see tested.
>> 
>> -Forest
>> 
>> 
>> 
>>> On Sun, Feb 26, 2023, 11:45 AM Forest Simmons <forest.simmons21 at gmail.com> wrote:
>>> 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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