Complexity of Condorcet (was Re: Reply on EM to Mike's Reply on

[Steve Eppley]

Bruce Anderson:
>> >I don't think I ever said that (in my notation) Smith//Condorcet is
>> >too complicated to propose to the public.  I said that it is
>> >significantly more complicated than just Condorcet.
>
>I also said that it is significantly better than just Condorcet. 
>I also said that I thought that there are still other voting
>methods that are both better and simpler than Smith//Condorcet. 
>None of these statements mean that Smith//Condorcet is too
>complicated for public use.  In fact, I think that it is too
>simple, and that the public could accept the slightly more complex
>method I denote by Smith//Condorcet//Plurality.  I presume that
>Mike thinks that Smith//Condorcet is slightly more complicated and
>slightly better than just Condorcet.  Maybe we'll never reach
>agreement on "slightly" versus "significantly," but maybe we can
>agree on the unmodified "more."
>
>Steve Eppley asked:
>> How do you define significantly more complicated, then?  Does your
>> definition relate to whether the public can be persuaded to approve
>> it when it's offered in state initiatives?  (To me, that seems the
>> practical benchmark.)
>
[snip]
>First, Smith//Condorcet involves two fundamentally different
>methods, so it involves explaining and asking people to understand
>and accept two different methods.  Condorcet alone, of course, does
>not.

So that *is* related to public understanding.

>Second, in attempting to advocate Smith//Condorcet, one has to
>explain, if asked, why Condorcet alone is not good enough.  In my
>opinion, such an explanation is quite simple, for example, for
>either Condorcet//Plurality or Copeland//Plurality; namely,
>Plurality is quite poor as an initial voting method, but it serves
>well as a tie-breaker.  In my opinion, the corresponding statement
>for Smith//Condorcet (i.e., Condorcet is quite poor as an initial
>voting method, but it serves well as a tie-breaker) is much more
>difficult to accept.

That "corresponding statement" is not one that I've heard anyone
here make.  Of course it's easy to explain why Plurality by itself
would be poor.  It would be hard to explain why Condorcet by itself
would be poor, since that's not true.  You're pointing out a rather
odd dilemma:  The better the tie-breaker the worse the overall
method, since it's hard to explain why not just use the tie-breaker. 
I'm not very concerned about this dilemma.

There's some ambiguity in your last phrase: do you mean "is difficult
to accept" by you, Bruce, or that it would be difficult to accept by
the public when we're trying to teach them?

This Smith//Condorcet syntax is interesting.  I'm not certain that
it's going to be better to teach S//C by first teaching Smith,
though.  I'll have to ponder this.  After all, I'm the one who
suggested dropping the "beats-all-others" sentence from the Condorcet
explanation since it didn't change the result and added complexity.
Putting Smith first adds even more complexity, and for little gain
since the main issue is whether smallest worst pairwise defeat is a
suitable method.

Maybe it will be better to drop this S//C syntax and use something
which matches the outreach approach.  If we teach Condorcet before
Smith, maybe we should use a syntax like "Condorcet & Smith" to
indicate that Smith is a filter added to restrict Condorcet.  This
will also avoid Bruce's Dilemma ("the better the tie-breaker, the
worse the overall method").

>Third, while neither Smith nor Condorcet, alone, is extremely
>complex, neither one is all that simple either.  As evidence of
>this, note that erroneous or vague statements of these methods
>frequently appear on these lists.

That's typical of shorthand, bumper stickers, and sound bites, and
that's what we're looking for.  This is about outreach to the
public, not rehashing academic literature.

What we must keep in mind is that different audiences will have
different needs.  For some audiences, a vague but simple explanation
followed by a clear example will be more appropriate than a rigorous 
(and too-challenging) explanation.

>As far as Condorcet goes, it seems to me that most people on this
>list, except for Mike, say (as you imply below) that the Condorcet
>winner is the candidate whose worst pairwise defeat is the
>smallest, whenever there is no candidate that pairwise beats every
>other candidate.  

I've muddied it more than that!  :-)  
I omit your "whenever ..." clause since it doesn't affect the result
and adds complexity to the explanation.  (I suppose you'll object 
that this omits the beats-all-others candidate from contention, and if 
you take it literally that's true.  But examples can make clear that 
what is intended is that a beats-all-others candidate's worst defeats 
are smaller than any others.)

>There are two problems with this statement. First, there can be two
>or more such candidates, which this statement does not allow. 
>Admittedly, it is not hard to construct a more complex correct
>statement, but complexity is the issue here.

You mean tie-breaking the smallest worst defeat?  That hasn't been
the point here, since it would be extremely unlikely for large
elections. It's not more necessary to construct a more complex
statement here, or in our primary outreach, than it is to teach
people plurality by including its tie-breaker.  Truth is I haven't
the slightest idea what would happen in a plurality Senate race, for
example, if the two leading candidates tied.  I assume a 2-way
runoff... but the point is that this doesn't have to be taught at
the same time.  

I think this is missing the forest for the trees.  The time to
explain tie-breaking is *after* people grasp the basic method, not
while they're being taught the basic method.  If the basic method 
takes an instant to be grasped, then the tie-breaker can be presented 
in the next sentence.  If the basic method takes a day to be grasped, 
then the tie-breaker can be presented on day 2.

>Second, consider the following example.
>46:  A, (B & C tied)
> 6:  B, A, C
> 4:  B, C, A
>44:  C, B, A
>
>Then #(A>B) = 46 and
>     #(B>A) = 54, so B beats A by 54-46 =  8.
>Also #(A>C) = 52 and
>     #(C>A) = 48, so A beats C by 52-44 =  6.
>Also #(B>C) = 10 and
>     #(C>B) = 48, so C beats B by 48-10 = 38.
>
>A says:  "I scored 46 in my only defeat, while C only got 44, and B
>only got 10. So my worst pairwise defeat was the smallest."
>
>B says:  "My opponent only scored 48 in my one defeat, while C's
>opponent got 52, and A's opponent got 54.  So my worst pairwise
>defeat was the smallest."
>
>C says:  "I lost by only 6 in my one defeat, while A lost by 8, and
>B lost by 38.  So my worst pairwise defeat was the smallest."
>
>This is simple?  At best, it's understandable only if it's very
>carefully explained.  And the need for such a very careful
>explanation is part of my argument for complexity here. 

Your notation makes it appear much more complicated than mine, which
may be why you made some arithmetic errors.  (Or maybe you were just
sleepy, as you pointed out.)  You've needlessly included intermediate
terms in your example:

Try my notation:
  B beats A by  8                -46 +6 +4 +44
  A beats C by  4                +46 +6 -4 -44
  C beats B by 34                  0 -6 -4 +44

I hope you'll leave it to a method's proponents to teach it, and save 
the complexities for the rebuttal arguments.  :-)

I think that in order to teach Condorcet to an audience around the
high school level, much of the instruction will have to come through
the use of examples and step by step explanations.  If people used
to more expert audiences try presenting it with a more challenging or
needlessly complicated approach, that would be a disservice. 

I'm still puzzled by your use of "significantly more complex".  You
wrote above that the public could accept Smith//Condorcet//Plurality,
which I interpret as meaning that S//C and S//C//P aren't *too*
complex.  Perhaps we should drop vague phrases like "significantly
more complex" and use the standard which matters: too complex
to be approved. 

--Steve