[EM] Meta-criteria 6 of 9: Heuristics. #1, simplicity

[Juho]

On May 7, 2010, at 12:19 AM, Jameson Quinn wrote:

> In particular, for Schulze voting, here's the pitch: "The basic idea  
> is to elect the person who wins against all others. If there's no  
> such person, you try to eliminate the minimum number of ballots  
> until there is. But you don't want to have to bring the ballots to a  
> central location and then try every combination of ballots to  
> eliminate. So there's a process that is designed to almost always  
> give the same answer, but can be done using a local count..." (now,  
> if people ask, you can describe the method.)
>
> I know, the beatpath is not always the same as eliminating ballots,  
> if you have more than 3 candidates in the Smith set and [some other  
> improbable criteria which are too involved to state here]. But for  
> me, personally, I am more likely to support the Schulze method now  
> that I understand it as a summable approximation of minimal-ballot- 
> elimination. And for those who support it more than I do, I think  
> that pitching it as such is honest and useful.

The concept of ballot elimination is a bit complex since there could  
be different kind of ballots and some kind of ballots (that we would  
like to eliminate) might not exist. I guess you were saying that if we  
had all the original ballots available (not only the summable pairwise  
comparison matrix) then we could see how many voters we need to ignore  
to get a Condorcet winner.

Another closely related concept is how many additional voters we would  
need to make someone a Condorcet winner. The benefit of this approach  
is that it is simpler and now we are not bound to the actual votes. It  
is enough to just use simply additional bullet votes if we want to  
lift one of the candidates above the others (later preferences have no  
influence on the pairwise comparisons between the to be winner and  
other candidates).

With the additional votes concept the answer to what is the optimal  
method from this point of view is also obvious. It is minmax(margins).  
I don't know how much the result would change if we would use the  
concept of eliminating some (existing) votes instead. There are also  
other criteria like the clone criterion. Minmax(margins) fails that  
criterion (in some special situations). This means that the lowest  
number of additional votes criterion and the clone independence  
criterion are mutually exclusive.

One could discuss which rule should apply in those special cases when  
both criteria can not be met. In order to determine exactly when we  
have true clones in our hands we would need to have the original  
votes, and also the preference strengths to know if the candidates are  
closely related or not. (Actually also near clones should be treated  
as clones since we can not expect that all voters treat those  
candidates as clones.) The pairwise matrix contains only partial  
information. If we make a method 100% clone proof using the matrix  
information only we can not limit to the clone cases only but we are  
bound to influence the result also in other cases. One pairwise matrix  
can be obtained both from votes that have clones and from votes that  
do not have clones. It is for example possible that no voter ranks  
together those candidates that we must now deem to be (potential)  
clones since there is a possibility that in some other vote set they  
could be clones. The Schulze method uses path heuristics to eliminate  
all cases where clones could exist and influence the end result.

(Are there other strong reasons behind the use of paths? In real life  
the existence of the long beat paths maybe doesn't refer to any  
natural key target.)

I'll ask myself second time which rule should apply in those special  
cases when both criteria can not be met.

In minmax(margins) one could have three strongly looped clones that  
together as a group have 51% support, and one more candidate with 49%  
support. If one knows that all those circular preferences are weak in  
the sense that most sincere opinions (with utility information) of the  
clone supporters look like C1=99 C2=98 C3=97 A=1 then minmax(margins)  
maybe makes a mistake and one of the clones should be elected instead  
of A (whose worst loss was much smaller in terms of pairwise  
preference counts than the worst loss of any of the clones). On the  
other hand those clones could be severely fighting against each others  
and the strong circular opinions and resulting strong opposition could  
hamper the work of those clones (or "clone looking bitter enemies of  
one wing of the political spectrum") if elected, in which case A (that  
is anyway few votes short of being a Condorcer winner) might be a good  
winner.

In another possible situation the same pairwise matrix has been  
generated from votes where all voters rank one of the C candidates  
first and one last and thus never all together. In this case the  
argument that the candidate that needs only few additional votes to  
become a Condorcet winner should win gets stronger since there is no  
clone argument present. Any clone proof method that uses the pairwise  
matrix to make the decisions must pick one of the (non-clone) C  
candidates in this case.

Another characteristic feature of the Schulze method is the use of  
winning votes. My understanding is that the history behind winning  
votes is mostly based on strategic voting related concerns. Unless use  
of winning votes is considered ideal for sincere votes, this decision  
means some deviation from electing ideal winners wit sincere votes.

Yet another possible factor that may influence this discussion on what  
the basic idea behind Schulze method (and other methods) is is the  
concept of implicit approval cutoff after the ranked candidates. Some  
criteria and discussion on what the ideal winner is do refer to the  
assumption that voters have indicated that they support/approve those  
candidates that they have ranked and do not support candidates that  
they have not ranked. (Depending on the ballot type and number of  
candidates and interpretation that could mean truncation or candidates  
ranked equal last.) This interpretation of the votes is thus not  
purely ranking based but includes also additional information. One  
problem of this approach is that if voters behave this way then will  
not express their preferences on the preference between those  
candidates that they do not like, and that could mean high level of  
truncation and bullet voting. Implicit approval argumentation usually  
appears together with arguments on why winning votes are natural or  
how they work.

My understanding of the history of developments behind the Schulze  
method is that in addition to Condorcet compatibility one has aimed at  
summable matrix, 100% independence of clones, defence against some  
strategic voting patterns (=> winning votes), deterministic decisions  
(best candidate elected, no lotteries, except when exact ties). I  
don't think the interest to aim at minimum number of ballots that must  
be eliminated (or added) has been a key target, at least not the  
leading one. The strategic concerns must have been strong if one  
considers winning votes not to be optimal with sincere votes.

In addition I tend to think that Smith set compatibility (that is  
related to clones) and maybe some interest to serialize the group  
opinions (related to Smith) have played some role (not necessarily a  
good target). The evolution of the Condorcet methods has gone from  
simpler methods to more complex ones, with the intent to patch some of  
the identified problems. In such evolution process it is possible that  
some fixes may unintentionally cause more damage than they fix  
problems. For example in the area of strategies it is typical that a  
modification that helps in some set-up will make the vulnerabilities  
worse in some other set-up (e.g. winning votes). Similar balance  
related problems may appear also in performance with sincere votes  
(e.g. additional votes vs. clones).

Maybe Markus Schulze and others that have worked with and studied  
Ranked Pairs, River etc. can give some more light on the historical  
and current motivation.

Juho