[EM] Condorcet methods and Hare versus Median Ratings methods like Majority Judgement and Bucklin

[Chris Benham]

One of the candidates for the poll is Majority Judgement  (a "category" ).

A few months ago I posted something about a version of Bucklin that is 
also a Median Ratings method.

I reproduce it  here to explain why in the poll I will be voting it 
below Hare and all the reasonable Condorcet
methods:


I think it is now generally agreed that least bad is to use some sort of 
limited-slot grading/rating ballot,
with voters There has been more than one version of Bucklin that has 
been used or proposed. 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 ranking 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 Hare 
(aka IRV).

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 Hare'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.
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