[EM] Why I Prefer IRV to Condorcet

[Kristofer Munsterhjelm]

Juho Laatu wrote:
> --- On Thu, 27/11/08, Kristofer Munsterhjelm <km-elmet at broadpark.no> wrote:

>>> Minmax may elect the Condorcet loser only when there
>>> is no Condorcet winner. And only in situations where
>>> all other candidates are worse than the Condorcet
>>> loser from the minmax philosophy/utility point of
>> view.
>>
>> The problem is criterion compliance. Isolated, I think
>> passing Condorcet and failing Condorcet loser is a
>> contradiction, because this means you can possibly reverse
>> the election and get a "worst" that is the
>> "best". I know that there are weaknesses to my
>> argument (since others could make the same reasoning about
>> Consistency, for instance, and exclude all Condorcet
>> methods), but I think that inasfar as voting methods are
>> metrics of winners, and the reason for why one is supposed
>> to use this method is because of its criterion compliance
>> (which is really a way of saying certain ways of picking
>> winners/not picking winners is desirable), one should take
>> the reason to its full extent, which a method that fails
>> Condorcet loser doesn't do.
> 
> There are different kind of criteria.
> If one decides the winner based on one single
> vote a method that would elect the least
> preferred candidate would be bad. Things get
> however more complex with group opinions that
> may contain cycles. Then it is possible that
> some candidate loses to every other candidate
> but still is the most liked one in the sense
> that there is only a very weak interest to
> change that candidate to some other candidate.

Then you should advocate Minmax for being Minmax, not for being 
Condorcet compliant. If you do the latter, then people may argue that 
the system is inconsistent because it doesn't follow up the implication 
of Condorcet (Condorcet loser, etc). But to my knowledge, you want to do 
the former, so I won't comment on this.

  >> Smith isn't just a hardening criterion. In a sense, it
>> also assures voters that they can vote in a way they want
>> without having to compensate in order to get a candidate
>> from the Smith (or mutual majority, etc) set, if all other
>> voters are honest. In this way, it would be similar to
>> independence of clones: a cloneproof method tells voters
>> that now it matters much less whether candidates are loosely
>> spread or tightly clumped around an area, even if the
>> candidates were clumped/spread apart simply because of the
>> political environment (and through no adverse intent nor
>> strategic nomination).
> 
> I can see two kind of reasoning that people
> may use to justify the use of Smith set as
> a criterion that determines the best winner.
> 
> 1) Clone based. Smith set is some sort of an
> approximation of clone candidates. Smith set
> is however wider (wider than the set of
> candidates that are next to each others in
> every ballot). (Note also that candidates
> that are next to each others in every ballot
> need not be clones in the sense that they
> would be ideologically similar.)
> 
> 2) Drawing technique based. When drawing a
> graph that represents the results of the
> election one typically draws the Smith set
> candidates at the top of the paper, and all
> the other candidates below that group. Since
> people intuitively model also group opinions
> as linear preference chains this drawing
> technique may give them a false impression of
> the group preferences. The problem is that
> this drawing technique hides the defeats of
> the Smith set members to each others.

I would have two reasons as well, but none of those you mentioned. It's 
possible to be cloneproof without being Smith and vice versa..

1. Logical endpoint of mutual majority. A mutual majority set is one 
that a majority prefers to all else. Now consider a mutual dominant nth 
set. A mutual dominant nth set is a set that 1/n of all voters prefer to 
all the others, and where one of the candidates within wins, pairwise. 
Smith is just mutual dominant set with n->inf.

2. Condorcet for sets. Smith is Condorcet for sets. If a set can beat 
all those outside the set pairwise, it should win. If the set is of size 
one, well, that's just Condorcet. The only reason why it should hold for 
size one, but not, say, size two, is if some other heuristic (like the 
Minmax metric/utility heuristic) is more important. If it is, see my 
first paragraph; but if we want this method primarily because it's 
Condorcet, or because the Condorcet idea itself is a good one, then we 
should be consistent and take that Condorcet as far as possible.

>>> One could see Kemeny as a good definition of a good
>>> social ordering. That may or may not correlate with
>>> the definition of the best single winner.
>> If the concept of a social ordering is to have any use, I
>> think the winner must be first on it.
> 
> My statement was not quite accurate. I should
> have said only that the criteria for
> determining the social ordering and the best
> winner in some single-winner election may be
> different.

In what situations would the single winner and the social ordering 
differ? It does, for proportional completion (because that's 
proportional and thus PR-esque thinking appllies), but to majority 
methods... I can't quite see when that would be the case.

>> Say we were going to make a "Organization for
>> Condorcet Voting". Advocating multiple Condorcet
>> methods would probably "split the vote" as it were
>> (considering the usual state of things as Plurality).
>> That's what some IRV supporters say about Condorcet
>> itself (to my knowledge), that we should support IRV and
>> then possibly go to Condorcet later rather than fragment
>> electoral reform. So which will it be? What we have to go on
>> is, on one hand, the theoretical measures, and on the other,
>> a few pieces of data. It's not going to be easy...
> 
> I think it would be good to agree on the
> target first. For example the target of
> making U.S. a multi-party democracy is quite
> different from the target of removing the
> problem of small party spoilers in the
> presidential elections. And promotion of
> one's favourite method at all cost is yet
> another quite different target.

The CVD (FairVote) tries to do both. In one sense, that may be why they 
got stuck with IRV in the first place, though now I think that it's at 
least in part because of their (CVD's) partly undemocratic nature. In 
any event, it might be good to have a plan for both, or they could say 
"we want both multiwinner and singlewinner change, they have just one". 
Perhaps that's not very likely, but still... which of course leads to 
the question of what a multiwinner method that reduces to Condorcet in 
the single-winner case yet runs in polytime would look like.

In any event, if we want to "hit the ground running", then you're right, 
instead of inventing a united method, we should focus on one or the 
other. I think QPQ would be a good multiwinner method, but to my 
knowledge, it reduces to IRV. I also haven't checked if there are any 
"automatically disqualifying failures", like MMPO's Plurality failure.

>>> One observation about clones. One can get the same
>>> pairwise matrix from ballots that contain clones and
>>> from ballots that do not contain clones. That means
>>> that (matrix based) clone proof methods will protect
>>> also other sets of candidates than sets of clones
>>> (e.g. Smith set may or may not consist of clones).
>> What do you mean by that the you can get the same matrix
>> from ballots with and without clones?
> 
> Here's an example of what I was thinking.
> 
> 2: A>B>C>D
> 2: B>C>A>D
> 2: C>A>B>D
> 1: D>A>B>C
> 1: D>B>C>A
> 1: D>C>A>B
> 
> A, B and C are clones in the sense that they
> are next to each others in every ballot. A, B
> and C also form a Smith set.
> 
> 3: A>B>D>C
> 3: B>C>D>A
> 3: C>A>D>B
> 
> With these ballots the resulting matrix (and
> Smith set) is exactly the same. But A, B and
> C are not next to each others in any of the
> ballots.

Well, if we're cloneproof, it doesn't matter (from the point of view of 
the cloneproof criterion) which of the clones we pick when we pick among 
the clones, if the clone set won. So the method might pick the candidate 
that would be the most suitable one if the set wasn't a clone set, and 
that way it only goes against the objective of picking the most suitable 
winner when to do otherwise would make it no longer cloneproof -- kind 
of like Schulze's weak invulnerability to Hylland free riding, saying in 
effect "a method passing this is only vulnerable to Hylland free riding 
when not being would make it fail the DPC".

> As demonstrated above clones (as typically
> defined) can not be derived from the matrix
> alone. Also ballots are needed.
> 
> Yes, one could replace sets of clones with
> some virtual candidate. If that virtual
> candidate wins then one can use some further
> "completion method" to determine the winner
> within that clone set.
> 
> I have also played with the idea of allowing
> the candidates themselves to indicate which
> of them should be treated as clones. That
> would guarantee that all clones and only
> clones are treated as clones.
> 
> (One could go also further and allow
> hierarchies of clone groups.)

It's easy to make a voting method "cloneproof" in this manner. Just have 
a prefix that collapses clones down to single candidates, then if any of 
those pseudocandidates win, pick the one that's closest to winning were 
the clone candidates not collapsed. But that would be a very fragile 
cloneproof condition indeed.

Consider something like this:

1000: A>B>C>D>E
1021: D>A>B>C>E
  874: E>D>A>B>C
  760: C>B>A>D>E

(Schulze gives D > A > B > C > E)

{A,B,C} are clones, and the prefix method would replace them with a 
candidate of its own. But now consider this:

1000: A>B>C>D>E
1021: D>A>B>C>E
  874: E>D>A>B>C
  760: C>B>A>D>E
    1: C>D>B>E>A

(Schulze still gives D > A > B > C > E)

Now they're not strict clones anymore. A good method should recover 
gracefully from this condition, since in real world elections, it's very 
unlikely that all voters would vote the clones exactly in the way to 
make them obvious as clones. The prefix wouldn't do that.

> In Condorcet vote management could be the
> most probable path leading to "too high
> levels" of strategic voting. In large public
> elections with independent voters the risks
> are at rather low level.

Do you mean the risks from vote management, or non-vote-management strategy?

>> Though there's always the chance that if we were to set
>> up an Organization for Condorcet Voting, IRV or FPTP
>> supporters would say something like "they say IRV is
>> nonmonotonic, well, this thing can't even make up its
>> mind what the true winner should be!" (regarding
>> Reversal symmetry). That's one way theoretical issues,
>> even those that don't really matter in real life
>> elections, could come into play. (Of course, one could then
>> respond that "IRV squeezes the center and FPTP explodes
>> said center, but Condorcet supports the center", for
>> instance. I'm using general statements here - they may
>> not fit completely, but you see the idea.
> 
> My theoretical approach to the problem of
> having many different Condorcet (and other)
> methods is that there may be many utility
> functions that one may choose. In some cases
> there might also be a need to strengthen the
> methods and make them more strategy resistant
> (at the cost of not always electing the best
> winner according to the agreed utility
> function).
> 
> My practical approach might be to pick a
> representative set of Condorcet methods and
> say that they are all good.
> 
> These election method evaluation questions
> are tricky. It is very difficult to explain
> all the relevant factors. And on the other
> hand it is easy to develop various threat
> scenarios that can be used against other
> methods.
> 
> A unified front of respected experts could do
> a lot. Unfortunately all the experts seem to
> have their own favourite methods and
> corresponding campaigns :-).

That was a reference to Minmax. If you throw nonmonotonicity at IRV, 
they might throw reversal symmetry failure at you in return. This could 
happen even with the kind of criteria we think are not very important 
(Consistency, Participation, or LNH* for a method that fails both LNHs), 
but it would carry greater weight for those criteria that are meaningful 
("they say IRV spoils candidates, but with Nanson, it's not even 
cloneproof!" or whatever).

As for experts, again we hit the problem of estimating how much strategy 
would happen. Ideally, we'd either have that data or we'd have some way 
of saying "all we mean is that Condorcet is good: if you want something 
good but possibly complex, choose this, otherwise..", and unite under 
Condorcet. Perhaps some sort of "here's the criteria the different 
methods pass, pick what you think would be best", but I think knowing 
real world strategy so we could find a single Condorcet method would be 
better.