[EM] MDD,ER-Bucklin (whole)

[Chris Benham]

Participants,
I've recently had the idea that  ER-Bucklin (whole)  could be improved 
by combining it with the
"Majority-Defeat Disqualification" (MDD)  component from  MDD//Approval 
(MDDA).

So  I  suggest  MDD,ER-Bucklin(whole)  as my favourite method that meets 
FBC.

"Voters rank the candidates,  truncation and equal-ranking allowed.
If  not all candidates are majority-strength pairwise beaten 
(i.e."dominated") by some other candidate,
then "disqualify"  (but  not  drop from  the  ballots)  those  that  are.
If only one candidate remains, that candidate wins.
If  more than one candidate remains, commence the ER-Bucklin(whole) 
process  until  (at least) one of
the not-disqualified candidates has a vote tally greater than (or equal 
to) half  the total number of  valid
ballots.
At that point elect the not-disqualified candidate with the highest vote 
tally."

[The "(or equal to)"  bit isn't traditional, but I suspect as a fine 
point it is good.]

To explain  "the ER-Bucklin(whole) process":
"In the first round each ballot contributes a whole vote each to  the 
tallies of those candidates they rank top or
equal-top.
In the second round, ballots that have contributed to the tallies of  
less than two candidates each contribute
a whole vote each to the tallies of  candidates they rank second or 
equal-second.
(Ballots that in the first round contributed to the tallies of  more 
than one candidate do nothing in the second round)
In the third round, if there is one, ballots that have contributed to 
the tallies of  less than three candidates now
contribute a whole vote each to the tallies of candidates they rank 
third or equal-third.
And so on."

Or as it is better explained at the Electowiki:

> /If a ballot lists n candidates as tied in kth place, count that 
> ballot as a whole point for all n candidates beginning in the kth round./
>
> Note: A candidate is ranked in kth place on a given ballot if there 
> are k-1 candidates who are ranked strictly higher. For exampe, a 
> ballot marked A>B=C=D>E>F=G=H=I>J should be considered to rank A to in 
> 1st place, B, C, and D in 2nd place, E in 5th place, F, G, H, and I in 
> 6th place, and J in 10th place. Thus, the ballot would not count in 
> favor of E until the 5th round, and it would not count in favor of J 
> until the 10th round.
>
> This rule is perhaps unique in that it satisfies both the "favorite 
> betrayal </wiki/Favorite_betrayal>" criterion and the Majority 
> criterion for solid coalitions 
> </wiki/Majority_criterion_for_solid_coalitions>.
>
 If  the majority-disqualified candidates were dropped from the ballots, 
then  Bucklin's compliance with Mono-raise would (I presume) be lost.
(If  I'm wrong about, then there probably isn't any reason not to just 
first drop them from the ballots, making the method the simpler and more
"intuitive" MDD//ER-Bucklin(whole).)

I think the only criterion compliance of plain ER-Bucklin(whole)  that  
we lose is  "Later-no-Help", which I don't expect anyone will miss.

What we gain is "Smith-Condorcet (Gross)"  (which means that the winner 
comes from the smallest non-empty set of candidates  that all
majority-strength pairwise beat any and all outside-the-set candidates)  
and the  Strategy-Free Criterion (SFC) .
Electowiki  definition of  SFC:

> /If a Condorcet candidate exists, and if a majority prefers this 
> candidate to another candidate, then the other candidate should not 
> win if that majority votes sincerely and no other voter falsifies/ any 
> preferences.
>
> In a ranked method, it is nearly equivalent to say: /If more than half 
> of the voters rank /x/ above /y/, and there is no candidate /z/ whom 
> more than half of the voters rank above /x/, then /y/ must not be 
> elected./
>



The method meets  FBC/Sincere Favourite, Majority for solid 
coalitions/Mutual Majority, the Plurality criterion, Smith(Gross), SFC
(and GSFC?).

It fails  Clone Independence (doubtless both Clone-Winner and 
Clone-Loser),  Later-no-Harm,  Independence from Irrelevant Ballots,
No Zero-Information Strategy.

Compared to plain ER-Bucklin(whole), this method would have a less 
severe Later-no-Harm problem. In ER-Bucklin(whole), if the voter
has a big sincere approval cutoff, the s/he should equal-rank above it 
and truncate below it. 
In MDD,ER-Bucklin(whole), if  the voter's sincere approval cutoff is 
above the most preferred member of  the anticipated sincere Smith
set  then it is probably safe to truncate below that candidate.

In my view the worst feature of this method is its bad clone problem. 
Also I dislike methods that fail  Irrelevant Ballots (in the same spirit
as  the "Blank Ballots Criterion").

But combining FBC with Majority for solid coalitions and  Smith(Gross) 
in my view makes it an ok package,  better than MDDA.


Chris Benham







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