Bruce's part 2 reply to Steve

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Bruce's part 2 reply to Steve

Craig Carey-2
Bruce Anderson writes:

[I'm replying to parts of this letter, farther down in the letter--Mike]

> Part 2:
> On Apr 3,  8:25pm, Steve Eppley wrote:
> > Subject: Re: Reply on EM to Mike's Reply on ER
> > 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.)
> In part 1 of this reply, I gave three reasons why it is my opinion that
> Smith//Condorcet is significantly more complicated than just Condorcet, and I
> promised to give two more reasons.  These (numbered fourth and fifth) follow.
> Fourth, Mike wrote me a while ago saying that he was meeting with opposition
> when he tried to suggest using Smith//Condorcet instead of just Condorcet, and
> that the opposition was based on the increased complexity.  I don't think that
> Mike would have said this if he felt that the opposition was insignificant.  In

As you mention later in this letter, the complicatedness objections
that I got tended to be about the fact that it was about "sets". If
I'd avoided using the word "set", there may well have not been a
problem. I told you that in a letter to you, Bruce, that the use
of the word "set" was probably the cause of the complicatedness objections.
I also said that I believe that a simply & familiarly worded statement
of the Smith Criterion will be more easily accepted by people than
a definition of the Smith set, or the "Smith method".

The way I worded the Smith Criterion in my recent letter would never
be called complicated. The only thing that could draw flak is the
overall length of the resulting definition of Condorcet's method. But
people read & understand things longer than that, like Baseball &
football rules, racing forms, etc.

> any event, I judged from Mike's comment that, whoever he was talking to, they
> felt that the increase in complexity was both significant and not worthwhile.  
> I'm convinced that it is worthwhile, but I am certainly willing to listen to the
> opinions of anonymous others concerning their views of relative complexity.  Are
> you they?

No. No one in this list were among those who rejected Smith//Condorcet
when I was proposing it. It was other people I was corresponding with
at that earlier time, and some people in newsgroups.

> Fifth, in the short time that I have been on this list, it seems to me that the
> only mentions of Smith//Condorcet before my examples started to be circulated
> were either of the form "Mike suggested it, but we thought it was too
> complicated," or were comments by Mike of the form: [complicated]...because
> I used the word "set" of candidates... .

And, as I said, without the word "set", it certainly isn't too complicated.
You've seen my recent familiarly & simply worded statement of the
Smith Criterion, in my letter entitled "My public Smith//Condorcet

> All in all, I think that I have pretty sound reasons for reaching my opinion
> here.  But it's still just an opinion.
> > >Determining which standards are most desirable would be a good
> > >topic to debate--too good to try to rush out in a "quick-and-dirty"
> > >e-mail posting.
> >
> > Did you receive the Glossary of standards I posted about a week ago?  
> > It's overly inclusive of standards.  If you have more standards you
> > want added, though, I'd like to hear them so I can add them.
> >
> > Several weeks ago we had two proposals for how to proceed here.
> > Both involve our voting on the order in which we will discuss
> > standards (and how well the methods score on each standard).  
> > Maybe we'd vote all at once on the complete order, or maybe just
> > periodically pick the next few.  To be considered in the voting,
> > they should be in the Glossary.
> I intentionally, but (in hindsight) perhaps mistakenly, used the word
> "standards" because I was responding to Mike and he used that word.  I would

No, "standard" is a really basic word, and the right word for something
that we judge things by. A way of judging things. "Attribute" or
"property" are words that I sometimes use too, but they don't have
the same meaning as "standard". "Standard" is the main word in the
English language for the purpose we're using it for. I don't think
we'd consider replacing it.

"Criterion" is a word that means to me pretty much what it means
to you: A yes/no test that can be applied to a method. Maybe that
doesn't sound exactly like your definition, but we seem to use
the word the same way. A criterion can be a standard, if you really
believe that that criterion is important enough. But a standard
needn't be a criterion. A standard needn't be stated in the way
that you (Bruce) say that criteria must be stated. A standard is
valid without being stated in that way. I repeat that this isn't
a mathematical journal, and the public aren't a mathematical journal
readership. Standards can be perfectly valid without being said
the way that mathematicians like things to be said.

English is the language at which we should meet. You shouldn't
ask us all to talk in mathematical journal language. By "vague",
you probably mean not stated as a precise yes/no test. But
when I said that a majority should be able to get what it wants
(elect someone or deny election to 1 or more candidates) without
any use of defensive strategy, including ranking a less-liked
candidate equal to or over a more-liked one, or truncation, and
that we'd like that majority to be able to do that under as
broad a range of common conditions as possible, the standard
containing that wording is a valid standard, with a definite
meaning, even though it isn't a yes/no test. It says something that
we want. It's a standard.

After saying that, in a previous letter, I re-worded that paragraph
to conform to what I mean by a "criterion", & maybe qualify as a
criterion as you use the word.

> prefer the words "criteria" and "attributes."  A criterion must be able to be
> applied to every voting method in a class specified by the criterion, such as
> the class of ranked-ballot single-winner voting methods, and it must be worded
> sufficiently precisely that it is mathematically provable whether or not any
> given well-defined voting method in that class satisfies that criterion
> according to (and only to) that wording.  Conversely, an attribute (feel free to
> suggest an alternative word, if you'd like) is an intuitive but vaguely defined
> concept, which reasonable people can express opinions on, but which any
> particular voting method cannot be mathematically or scientifically proven
> either to possess or to fail to possess.

A standard needn't be something that a method can be scientifically or
mathematically proven to possess or fail to possess. On the other
hand, of course a standard could be worded that precisely. But it
could also be merely a statement about what we'd like in a method.

If a standard is in the form of a yes/no test, then I can say
that Condorcet meets a certain standard & Copeland doesn't. If a
standard isn't of that type, then I can still say that Condorcet
does better by that standard than Copeland, though, due to the
wording of the standard, that can't be mathematically proven, but
may still be quite obvious. But a standard, as I said, could also
be a precise yes/no test.

> Most, if not all, of the standards in the glossary seemed to me to either be
> attributes, which is fine for what they're worth, or be imprecise statements of
> purported criteria, which is not fine with me.

You have cause to object if I call something a criterion when it
isn't an exact yes/no test. But if I don't call it a criterion, and
don't try to use it as a yes/no test, I don't know what there is to
object to.

Maybe a standard that isn't a criterion could be called a

But sometimes things seem imprecise to a mathematician merely because
they aren't said in mathematician language, even though their
plain-English meaning is unmistakeable.

> > >As an aside, in my notation X-Y either means a particular voting
> > >method jointly proposed by X and Y, or it means a particular voting
> > >method where X and Y alone have no intrinsic meaning.  X//Y means
> > >that X and Y are both ranked-ballot single-winner voting methods
> > >where, if X produces a single winner, then that winner is the X//Y
> > >winner; while if X results in tied winners, then the result of
> > >applying method Y to this set of tied winners yields the overall
> > >X//Y winner(s).
> >
> > So with this syntax the order of the two (or more) methods is
> > important.  Smith//Condorcet and Condorcet//Smith would be two
> > different systems.  In Smith//Condorcet, if the Smith set doesn't
> > have exactly one candidate, then smallest worst defeat breaks the
> > tie.  
> First, you are quite correct in saying that "the order of the two (or more)
> methods is important.  Smith//Condorcet and Condorcet//Smith would be two
> different systems."  In mathematical terms, the operation "//" is not
> commutative.  Many examples of this failure to be commutative can be constructed
> here.  However, note that if "L" and "M" are any two (not necessarily different)
> ranked-ballot single-winner voting methods, then both "L//M" and "M//L" are
> valid ranked-ballot single-winner voting methods.  Also note that "//" can be
> proven to be associative in that, if "L", "M", and "N" are any three
> ranked-ballot single-winner voting methods, then both "L//(M//N)" and
> "(L//M)//N" produce the identical ranked-ballot single-winner voting method.  
> Accordingly, "L//M//N" defines an unambiguous ranked-ballot single-winner voting
> method.
> Second, as my example in part 1 shows, "smallest worst defeat" is not, by
> itself, well defined.

And, as has been pointed out, that wasn't intended as an official
definition, but was proposed as an introductory description, to
be followed by examples, for members of the public who might not
have patience for a precisely worded definition.

> > Smith//Condorcet has three cases:
> >
> > 1. Smith set is empty:  Condorcet picks from the candidates not in
> > the set (i.e., picks from all the candidates).
> >
> > 2. Smith set contains exactly one candidate:  This is the winner.
> >
> > 3. Smith set contains more than one candidate:  Condorcet picks from
> > the candidates in the set.
> As it is defined in part 1, the Smith set cannot be empty.  So there are only
> two cases here.
> > >In fact, I have never used and I strongly oppose the use of the
> > >term "Instant Runoff" to mean MPV/STV/Hare voting.  Runoff has an
> > >established meaning, at least it does in the United States--it's
> > >the pairwise match between the top two plurality leaders, if
> > >neither has a strict majority.  So an "instant runoff" should be
> > >that pairwise match based on the voters ranked ballots.
> >
> > How about "Instant Runoffs"?
> >
> > But I'd rather reserve this moniker for pairwise methods, since the
> > pairings are just as much (maybe more so) "runoffs" as MPV's
> > attrition runoffs.  And maybe reserve it for one particular pairwise
> > method, so it can be used before the public when it's campaign time.
> >
> > --Steve
> >
> >-- End of excerpt from Steve Eppley
> I'd rather reserve "Instant Runoff" for something else also.  But it would be
> for "pairwise methods" until this term was defined.  In response to a similar

> statement by me, Mike said:
> "The term "pairwise system" has long been used in ER & EM
> to refer to the methods that compare the alternatives pairwise (hence
> the name) to determine which is ranked over the other by more voters."

All that's missing from this definition is the fact that Pairwise methods
also elect a candidate if, when compared separately to each of the other
candidates, he is ranked over that candidate by more voters than vice-versa.

With that added to the paragraph that you quoted directly before it,
that's the complete definition of a Pairwise method.

> Literally, this would mean that any method that compares the alternatives
> pairwise, to determine which is ranked over the other by more voters, in any
> part of its calculations would qualify as a pairwise system.  Somehow, I doubt
> that this is what anyone on ER or EM means either by pairwise method or pairwise
True. As I said, I'd add that 2nd paragraph to the definition, referring
to picking the candidate that beats each one of the others, if such
a candidate exists.

> system.  Let me suggest two quite different possible definitions.

The following is an example of what I mean by "formulese", though not
a bad example of it, as formulese goes. Things that would be clear
in English can be made to sound complicated or involved when said
in formulese.

> DEFINITION 1:  Let p(x,y) be the number of voters who rank x over y, and let
> q(x,y) be the sum of the number of voters who rank X as tied with y plus the
> number of voters who do not explicitly rank either x or y, and let p and q be
> the corresponding arrays of values of p(x,y) and q(x,y).  Then a ranked-ballot
> single-winner voting method is a "pairwise method" if and only if its set of
> winners can be calculated from the values in p and q.

No, because that definition doesn't say that any alternative which,
when compared separately to each one of the others, is ranked over it
by more voters than vice-versa shall be declared the winner.

But that omission is my fault, since I unintentionally left it out
of my definition too.

> DEFINITION 2:  A ranked-ballot single-winner voting method is a "pairwise
> method" if and only if it satisfies the Condorcet (winner) criterion.

That would be stretching it a little, since there's at least 1 Condorcet
Criterion method that doesn't explicitly do pairwise comparisons.

In informal brief language, a Pairwise method is a method that does
pairwise comparisons of what is ranked over what by how many voters,
and what beats what, and declares as winner any alternative that
beats everything else.

Maybe I should say it a little more carefully:

A Pairwise method is a method that, for pairs of candidates, A & B, counts
how many voters have ranked A over B, & how many have ranked B over A; and
which determines whether A is ranked over B by more voters than rank
B over A; and which gives the election to any alternative which, when
compared separately to each one of the other alternatives, is ranked
over it by more voters than vice-versa.

That definition doesn't say that each pair must be compared, because,
or course, an alternative that beats each of the others might be found
before all the pairs are compared.

This is what we've meant by Pairwise methods, though we didn't
explicitly state it.

> Is it either one of these, or is it something else?

The 1st definition is close, and needs only that addition about
picking as winner an alternative if it beats everything else.

Pairwise methods, of course, have different ways of solving situations
where no 1 alternative beats each one of the others. Some of them,
after determining what beats what, ignore people's votes thereafter.
Pairwise methods differ radically in their merit. It makes all the
difference how a Pairwise method deals with the situation where
no 1 alternative beats each one of the others.

> Bruce
> .-

By the way, since the really important & basic thing about a
round robin is that any team that beats each of the others wins,
Instant-Round-Robin does seem an acceptable name for Pairwise methods,
regardless of how they solve the situation where no 1 alternative beats
each of the others.

But I feel it's important to distinguish them by calling them by
their own separate names. Not much can be said about all the Pairwise
methods, taken together. As good as the best ones are, I don't know
if the worse ones are as good as  Approval. In terms of merit &
desirability, the various Pairwise methods differ so much that
to speak of them as a class is an insult to the better Pairwise