[EM] Condorcet strategy and weighted pairwise method

[Dave Ketchum]

Is this method worth the pain?

Could be for some groups outside of public elections.

But think, for public elections:
      This is a variation on Condorcet which, if worthy, would be suitable 
for electing governors, where there can be thousands of polling places 
(precincts).
      Does not matter unless you have more than two candidates evenly 
matched enough to produce cycles - often enough to justify the pain 
(because cycles can happen they must be attended to - question is how much 
complexity to build in).
      Voters must understand marking these ballots (topic is mixing 
ranking and rating - stepping from Plurality to Ranked ballot ranking 
seems clearly understandable and worth its pain).
      Condorcet precinct results are an array, with the arrays summable 
for the whole district or any subdistrict.  IRV is not so simple.  Where 
does this method fit in as to complexity of doing and understanding?

Dave Ketchum

On Mon, 19 Jul 2004 21:30:49 -0400 James Green-Armytage wrote:

> 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 wouldnt 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 couldnt 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 wont win. The C>B>A voters cant do anything to
> change the result, and although the C>A>B voters could cause a cycle by
> voting C>B>A, theres 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 cant 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 cant 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 cant 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? Thats 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 As victory, but they probably wont
> want to do that if theyre not yet sure who will win the sincere A-B
> pairwise contest. So theres 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 doesnt 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.

-- 
  davek at clarityconnect.com    people.clarityconnect.com/webpages3/davek
  Dave Ketchum   108 Halstead Ave, Owego, NY  13827-1708   607-687-5026
            Do to no one what you would not want done to you.
                  If you want peace, work for justice.