[EM] Condorcet strategy and weighted pairwise method

[James Green-Armytage]

Dear election methods fans,
	Heavens, this is strange... this is the longest gap I've experienced
since I've been on the list. Is everyone at the beach or something? I hope
so, I guess; that's a nice place to be in the summer.
	Anyway, in case anyone's still out there, I'd like to share my analysis
of strategic manipulation in the weighted pairwise method, compared to
strategic manipulation in winning votes and margins. My overall thesis is
that although of course weighted pairwise doesn't totally obviate any use
of strategy, it is more resistant to manipulation than both winning votes
and margins Condorcet. That is, in some situations, highly-damaging
manipulation is quite possible in winning votes and margins, but
impossible in weighted pairwise. In other situations, manipulation is
possible in all methods, but the counter-strategies which can be employed
in weighted pairwise are much less risky and unstable than their
counterparts in winning votes and margins.
	Therefore, I like the weighted pairwise method for two main reasons: One,
it offers more meaningful solutions to sincere majority rule cycles, and
two, it makes it harder for strategic manipulation to get out of hand. The
fact that it kills these two rather significant birds with one stone makes
me believe that it is a very good method, worth the extra complexity of
having ratings ballots.
	Anyway, without further ado, here is the my analysis. Actually this
section is currently a part of my overall weighted pairwise proposal
paper, which can still be found at
http://fc.antioch.edu/~jarmyta@antioch-college.edu/voting_methods/weighted_pairwise.htm

my best,
James
________________


IV    Coping with strategic manipulation:
            The primary purpose of the weighted pairwise method is to
provide more meaningful solutions to majority rule cycles when sincere
votes are cast. However, in addition to this, there is some reason to
believe that weighted pairwise is actually less vulnerable to strategic
manipulation than most other Condorcet-efficient methods. I will try to
illustrate this using a few simple examples, which I believe can provide
insight into a variety of more complicated situations. In these examples,
there are two candidates A and B who are relatively similar to one
another, and account for 53% of the first choice vote, and a candidate C
who accounts for the remaining 47%. 
            Case 1.0, sincere preferences:
23: A>B>C
5: A>C>B
23: B>A>C
2: B>A>C
25: C>A>B
22: C>B>A
	Pairwise comparisons:
A>B = 53>47
A>C = 51>49
B<C = 48<52
            Case 1 deals primarily with the possibility of strategic
incursion by the voters who favor candidate C. If sincere votes are cast,
A beats both B and C, while C beats B. However, those whose preferences
are C>A>B have an opportunity to gain an advantage by insincerely voting
C>B>A. If we are using a version of minimax that is based on margins, then
the C>A>B voters can achieve them simply by truncating their ballot,
effectively voting C>A=B. If 9 of them do this, we have 
            Case 1.1, altered preferences:
23: A>B>C
5: A>C>B
23: B>A>C
2: B>C>A
16: C>A>B
9: C>A=B (sincerely C>A>B)
22: C>B>A
	Pairwise comparisons:
A<B = 44<47
A>C = 51>49
B<C = 48<52
            The resolution to this cycle depends on whether we are
focusing on margins or winning votes. The defeat of least margin is A-->C
(51-49), but the defeat of least winning votes is B-->A (47-44). The
latter defeat is artificial, and the former genuine. In general,
margin-based versions allow a group of voters to change the result from a
Condorcet winner to another candidate they prefer simply by truncating. In
winning votes versions, it is necessary for voters to actually reverse
preference orders to achieve this result. In this way, strategic
manipulation is generally easier in margins-based versions. Therefore, for
the rest of my analysis here I will focus on winning votes Condorcet, and
compare it to the weighted pairwise method. (We can assume that the
completion method is minimax, since it does not differ from beatpath or
ranked pairs in three-candidate examples.)
            So, using winning votes Condorcet, C>A>B voters can gain an
advantage from case 1.0 by reversing their preferences and voting C>B>A.
For example...
              Case 1.2, altered preferences:
23: A>B>C
5: A>C>B
23: B>A>C
2: B>C>A
19: C>A>B
28: C>B>A (6 of these are sincerely C>A>B)
	Pairwise comparisons:
A<B = 47>53
A>C = 51>49
B<C = 48<52
            Now, the weakest defeat is A-->C (51-49), and dropping that
leads to the election of C. As it stands, the strategy has been
successful. Now, if the method is winning votes Condorcet, and it is
suspected ahead of time that the B voters will use this strategy, what can
the other voters do about it? 
	Voters who have already ranked A first cannot do anything more to help A;
actually the only change they can make is for the 5 A>C>B voters to change
their vote to A>C=B, which reverses the C-->B defeat and makes B a
Condorcet winner. Of course, this change is not an improvement for an
A>C>B voter. Hence the only possible value of the possibility is that
perhaps if the A>C>B threaten to do this before the election, they might
be able to deter the C>A>B voters from trying their strategy. However,
such a counter-strategy is fraught with instability, since the different
groups of voters have no way of knowing for sure how the other groups will
vote until it is too late to make a change.
            Another possibility is for the B>A>C voters to compromise by
voting B=A>C. If at least 3 of them do this, the B-->A defeat will be the
weakest in the cycle, and will be dropped. The problem with this
(counter)-strategy is that voters will not have perfect information before
the election. Instead, they will face the possibility that B is a sincere
winner (even a sincere Condorcet winner), and that voting B=A>C instead of
B>A>C would be to needlessly hand the victory over to A. Hence the B>A>C
voters will face a dilemma between voting B>A>C, leaving open an
opportunity for the C voters to steal the election, and voting B=A>C,
giving up the hope of getting their first choice elected.
            However, when the weighted pairwise method is used instead, it
seems that this dilemma can often be avoided. In many cases, I think that
the initial strategy of the C>A>B voters switching to C>B>A wouldn’t even
be effective in the first place; hence counter-strategy would be
unnecessary. That is, going with the assumption that candidates A and B
are relatively similar, it is likely that the ratings gaps between A and B
will be relatively small, and the ratings gaps between {A, B} and C should
be relatively large. For example,
            Case 1.0 revisited, sincere preferences, with ratings
exaggerating the largest sincere utility gap:
23: A 100 > B 100 > C 0
5: A 100 > C 0 > B 0
23: B 100 > A 100 > C 0
2: B 100 > C  0 > A 0
25: C 100 > A 0 > B 0
22: C 100 > B 0 > A 0
	Pairwise comparisons, with weighted magnitudes:
A>B = 53>47, 500
A>C = 51>49, 5100
B<C = 48<52, 4700
            Again, A is a Condorcet winner, and C beats B. The C voters
cannot change the fact that A beats C. Also, even if they reversed the A>B
defeat to make a cycle, they couldn’t do anything to get the A-->C defeat
anywhere close to being the weakest in terms of weighted magnitude. Hence,
no matter what they do, C won’t win. The C>B>A voters can’t do anything to
change the result, and although the C>A>B voters could cause a cycle by
voting C>B>A, there’s no reason that they would want to do this, since the
cycle would be very unlikely to resolve in favor of C, likely to resolve
in favor of A, and possible that it would resolve in favor of their last
choice, B. The latter would happen if the C>B>A voters put most of their
ratings gaps between B and A. So, it is nice to see that the C voters who
are needed to cause the cycle are precisely the ones who have the least to
gain from it. Also, of course, the B>A>C voters as a group can’t do
anything to get a better result, since voting B>C>A will simply make C a
Condorcet winner instead of A. The B>C>A voters cannot change anything
either, since they are already voting in retrograde to the full transitive
ordering of A>C>B. And of course, none of the A>B>C or A>C>B voters are
interested in changing the result. So, this is a relatively stable outcome
when weighted pairwise is used. 
	In general, I think that in weighted pairwise, when there is one
candidate who is a sincere winner, it is usually impossible to steal the
election in favor of another candidate, when those who favored the sincere
winner over the artificial winner put a large ratings differential between
the two candidates. To be more specific, if any defeat has more than 1/3
of the highest possible weighted magnitude, it can’t be dropped. For
example, there are 100 voters and the ratings ballots are 0-100, and a
given defeat of candidate Y by candidate X has a weighted magnitude of
3334, candidate Y cannot win. If a majority (1/2+) prefers X over Y, and
they assign at least 2/3 of their rating differentials to the gap between
X and Y, then you already have the weight needed to assure that Y will not
win. And in most cases, it is unnecessary to have the full 1/3, as it
should be rare for the tally to drop a defeat with close to 1/3 of maximum
magnitude.
	So, a candidate who suffers such a defeat cannot be elected, no matter
how those on the losing side of the defeat manipulate their ballots.
Hence, while election results can be manipulated by voter strategy, such
manipulations are extremely unlikely to award a victory to a candidate who
is polar opposite to the sincere winner, which is unfortunately not true
of most other Condorcet methods.
            However, what about strategic manipulation by supporters of a
candidate who is relatively similar to the sincere winner? This question
brings us to...
            Case 2.0, sincere preferences, with ratings exaggerating the
largest sincere utility gap:
28: A 100 > B 100 > C 0
25: B 100 > A 100 > C 0
24: C 100 > A 0 > B 0
23: C 100 > B 0 > A 0
            Case 2 is pretty similar to case 1, the only major difference
being that in case 2, B has a pairwise win against C rather than the other
way around. A is again a sincere Condorcet winner, but in this case there
is a possibility of strategic incursion by the B voters. Using winning
votes Condorcet, some B>A>C voters can reverse their later preferences to
get this result.

            Case 2.1, altered preferences:
28: A>B>C
18: B>A>C
7: B>C>A (sincerely B>A>C)
24: C>A>B
23: C>B>A
            Pairwise comparisons:
A>B = 52>48
A<C = 46<54
B>C = 53>47
            Now the A-->B defeat is dropped and B wins instead of A,
thanks to his underhanded supporters. How can the other voters respond to
this in winning votes? 
            As in case 1, although the A voters can’t do anything further
to elect A, they can threaten to punish the B voters by truncating. If the
B voters order-reverse as above, and at least 2 of the A>B>C voters
truncate (vote A>B=C instead), then C will win. So if the A voters
threaten to truncate, the B voters may repent and vote sincerely. However,
again, this is a fundamentally messy and uncertain counter-strategy. What
if the a spokesman for the A voters threatens to truncate, and a spokesman
for the B voters promise to vote sincerely, so the A camp agrees not to
truncate... but then on election day a contingent of B voters take matters
into their own hands and truncate anyway, stealing the election for B
after all? That’s bad news. Or, what if a group of A voters decides to go
ahead and truncate in case the B voters try a reversal strategy, and a
group of B voters truncate in case B is the Condorcet winner but the A
voters try a reversal strategy? If a small fraction of voters from each
camp truncates, then they will hand the election directly over to C, which
is the last thing they want. 
            The C>A>B voters could take it on themselves to settle the
issue by voting C=A>B and ensuring A’s victory, but they probably won’t
want to do that if they’re not yet sure who will win the sincere A-B
pairwise contest. So there’s a fair amount of risk-taking and guesswork,
with a chance for disaster if people guess wrong.
            Can the weighted pairwise method improve the situation at all?
Well, a bit, perhaps. It might be more necessary to use conscious
counter-strategy than it was in case 1, but the counter-strategies that
can be applied are arguably less risky than their counterparts in the
winning votes system. 
	First, the “deterrent” strategy used by the A supporters in winning votes
Condorcet. In weighted pairwise, they can get a similar effect by
maintaining their full sincere rankings but altering their ratings
information to increase the gap between A and B, for example, voting A 100
> B 0 > C 0 instead of A 100 > B 100 > C 0. Hence, they decrease the
chances that an A-->B defeat will be dropped, thereby decreasing the
incentive for the B voters to create a cycle. This counter-strategy is
somewhat more stable than the deterrence-by-truncation strategy. That is,
even if lots of A voters and lots of B voters mistakenly employ it in the
absence of actual order-reversal strategy, it doesn’t affect the direction
of the pairwise defeats. Hence, in this example, it would have no impact,
since C is a Condorcet loser and hence none of the C voters have an
order-reversal incentive.
	Second, the “compromising” strategy used by the C>A>B voters in winning
votes Condorcet, voting C=A>B in order to protect A. In weighted pairwise,
they can get a similar effect by voting C 100 > A 100 > B 0 instead of C
100 > A 0 > B 0. Thus, in case some of the B>A>C voters switch to B>C>A,
creating a false A-->B-->C-->A cycle, this strengthens the A-->B defeat,
and increases the chance that A will win rather than B. Again, this
strategy is less likely to be regretted than the equal-ranking compromise
in winning votes Condorcet, since it doesn't affect the direction of the
pairwise contests, and hence can't change a sincere pairwise win by C into
a pairwise loss.
	Of course, none of this is to say that the weighted pairwise method is
totally invulnerable to strategic manipulation, or that strategy will
never lead to an unfair or unstable outcome. However, I do think that it
is less vulnerable to strategy than most other Condorcet methods, whether
based on margins or winning votes. In the highly-unlikely event that a
group of voters will be able to manipulate the result to steal the
election for their favorite candidate, there is a significant limit as to
how different that candidate can be from the sincere winner. Also, when
people anticipate strategy and engage in counter-strategy, there are
effective counter-strategies available in the weighted pairwise method
which are more stable and less risky than their counterparts in margins
and winning votes.