[EM] Defensive strategy for Condorcet methods

[Juho Laatu]

On 13.6.2011, at 17.33, Kristofer Munsterhjelm wrote:

> Kevin Venzke wrote:
> 
>> Is Condorcet//FPP a bad method?
> 
> I agree with Jameson Quinn, the gap is too far and so it could be quite tempting to compromise as in FPTP (and failing that, to engineer a cycle if your candidate has great first place support).
> 
> Smith,FPP... perhaps better, but there's still a gap between the Condorcet and the FPP part.
> 
> If you want something that deters burial strategy, how about what I called FPC? Each candidate's penalty is equal to the number of first-place votes for those who beat him pairwise. Lowest penalty wins.
> Burying a candidate may help in engineering a cycle, but it can't stack more first-place votes against him. Unfortunately, it's not monotone.
> 
> Finding the most strategy-resistant monotone Condorcet method is an interesting problem. If you permit approval cutoffs, UncAAO and C//A are probably quite good, but if not... what, I wonder? Perhaps some Ranked Pairs variant where winning contests are sorted ahead of losing contests, and then sorted further by FPP score of the first person in the ordering (e.g. A for A>B and B for B>A)? Or some Maxtree generalization. Who knows?

Yes, this is an interesting problem and the FPC approach is an interesting approach. Maybe the number of problem and potential for improvements could be the burial strategy. One can study the resulting cyclic preferences and try to identify who are strategists and not let them win anyway. I see the FPC philosophy coming from this direction.

So, let's focus on burial and the most typical cases there. Let's study the simplest and probably most common case where there are three candidates and they form an artificial loop as a result of someone using the burial strategy.

There are three candidates (A, B, C). A is the sincere Condorcet winner, B is the strategist, and C is the candidate that B supporters use to bury A.

FPC could reduce strategic behaviour if C is a weak candidate that does not have as many first place supporters as A and B. In that case A gets only  a small number of penalty points. And as a result the sincere Condorcet winner wins. This is a good result from strategy avoidance point of view. One problem is however that not all burials follow this pattern. One could have e.g. votes 35: A>B, 25: B>A (=> strategic B>C), 40: C. Now C has the highest number of first place preferences.

First preferences is thus one way to analyze the loop of three in order to find the "A", "B" and "C" roles there. Another possible (but far less common) problem with first preferences is that some of the candidates might have clones. There could be two candidates A1 and A2 instead of one A. In that case the first preference support of A1 and A2 would be lower. Also number of minor candidates and their position on the political map may have an impact since they all tend to steal some first preference votes from A, B and C.

Another approach to analyzing the cycle would be to check the defeat strengths. Thera are problems also in this approach. In some typical scenarios the B>C pairwise victory os strong, but not in all scenarios.

A third approach would be to check the number of voters that gave an opinion on each pairwise comparison (instead of typically ranking them equal last). In some scenarios comparison A vs. B could have low number of indicated opinions (e.g. in my example above). But again, not all burial scenarios will follow this pattern. (The expected overall popularity and voter's distance to different candidates have an impact on the probability of giving a pairwise opinion on some pairwise comparison.)

One could also analyze the actual ballots to see which how near clones those three candidates are. A and C are not near clones since B could not bury A under a clone of A. B and C are not near clones since B could not bury A under a clone of B. So, if there are near clones, they must be A and B. In my example above A and B are indeed sincere clones. But in the strategic votes the clone relationship is lost.

It seems that it is not very easy to draw reliable conclusions from the matrix or the actual ballots. Maybe the probability of different burial scenarios is different and therefore we could make some statistical guesses on which candidates are in which roles. Our guess should be such that the winner is either the sincere Condocet winner A (=> no harm done) or C (=> disincentive to bury). We should thus just avoid electing B and rewarding the strategists. Of course it would be good if the method would elect a decent winner also in the case of a sincere loop.

There are also cases where we can not draw any conclusions. One basic example is a symmetric loop. Maybe the strategic / actual votes are 33: A>B>C, 33: B>C>A, 33: C>A>B. The sincere opinions of the B supporters were maybe 33: B>A>C. We know that B is the strategist (according to our naming convention) but the method can not make a difference between the three candidates. So this means that there are many near tie / near balanced cycle cases where we can make only rough guesses on who the strategist is. And of course it is always also possible that there are no strategists but all the votes are sincere.

One additional problem is that, depending on the chosen method, there may be also some defensive strategies like truncation involved. Such truncation may have impact also e.g. on the low participation analysis above. One must also assume some interest in bullet voting (especially among the supporters of the strongest candidates) (also without any defensive plans).

The question is if one can draw some statistically meaningful conclusions from the matrix and ballots (to determine which candidates are "A", "B" and "C"). Maybe the best approach would be to do a complete analysis on all the burial scenarios and then estimate the probability of each one. One should check also the impact of counterstrategies, multiple strategies and disincentives on each method. What other useful data (input data for the method) can we derive from the matrix and ballots than the few points discussed above? Would ratings help (ref. James Green-Armytage's cardinal-weighted pairwise)?

Juho