If there is a polyspace summable method to determine the mutual majority
set, that method can't be using the pairwise matrix alone. After
tinkering a bit, I found an example where two different sets of ballots
give the same pairwise matrix, yet have different mutual majority sets;
since a method that only knows the pairwise matrix can't distinguish
between the two and thus will have to make at least one mistake.
The two ballot sets are
1: A>B>C
1: B>C>A
1: C>B>A
A>B: 1 A>C: 1 B>C: 2, mutual majority set: {BC}
and
1: B>A>C
1: B>C>A
1: C>A>B.
A>B: 1 A>C: 1 B>C: 2, mutual majority set: {B}.
Note that there's always at least one common member of the two mutual
majority sets - in this case, B - or no methods using the pairwise
matrices alone could hope to satisfy mutual majority.
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I also found out that it's possible to detect the mutual majority set in
a three candidate election using the positional matrix. The positional
matrix is an n*n matrix where entry [a][b] is how many voted the ath
candidate in bth place.
It's rather obvious in retrospect, and since there are 6 entries in the
positional matrix for three candidates and 6 possible (non-truncated,
non-equal-rank) rank orderings, we aren't really compressing the data by
any amount, but it surprised me at the start.
The key observation is that for three candidates, a mutual majority set
is either:
- a single candidate,
- two candidates, i.e. everything except a single candidate, or
- everybody.
To check if a single candidate is in the set is easy: simply check if a
majority voted that candidate first. To check if it's a two-member
mutual majority set is also easy: check if a candidate was voted last by
a majority. If he was, the mutual majority set consists of the other two
candidates. If neither is true, the mutual majority set is the set of
all candidates.
Because it's possible to get the number of voters who voted, say, {AB} >
C, directly, that would permit a three-candidate method that satisfies
Droop proportionality both for one and two candidates. For two
candidates out of three, the constraints would be:
- if a candidate is voted last by > 2/3, elect both but him
- if a candidate is voted first by > 1/3, you have to elect him
- otherwise, do what you want.
I don't think it will be easily transferred to more than three
candidates, but it suggests I could make my set Bucklin method more
proportional (if not fully DPC) by considering both "exclusions" from
the last place towards the first, as well as "inclusions" from the first
place towards the last.
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