[EM] Obvious Elimination

[Forest Simmons]

Another possibility:

Until only one candidate remains ...
Eliminate from among the remaining  ...
Friends(PL(MaxMaxPO,minminPO))
EndUntil
Elect the only remaining candidate

Example1
48C
28 AB
24 B
MaxMaxPO=MaxTop=MaxMinPS=C
minminPO=MaxBot=MinMaxPS=A

PL(C,A)=A

Friends(A)=A,B

Eliminate A,B --> Elect C

Example 2
a ABC
b BCA
c CAB
Suppose a>b>c.
C=MaxBot=mMPS=MMPO
A=MaxTop=MmPS=mmPO
PL(A,C)=A, Friends(A)=B,A --> C wins

Check me ... this is tricky!

-Forest



On Sun, Mar 19, 2023, 9:31 AM Forest Simmons <forest.simmons21 at gmail.com>
wrote:

> Here's an even simpler method in the same spirit that accomplishes the sam
> thing.
>
> First some notation:
>
> Friends(X) is the set of candidates that do not beat X . the complement of
> Enemies(X).
>
> Note that Friends(X) includes X itself ... "Love thy neighbor as thyself."
>
> PW(X,Y) is the Pairwise Winner in the contest between X and Y.
>
> Similarly, PL(X,Y) denotes the Pairwise Loser of the same contest.
>
> Here's the method:
>
> While more than one candidate remains ... eliminate
> Friends(PL(MaxTop,MaxBot))
> EndWhile
> Then elect the remaining candidate.
>
> If a burial has taken place, at some point the respective burier and
> buried are likely to take on the roles of MaxToo and MaxBot, respectively.
>
> If so the burier will lose the pairwise contest, and be eliminated along
> with the rest of the burier's friends, including the one under whom the
> buried candidate was buried.
>
> Remember MaxTop and MaxBot are the respective candidates with the greatest
> Top and Bottom counts.
>
> MaxTop=Argmax TopCount(X)
> MaxBot=Argmax BotCount(X)
>
> TopCount(X) is the number of ballots on which no other candidate outranks
> X.
>
> BotCount(X) is the number of ballots on which X outranks no candidate.
>
> It is easy to see that this method is Landau efficient because every pivot
> candidate is among its friends, and all of the eliminated candidates are
> friends of the pivots of the stages that eliminated them. In other words,
> each candidate is a friend of a friend of the winner when not a direct
> friend of the winner itself ... the very definition of Landau efficiency in
> the language of friends!
>
> In fact, the set of pivot candidates forms a totally ordered chain that
> covers every non pivot candidate ... it cannot be extended further above
> the winner ... which means the method is Banks efficient.
>
> This message is not intended for the lay person. Somebody with extra time
> on there hands can write it up for general audiences. ... leaving out the
> technical information that means nothing to voters ... and putting in
> things helpful to voters that technical readers take for granted.
>
> That's plenty for now. Next time a tournament friendly version that is
> efficiently precinct summable ... or why not now?
>
> While more than one candidate remains ... eliminate
> Friends(PL(MinMaxPS,MaxMinPS))
> EndWhile
> Elect the remaining candidate.
>
> MinMaxPS is the candidate whose Maximal Pairwise Support is minimal.
>
> MaxMinPS is the candidate whose Minimal Pairwise support is maximal.
>
> -Forest
>
> On Sat, Mar 18, 2023, 7:02 PM Forest Simmons <forest.simmons21 at gmail.com>
> wrote:
>
>>
>>
>> At each stage, among the remaining candidates let MaxTop and MaxBot,
>> respectively, be the candidates with the largest Top and Bottom counts ***.
>>
>> Let COACC be the candidate with the greater of these two max counts.
>>
>> Eliminate every candidate X  that is outranked by the COACC candidate on
>> a majority of those ballots that show any preference between them ....
>> between X and the COACC, that is.
>>
>> Also eliminate COACC itself if its Bottom Count is greater than its Top
>> Count ... otherwise keep it.
>>
>> ***[At any stage a candidate's Top count is the number of ballots on
>> which it is not outranked by any remaining candidate. Similarly, its Bottom
>> count is the number of ballots on which it outranks no remaining candidate.
>> These Top and Bottom counts are supposed to be updated (by vote transfers
>> from eliminated candidates to those remaining) between all elimination
>> stages.]
>>
>> Example
>>
>> 48 C
>> 28 A>B
>> 24 B
>>
>> MaxTop and MaxBot are C and A, respectively, with counts of 48 and 72.
>>
>> Since 72 is larger than 48,  the obvious approval cutoff COACC is A, and
>> because A's bottom count is greater than its top count, A itself will be
>> eliminated along with the candidate B, that is disapproved because it is
>> outranked by A=SOACC on a majority (28 to 24)  of the ballots that show any
>> preference ... leaving C as the winner.
>>
>> Note that the cutoff candidate SOACC will always be someone that should
>> obviously be approved (most Top) or else someone obviously disapproved
>> (most bottom).  Of these two obvious choices, the more obvious of the two,
>> the one with the greater max count, is the SOACC cutoff.
>>
>> This completely resolves the two most difficult approval questions ...
>> where to place the cutoff between approved and disapproved, and whether or
>> not to approve candidates ranked exactly on the cutoff boundary. The voters
>> don't have to worry about it at all unless they want to over-ride the SOACC
>> cutoff for some personal reason.
>>
>> Anybody find any significant fault with this method compared to their
>> previous favorite?
>>
>> Easy or hard to understand?
>>
>> Likely or unlikely to elect the "best" candidate?
>>
>> Easy to sell or hard to sell?
>>
>> Best selling point?
>>
>> Biggest drawback?
>>
>> Thanks!
>>
>> -Forest
>>
>>
>>
>>
>>
>>
>>
>>
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