[EM] IRV vs. approval: dominant mutual third

[James Green-Armytage]

Kevin Venzke wrote:
>Clones, and a majority-strength coalition for a set of more than one
>candidate, hardly ever exist. I would love to see a real IRV election
>where they did. 

	In a real public election, there is a huge difference between the
probability of a clone set and the probability of a mutual majority set.
	Consider the definition of a clone set: *Every single voter* needs to
rank the members of the clone set alongside one another. For example,
given would-be clones C1, C2, and C3, and other candidates X, Y, and Z, if
only one voter out of millions votes X>Y>C1>C2>Z>C3, then it is no longer
a true clone set. Thus I think that the probability of a true clone set in
a large public election is negligible. (This isn't to say that clone
independence isn't an important criterion, however. For example, lack of
clone-independence is a very severe problem in the Borda count, where the
existence of clones can be a huge advantage to a given faction.)
	Mutual majority sets, on the other hand, shouldn't be terribly unlikely
to occur in a public election. It's hard for me to really quantify the
probability, but I definitely don't think that it's negligible. But, then
again, I wouldn't argue that it's going to happen every time, either. 
	I do think, though, that when a mutual majority set does occur, then it
is pretty important for a candidate from that set to win. If another
candidate wins, then you're violating majority rule in a pretty flagrant
way.

	I want to introduce a definition for another kind of set, similar to the
mutual majority set, but one with a higher probability of occurrence. I
call it a dominant mutual third set.

Dominant mutual third set: A set of candidates such that every candidate
within the set pairwise-beats every candidate outside the set, and more
than one-third of the voters prefer the members of the set over every
non-member of the set.

	This is specifically designed to be a more-probable version of the mutual
majority set, which IRV still always selects a candidate from when such a
set exists. (That is, it selects a candidate from the minimal mutual
dominant third set--blagh, that's a mouthful.)
	So yes, I am inventing this criterion specifically so that IRV can pass
it, and I'm not trying to conceal that fact at all.
	Notice that a mutual majority set is always a dominant mutual third set,
but a dominant mutual third set is not always a mutual majority set. (A
mutual majority set is always also a dominant mutual majority set in that
every member will beat non-members in pairwise comparison.) Hence the
probability of a dominant mutual third set occurring is higher.

	Axiomatically, I think that if there is a dominant set, then
single-winner methods should always take the winner from that set. (Note
that the Smith set is the minimal dominant set, that is, the smallest one
that can be defined given a set of preferences.) Unfortunately, IRV does
not always choose a member of the minimal dominant set. However, it does
at least choose from a dominant set when members of the set are preferred
by non-members by at least a third of the electorate.
	Which, of course, is more than approval voting can offer. 

	Making approval failure examples is a bit like shooting fish in a barrel,
but I suppose that I should make one here to illustrate my point.

Sincere preferences
18: A>B>C>D
15: A>D>C>B
20: B>C>A>D
1: B>A>C>D
15: C>B>D>A
19: D>C>B>A
12: D>A>B>C

(By the way, B and C in the example above aren't a true clone set,
although they would be if not for that lone B>A>C>D voter.)

Sincere preferences, plus approval cutoffs
13: A>>B>C>D
5: A>B>>C>D
15: A>>D>C>B
13: B>>C>A>D
7: B>C>>A>D
1: B>>A>C>D
10: C>>B>D>A
5: C>B>>D>A
14: D>>C>B>A
5: D>C>>B>A
12: D>>A>B>C
	
approval score
A: 33
B: 31
C: 27
D: 31

IRV tally
A		B		C		D
33		21		15		31
				-15
		+15
33		36				31
+11		+20				-31
44		56

pairwise comparisons
		A		B		C		D
A				45		46		54
B		55				51		54
C		54		49				53
D		46		46		47

	So, B is a Condorcet winner, and B also happens to win with IRV. The
point as far as IRV goes, though, is not that it produces a Condorcet
winner, but that it chooses a winner from the dominant mutual third set,
which in this case consists of B and C.
	Approval happens to elect A, although you can adjust the approval cutoffs
so that any one of the four candidates is elected.

	How exciting is the dominant mutual third set? Maybe not that exciting.
All I'm trying to do is make a small response to those who say that mutual
majority cases are IRV's only shining glory, by defining a slightly more
probable situation where it does relatively well. Dominant mutual third is
basically a weaker version of mutual majority.

sincerely,
James