[EM] Compatibility

[Juho]

On May 5, 2010, at 10:28 AM, Kristofer Munsterhjelm wrote:

> Juho wrote:
>> Let's say that there are three different criteria that an election  
>> method should meet. They are related to three different strategic  
>> voting related problems. As in the world of security also election  
>> methods are as vulnerable to strategic voting as their weakest  
>> link. In this situation there may be a solution that almost meets  
>> all the three criteria but does not meet any of them fully. That  
>> solution is often (but not always) the best solution.
>> So, there are patterns where a good method should meet only a  
>> strict subset of the requirements that some competing methods meet.  
>> One may thus improve a method by making it incompatible to some  
>> (good) criterion that it earlier met. The point is that  
>> compatibility with various criteria should often not be an on/off  
>> comparison but a richer analysis where also partial and almost  
>> complete compatibilities are counted.
>
> This seems to be an issue of the criteria being too demanding.  
> Consider, for instance, independence from covered candidates. I  
> think Forest Simmons said (in private correspondence) that IFCC  
> implies nonmonotonicity.
> If one considers monotonicity important, but also would like to have  
> the method be independent of as many types of non-winning candidates  
> as possible, the natural next thing to do is to ask: how strong can  
> we make this independence criterion and still satisfy monotonicity?

Yes, either the criteria are too strict or alternatively the laws of  
nature don't allow us to build perfect methods but we must always  
violate some important criteria. In the pairwise comparison based  
methods we must also take into account the fact that we do not measure  
the strength of personal preferences, and therefore the decisions that  
we make are often simplified assumptions (i.e. one set of votes may  
refers to a wide range of different possible sincere preference  
strengths). The pairwise matrix for example can not be used to  
determine which candidates are clones (or almost clones) (same matrix  
can be derived from ballots with clones or from some very different  
set of votes). Performance with sincere votes is one step easier to  
handle and define than performance when also strategic voting is  
included.

In some earlier discussion I proposed concepts like IIA-IAC (where IAC  
means "in the absence of cyclic preferences"). That approach gives us  
in one rough lower bound that for example in the case of IIA could be  
set as a requirement. After that one could either try to fine-tune the  
criterion to more exactly describe the level that we can reach, or  
alternatively we could take a practical approach and just estimate how  
often and how serious problems the "irrelevant" candidates can cause  
when there are cycles. It may be that often the optimal level of  
meeting IIA is not at some level that can be clearly defined as a new  
criterion but at some more vague level where the resistance to these  
problems just happens to be so small that we need not care about that  
problem any more.

Also monotonicity is a criterion that could be met only "in most  
cases". For example in multi-winner STV the non-monotonic features  
typically fall below the noise level and do not cause any complaints  
(or are not even noticed or can not be be proven). (There is however  
no reason to deviate from monotonicity if one is not forced to so for  
other reasons.)

>
> Similarly for your own situation; say that Smith compliance breaks  
> something else you want. Then you can try to find out what that  
> thing that you want is, and afterward attempt to find the closest  
> thing to Smith compliance that doesn't break it.

My use of the Smith set criterion (at least in some environments /  
societies / needs if not all) is such that Smith-IAC is obvious (same  
as Condorcet criterion). When there are cycles then I try to see what  
I'd like to happen with sincere votes. Most Condorcet based approaches  
try to minimize the opposition against the elected winner. In 99% of  
the cases (vote sets) it makes still sense to elect one of the Smith  
set members. But there are cases (<1%) where opposition against the  
Smith set members is (according to some definitions) is higher than  
against some canididate outside the Smith set. In that case and with  
these criteria (on who should be elected with sincere votes) it makes  
sense to elect outside the Smith set. The border line and definition  
of those cases is very clear if we use the amount of additional  
support needed to overcome the opposition to measure the level of  
opposition. We thus have a clear definition of the modified (less  
strict) Smith set criterion (but it is easier to define that criterion  
as the number of required additional votes than to modify the Smith  
criterion although the new criterion is close to the Smith set  
criterion).

Although that border line is very well defined there are still some  
questions left. Many different sets of sincere opinions and actual  
ballots may lead to one pairwise comparison matrix. In some situations  
the Smith set members could be clones (maybe actual clones from the  
same party with minor differences in preferences, maybe just  
candidates that look like clones but are actually highly competitive  
and against each others). Or one may find a mutual majority that would  
have strategic interest not to elect the candidate outside the Smith  
set. Or the cyclic preferences could be a result of some strategy. But  
on average and as a general rule and for practical use the default  
rule of measuring the strength of opposition after the election (from  
the pairwise comparison matrix) could be used. (Except that in real  
life these situations are probably so rare that most Condorcet methods  
lead to the same results anyway (with full rankings at least).)

>
> (For the record, I think Smith compliance is a good thing, and that  
> perhaps one should go even further - thus my interest in uncovered  
> set methods, IPDA, etc.)

Is that a rule for all societies / elections? Or a rule for most  
societies / elections?

Are you saying that there are no such elections where it would be in  
the interest of the society to elect the ("most agreeable") candidate  
with least opposition (to change the winner to one of the others)? Or  
maybe only that it would be better to limit the winners to the Smith  
set (and further) if if one must elect one method as a method that  
will be recommended as the best general purpose single-winner method  
for all societies and in all (or most) single-winner elections of the  
world.

Juho