Reply on EM to Mike's Reply on ER

Bruce Anderson landerso at
Sat Apr 6 00:46:56 PST 1996

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 
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?

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... .

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 
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.

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.
> >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 

Second, as my example in part 1 shows, "smallest worst defeat" is not, by 
itself, well defined.

> 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."

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 
system.  Let me suggest two quite different possible definitions.

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.

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

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


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