[EM] Bucklin

[C.Benham]

There has been more than one version of Bucklin that has been used or 
proposed.

I think it is now generally agreed that least bad is to use some sort of 
limited-slot grading/rating ballot,
with voters free to give as many or as few candidates as they like the 
same grade and to skip grades if
they want.  (So it is just a version of Average Ratings)

If ratings ballots are used then above-bottom equal ranking should get 
the whole (not fractional) votes
interpretation, but if say a ballot equal top-ranks 3 candidates that 
ballot gives a whole vote to each in
the first "round" but then "sits out" rounds 2 and 3.

(Not doing that was shown to make the method fail mono-raise).

If we are talking about one of these versions then we are talking about 
a method that meets Favorite Betrayal
and Majority for Solid Coalitions and Later-no-Help.

But there is a very strong incentive for voters to just submit approval 
ballots, and in a competitive election with
informed voters the extra complexity versus simple Approval doesn't seem 
to buy much.

40 A>B
30 B
09 C
02 X

81 ballots.

This example highlights the method's disadvantages compared with IRV/RCV.

I don't consider meeting Majority for Solid Coalitions to be an adequate 
standard of majoritarian representative goodness.

I propose the "Dominant Coalition" criterion:

*If a the number of ballots on which a set S of candidates is 
ranked/voted all below no outside-S candidate is greater than the maximum
pairwise opposition that any inside-S candidate gets from any outside-S 
candidate, then the winner must come from set S.*

The single-candidate version (that could be relevant for a method that 
fails Clone-Winner) is "Dominant Candidate".

*If the number of ballots on which candidate X is ranked/voted below no 
other candidate is greater than X's maximum pairwise opposition,
then X must win.*

Another criterion I like (and I think I coined) is Irrelevant Ballots 
Independence: adding or removing ballots that contain no information
relevant to any of the remotely competitive candidates should not change 
the result.

Another criterion met by IRV/RCV but not Bucklin is Mutual Dominant 
Third : "if a set S of candidates that pairwise beat all the outside-S
candidates are voted above all the outside-S candidates on at least one 
third of the ballots then the winner must come from S."

In the example A is the Dominant Candidate and the Mutual Dominant Third 
candidate (and so of course the CW) but the Bucklin winner
is B.
(A lot of people like IRV's compliance with Later-no-Harm. Of course 
here if the A>B voters had truncated then A would have won.)

But if we remove the 2 X ballots the winner changes from B to A, 
(because the majority threshold lowers from 41 to 40, so now there is no
second round) a failure of Irrelevant Ballots independence.

Chris B.