[EM] Unmanipulable majority and Condorcet

[Kristofer Munsterhjelm]

A long time ago, I said that my fpA-fpC three-candidate method passed UM 
and Condorcet. I've now found out that that's not the case.

Consider this election:

2: A>B>C
2+x: B>A>C
1-x: B>C>A
3: C>A>B

With x=1, A is the CW; with x=0, we have a tie-cycle (A>B, B>C, C=A). 
With an x slightly less than zero, the election is a proper ABCA cycle.

The candidate scores are:
	A: fpA - fpC = 2 - 3 = -1
	B: fpB - fpA = 3 - 2 = 1
	C: fpC - fpB = 3 - 3 = 0

so the ordering is B>C>A no matter what x is within the allowed range.

I've been thinking about particular types of three-candidate Condorcet 
methods. What I've been doing suggests that for a continuous method to 
pass UM and Condorcet, it must return an A=B=C tie for every pairwise 
tie election (i.e. where A=B, A=C, B=C). Furthermore, adding ballots to 
turn A=B and A=C into A>B and B>C must make A win with certainty.

The reason is that if we're somewhere on the "A is the CW" side of the 
(A>B, B>C, A=C) boundary, there exists an election where changing an 
epsilon of a B>A>C ballot into B>C>A will get us *onto* that boundary. 
Such a change leads to an UM violation if A doesn't win with certainty 
afterwards.

(It also looks like the number of elections near the boundary with no 
B>A>C ballots is so small compared to the number of elections with, that 
any election that makes A win with certainty on the area of the boundary 
you can reach by turning B>A>C into B>C>A must make A win with certainty 
along the whole boundary. At least if it's continuous, although I've 
only properly investigated a form of linear method.)

"Center-squeezy" methods like fpA-fpC fail by not returning an A=B=C tie 
on every pairwise tied election. Methods like minmax pass the boundary 
condition, but fail inside the ABCA cycle region itself.

What I'd have to show to have an actual impossibility proof is that it's 
impossible to construct a method that both respects UM when going from 
the "A always wins" region to the ABCA cycle region, and also respects 
UM inside the region. I don't have that (yet, at least). But if I were 
to guess, I think Condorcet and UM are incompatible. This is unfortunate 
because it would otherwise be a good criterion to use to find 
manipulation-resistant Condorcet methods.