precise definitions

[Mike Ossipoff]

Bruce said that there haven't been any precise definitions
of criteria that Condorcet's method meets. Since Bruce has been
on this list, I've posted a criterion that I called the Generalized
Majority Criterion. I've pointed out several properties of Condorceet's
method, any one of which could be called a "criterio", since they're
yes/no tests to apply to methods. Did I prove them? No, but I will
later in this letter, or maybe in a subsequent one. Anyway, some
of the properties/criteria that I named are quite obvious without
proof.

For instance, I pointed out that, with Condorcet's method, an alternative
with a majority against it can't win unless every alternative in the
set from which the Condorcet choice rule is to choose has a majority
against it.

(The set from which the Condorcet choice rule is to choose is the
set of all the alternatives in plain Condorcet, anb is the
Smith set in Smith//Condorcet)

X has a majority against it if there's another alternative ranked
over it by more than half of the voters.

This property of Condorcet's method should be quite obvious without
proof. Condorcet scores according to votes-against. It elects
the alternative over which fewest voters have ranked the alternative
which, among the alternatives that beats it, is ranked over it by
the most voters. Plainly if a majority of the voters rank A over
B, but no other alternative has a majority ranking something over
it, B is a big loser by Condorcet's choice rule. It follows directly
from the rule.

So let's define a criterion based on that property. I'd like to
borrow & re-use the name "Generalized Majority Criterion" for this:

A method meets the "Generalized Majority Criterion" if & only if
it has the property that an alternative with a majority against it
can't win unless every alternative in the set from which Condorcet's
choice rule is to choose has a majority against it.

That property is so simple & briefly stated that it deserves
the name "Generalized Majority Criterion". The other criterion
that I'd previously given that name to, the much wordier one,
I'll now call the "Defensive Strategy Criterion", a name that
more specifically tells what the criterion is about.

I've been talking about the abovementioned property on this list
for so long, and have repeated it so many times, that it almost
makes me ask Bruce where he's been. 

In individual e-mail with Bruce, I sent him the definition of
"Trunction Resistance":

A method is "truncation-resistant" if truncation can never gain
the election of an alternative over which a majority have ranked
the Condorcet winner.

>From this property, we could define the Truncaction Resistance Criterion:

A method meets the Truncation-Resistance Criterion if it is truncation-
resistant (as defined in the paragraph before last).

Why is  Condorcet's method truncation-resistant? Because, as I
said in a recent message, truncation can't change the fact that
no alternative is ranked over the Condorcet winner by a majority.
Truncation can only reduce the number of ballots ranking the CW
over another alternative; it can't increase the number ranking
another alternative over it. That could only be done by order-reversal.
Therefore, since the CW doesn't have an alternative ranked over it
by a majority, and, by assumption, that other alternative does,
that other alternative can't possibly win, when scoring is by
Condorcet's method.

***

In a recent post here, I stated the property of Invulnerability
to Mis-Estimate. That property too could be made into a criterion,
and will be, in this letter. But first let me repeat the propoerty,
since Bruce seems to have missed it, in all the times I've posted
it:

With Condorcet's method, even if every voter mistakenly believes
that a certain alternative is the best he/she can get, and
everyone includes it in their ranking, and no one includes in
their ranking anything that they like less than it, that can't
give the election away to it if there's a Condorcet winner which
is ranked over it by a majority.

Obviously some of the wording in that statement is superfluous,
and someone could say the superfluous part is too informal. So
I'll word it without that part:

With Condorcet's method even if everyone includes an alternative
that is not the Condorcet winner in their ranking, and no one 
includes in their ranking any alternative that they like less
than it, that can't result in victory for that alternative
if there's a Condorcet winner which is ranked over it by a
majority.

I call this propoerty Invulnerability to Mis-Estimate. It's
obviously important. Say there's a progressive candidate, like
Nader, and say that, for some reaon, he isn't gettng good media
coverage. Say that makes it look like Clinton is Condorcet winner,
the candidate who is the best result that the progressives can
get. So say the Nader voters consequently all include Clinton
in their ranking, but those who prefer Clinton to Nader don't
bother ranking Nader, since they are sure from the media that
they don't need him. This can create a circular tie. With
Copeland or Regular Champion that can give the election to
Clinton. With Condorcet's method, either plain condorcet or
Smith//Condorcet, that can't happen.

This is a re-statement of the lesser-of-2-evils standard, in
the form of a precisely-defined property. (There are, of course,
other precise ways of putting it).

Why does Condorcet's methd have that property? Because it's
also a re-statement of truncation-resistance, and I've just
shown why Condorcet's method is truncation-resistant.

So here, then, are some precisely-defined criteria that
Condorcet's method meets, and Copeland fails:

Truncation-resistance
Invulnerability to Mis-Estimate
Generalized Majority Criterion
Defensive Strategy Criterion

The Defensive Strategy Criterion couldn't be stated briefly,
and I'd have to look it up to re-state it. I will if asked to.
The Defensive Strategy Criterion has 2 separate parts. I said
that Condorcet strictly meets 1 of those parts of the criterion,
and meets the other for all practical purposes. Copeland
badly fails both parts.

If asked for a demonstration of that I'll give one. But the
part of the criterion that Condorcet strictly meets is veryk
close to being a re-statement of truncation-resistance, with
emphasis on need for defensive strategy.

As I was saying when I posted the Defensive Stratgegy Criterion
(which I at that time called the Generalized Majority Criterion),
the conditions under which order-reversal could succeed, so that
the order-reversers could achieve a violation of the not-strictly-
met part of the criterion would be so rare that the would-be
order-reversers could never count on them, and the practical
results is the same as if Condorcet strictly met that part of the
criterion. What if you told crook that there was a tiny chance
that he could get away with his crime? That wouldn't reduce
the deterrence appreciably.

Yes, I know there's something imperfect about using the 
Defensive Strategy Criterion, when Condorcet's method doesn't
strictly meet both parts of it. If that were really a problem,
of course I could divide that criterion into two separate ones,
so that the part that Condorcet strictly meets would he
a separate criterion.

Anyway, as we're all familiar with by now, everything isn't always
as we'd like it to be, with voting systems. Maybe a slightly
weaker version of the not-strictly-met part of the Defensive
Strategy Criterion could be defined, so that I could say
Condorcet's method meets both parts.  

But the important thing is that the criterion is precisely stated
(at least Bruce never said it wasn't), and Copeland fails both
parts of it in a big way, and Condorcet strictly meets 1 part
(the part about ordinary plausible conditions), and meets the
other part for all practical purposes. I'm sorry that "for all
practical purposes" doesn't sound mathematical enough. I would
remind Bruce (with Rob's permission, of course) that most people
who have joined this list did so because of interest in electoral
methods, not to seek mathematicl elegance for its own sake.

Bruce has been saying that there've been no precise statements
of criteria by which Condorcet is better than Copeland. I hope
I've shown that that isn't so. The criteria I've been descrsibing
here are about the lesser-of-2-evils standard. I'm not saying
that standard is important because I say so, but because
all electoral reformers who are interested in sw methods consider
it to be the important thing, the serious problem that Plurality
has, and the reason why a better sw method is needed. 

***

Would you agree with Bruce that the lesser-of-2-evils properties
have been over-sold? As I said, Bruce doesn't consider the lesser-
of-2-evils problem important, which greatly sets him apart from
electoral reformers. It doesn't set him apart from other
academic authors, however. It would seem that electoral reformers
& academic authors have conflicting goals for electoral reform.
I agree with the goals of the electoral reformers. Who knows what,
if any, the goals of the academic journal authors are.

***

Oh, do we want to talk about imprecise criteria?

Bruce defined the Majority Criterion, at one point, by saying:

"If a strict majority of the voters rank a particular alternative
as their unique 1st choice, then the voting method must select that
alternative as the unique winner."

By that definition, all the methods we've discussed (at least
all the ones that I've checked out) meet that criterion. Approval
meets it. Bruce said Approval doesn't meet it. Incorrect.

Obviously, in an Approval election, the only way to rank an
alternative as your unique 1st choice is to vote for it & it
only. If a majority do so, then that alternative can't lose.

But the alternative that Bruce calls the Generalized Majority Criterion,
and which I call the Mutual Majority Criterion, as Bruce defines
it, isn't met by any method. That's because it's worded in terms
of preference, not voting. It's possible, though not likely, for
any number of voters to vote the opposite of their preferences,
and so, strictly speaking, no method meets that criterion, as
it's worded. That majority who prefer every alternative in
set A to every alternative outside it might vote every alternative
outside set A over every alternative in set A. For all practical
purposes that criterion is met by Hare, Copeland & Smith//Condorcet,
but, strictly speaking, any method can do what that criterion
says not to do, and so no method meets that criterion.

I wrote about that Mutual Majority Criterion when I first
replied to the article Bruce sent to Steve & Rob. I pointed out
that, if we worded it in terms of votes instead of preference,
not only does Smith//Condorcet meet it, but in order to make
plain Condorcet fail it would require a circular tie among the
set A alternatives such that each alternative in set A has
another alternative ranked over it by a majority of all the
voters. Some mandate that would be for set A. For any set
A alternative, there's another alternative that more than half
the voters say is better than that alternative.

In my initial reply to Bruce's anti-Condorcet posting, some
time ago, I commented similarly about all the criteria that
Bruce used. I probably shouldn't repeat it all here, but
I'll re-post it if Bruce would like.

***

I emphasize again that it isn't just Bruce. Other mathematician
journal authors on the subject of voting systems have used
imprecise definitions of criteria. I bring this up because
when mathematicians tell us that our criteria aren't precise
enough, I now point out them that some of their own criteria
are imprecise.

For instance, the Condorcet Criterion, which mathematicians
(not just Bruce) tend to define according to preference.
It's often said, for instance, that any alternative that
would heat each of the others in pairwise constests should
win.

Oh yes? Then Copeland fails the Condorcet criterion. All it takes
is for a few people to truncate, and we get a circular tie, which
the tie-breaker gives to something other than the alternative
that would beat each of the others in pairwise contests. That
alternative, by the way, is called the Condorcet winner, and
mathematicians often mis-state the Condorcet criterion as saying
that the Condorcet winner must win. Again, no method can meet
the Condorcet criterion when it's defined that way, since it's
possible for everyone to vote the opposite of their preferences.

But I emphasize that it sure doesn't take anythign that unlikely
to make Copeland or Regular-Champion prove its failure of
Condorcet's criterion, as defined above. Mere truncation by
some voters will do it. The tie-breaker will often finish
the job. 

To continue a good point that Bruce made, some things that
are called criteria would be better named as "standards" or
"desiderata". So what Bruce & other mathematicians call the
Condorcet Criterion should be called, instead, the
Condorcet Standard. And methods could be judged according to
how good a job they do of meeting that standard. Mathematicians
don't like phrases like "do a good job of...", and they
prefer criteria, precise y/n tests. But those same mathematicains,
apparently without realizing it, have given us the Condorcet
standard, which isn't a crilterion. 

So then, how do the various methods do, according to the
Condorcet standard? Well, what would you say about a method
in which truncation can steal the election from the Condorcet
winner, and elect an alternative over which a majority have
ranked the Condorcet winner? If you like that, you'll like
Copeland & Regular-Champion.

How about a method in which media downplay of a candidate
can result in the election being given away to someone else?
That's Copeland  & Regular-Champion. I won't ever happen in
Condorcet if there's a Condorcet winner ranked over that
other alternative by a majority. For precise wording, I
refer you to my definition of Invulnerability-to-Mis-Estimate.

So, then, I would say that Copeland & Regular-Champion fail
the Condorcet standard in a big way, and that Condorcet,
by which I mean plain Condorcet or Smith//Condorcet, does
a much better job.

***

More later.

***

Mike




















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