[EM] AWP vs. S/WPO

[James Green-Armytage]

Dear election methods fans,

	I have previously claimed that AWP provides greater resistance to
strategy than S/WPO (added to WV), and I would like to lend some support
to this claim in the following post.

Example, sincere AWP votes:
26: A>>B>C
19: A>>C>B
7: B>>A>C
22: B>C>>A
4: C>>A>B
19: C>B>>A
3: C>>B>A

	B is a Condorcet winner. B and C are similar candidates, as measured by
the average rating differentials between them.

Average rating differentials (from 0 to 1):
A>B: 45/49 = .92
B>A: 48/51 = .94
A>C: 45/52 = .86
C>A: 48/48 = 1.00
B>C: 7/55 = .13
C>B: 7/45 = .16

	The only voters who have a strong interest in changing the result (i.e. a
change from a below-cutoff candidate to an above-cutoff candidate) are the
26: A>>B>C and 19: A>>C>B voters. However, in AWP, they do not have the
ability to change the result to A, because the B>A defeat has a weighted
magnitude of 48, and thus cannot be overturned as a result of a three
candidate cycle. Since those with the strongest interest in a burying
strategy have no opportunity to succeed, I say that this result is very
stable in AWP.
	In S/WPO, however, the A>>... voters actually do have an opportunity to
change the result to A. Assume that the sincere AWP translate into S/WPO
votes as follows:

26: A>>B>>C
19: A>>C>>B
7: B>>A>>C
22: B>C>>A
4: C>>A>>B
19: C>B>>A
3: C>>B>>A
	 
	Notice that these look somewhat different from the sincere AWP votes
above. For example, those who vote A>>B>C in AWP vote A>>B>>C in S/WPO.
Why do they vote a strong preference against C in S/WPO? If there is a 3
candidate A>B>C>A cycle, a strong B>C defeat can only help them. Of
course, they do not vote A>>B>>C in AWP because that is not a permissible
vote, and voting A>B>>C would undermine the strength of their A>B
preference, which they consider to be a higher priority than their B>C
preference. (Either way, their A>C preference is strong.)
	Similar logic applies to the ACB voters, the BAC voters, the CAB voters,
and the CBA voters. 
	Only the B>C>>A voters and the C>B>>A voters have a clear incentive to
express a weak preference in S/WPO. For example, consider the B>C>>A
voters. In the event of a B>C>A>B cycle, their top priority is that it
should not resolve in favor of A. Hence, weakening the B>C defeat makes
sense.
	Now, proceeding with the example, we see that the A>>... voters can gain
a victory for A by burying B under C, as follows:
 
45: A>>C>>B (26 sincerely A>>B>C)
7: B>>A>>C
22: B>C>>A
4: C>>A>>B
19: C>B>>A
3: C>>B>>A

	I use this example not as a proof of course, but as anecdotal data to
illustrate my point. In this example, strategic A>>... voters in S/WPO are
able to simultaneously add maximum strength to the A>C defeat and the C>B
defeat, enabling them to make the B>A defeat the weakest of the three. In
AWP, this is not possible; each A>>... voter needs to choose between
adding strength to A>C and adding strength to C>B, which means that no
matter how well they coordinate their efforts, the B>A defeat will not be
overruled.

	A more general statement of the argument is this: 
	Say that the number of votes cast is v, and a strong preference adds 1 to
the strength of a defeat. In AWP, a defeat with strength greater than v/3
absolutely cannot be overruled as a result of a 3 candidate cycle, and a
defeat with strength greater than v/2 absolutely cannot be overruled at
all. S/WPO cannot make either of these guarantees.  The fact that AWP can
make these guarantees gives it an anti-strategic stability that is not
shared by S/WPO.

	I await your questions, your comments, and (if needed) your corrections.

Sincerely,
James Green-Armytage
http://fc.antioch.edu/~james_green-armytage/voting.htm