Definition of Condorcet's method
Bruce Anderson
landerso at ida.org
Sun Apr 21 19:39:51 PDT 1996
This time with a subject (sorry):
On Apr 20, 2:00pm, Mike Ossipoff wrote:
> Subject: Re: Definition of Condorcet's method
> First, Bruce, would you mind explaining why you aren't willing to
> reduce your line-length so that it will be compatible with e-mail?
> Unless you do, it's difficult, & often impossible, as in the case
> of your letter that I'm replying to now, to include a copy of your
> letter in my reply. If you're going to write e-mail to this list
> then you should use a line-length that's compatible with e-mail.
> Shorten your line length, or say why you refuse to.
My e-mail software is "Z-Mail for Windows," by Z-Code Software of Novato CA. It
sets the line length for my messages.
> Regarding your list of my objections to your definitions of Condorcet's
> method, some resemble (distortedly) objectionss that I've stated,
> and some are news to me.
>
> But I'll very briefly summarize my objections: You're intentionally
> wasting this committee's time by dumping on us a big list of definitions
> for a method that some of us advocate. Why the hell should we be interested
> in a big collection of new definitions for a method that we advocate?
> You aren't saying that one of your definitions for Condorcet is better
> than, or even as good as, my Condorcet proposal, according to any
> specific standard or criterion. You're just dumping a proliferation
> of definitions on us, for the purpose of wasting this committee's time.
No one needs to read, much less respond, to anyone's postings.
> But I will reply regarding each of the objections that you list:
>
> 1. You say that I object to the use of the name "Condorcet's method"
> for any method that I don't approve of. On the contrary, I couldn't
> care less how many bad methods you define & call Condorcet's method.
> But they aren't of intererest to this committee.
You don't care, but THE COMMITTEE isn't interested?
> 2. You're right on with objection number 2: Yes, you've been
> misrepresenting my proposal, and you continue to do so in the
> letter that I'm replying to. You're, in fact, repeating the same
> misrepresentations that I'd previously commented on, further
> confirming my dishonesty theory, to explain your postings.
>
> After I told you that your definition of Condorcet's method counts
> votes-for, while mine counts votes-against, you continue to claim
> that your votes-for definition is the same method as the Condorcet's
> method that I propose.
>
> Also, my definition of Condorcet's method only looks at defeats.
> I said that the winner, if no 1 candidate beats each one of the
> others, is the candidate over whom fewest voters have ranked
> the candidate who beats him who is ranked over him by the most
> voters.
>
> I worded this in lots of ways for you, so let me repeat a few
> of the wordings:
>
> The winner is the alternative least beaten by any alternative that
> beats it, as measured by how many voters rank that other alternative
> over it.
>
> OR
>
> For each alternative, determine which alternative that beats it is
> ranked over it by the most voters. The number of voters ranking that
> other alternative over it is the measure of how beaten it is. The winner
> is the alternative least beaten by this measure.
>
> ***
>
> Careless or dishonest. I can't prove which it is, but I suspect
> the latter, due to repeated instances.
If you choose to respond to my posts, then you could do other things than
question my honesty. For example, you could answer my question:
To repeat, some relevant basic
definitions are as follows. Let's use Steve's definition of p and q:
Let p(x,y) be the sum of the number of voters who
explicitly rank x over y plus the number of voters who rank x and
leave y unranked. Let q(x,y) be the sum of the number of voters who
explicitly rank x as tied with y plus the number of voters who leave
both x and y unranked. Let p and q be the corresponding arrays of
values of p(x,y) and q(x,y).
Given these definition of p and q, define r = r(i,j;x) for 0 <= x <= 1 by:
r(i,j;x) = p(i,j) + xq(i,j).
Define i to be a Condorcet(1) winner if and only if
min/j of r(i,j;1) >= min/j of r(k,j;1) for every candidate k on the ballot.
Define i to be a Condorcet(1/2) winner if and only if
min/j of r(i,j;1/2) >= min/j of r(k,j;1/2) for every candidate k on the ballot.
Define i to be a Condorcet(0) winner if and only if
min/j of r(i,j;0) >= min/j of r(k,j;0) for every candidate k on the ballot.
Define i to be a Condorcet(M) winner if and only if i is a winner according to
your definition of the Condorcet voting method. Then I claim that:
i is a Beats-all//Condorcet(1) winner if and only if i is a Condorcet(M) winner;
and so I claim that Beats-all//Condorcet(1) is the same as Condorcet(M). I
think that we agree completely on the definition of "Beats-all." (I spell it
differently only to make it easier for me to type, but I have no real objection
to your spelling either.) Do you agree with this equivalence of
Beats-all//Condorcet(1) and Condorcet(M)?
It would seem to me that either: 1) You agree, in which case your criticisms of
my postings on this matter are completely unfounded. 2) You disagree and can
prove that Beats-all//Condorcet(1) and Condorcet(M) are not equivalent. In this
case, could you provide a proof (a simple example would suffice)? 3) Based on
your intuition, you disagree, but don't really know whether or not
Beats-all//Condorcet(1) and Condorcet(M) are equivalent. In this case, your
last paragraph above would seem to be a somewhat harsh criticism of statements
that, for all you know, are true.
> 3. You say that I object that some of your statements are true, but
> wouldn't be true if you defined your terms differently. I agree with
> you that this would indeed by a silly objection. But it isn't an
> objection of mine. If you'll cite a particular thing I said that
> you're referring to here then I'll explain what I meant in greater
> detail.
>
> Actually though, if something that you said is true only because of
> an original definition that you're using, and it isn't true or
> important in terms of the definitions that we've been using, then
> that is a valid objection. If I've used such an objection then
> I stand by it & I don't agree that it's silly. If something that
> you said only seems important based on definitions that only you
> use, then that's a valid objection to it. Whether I've used that
> objection, I don't know.
>
> 4. It seems to me that the 4th objection that you list is that
> you might at some future time say something that isn't true. I'm
> sure that I didn't post that objection.
Rather than rehash the past, let's say that #'s 3 and 4 are only potential
problems that we agree to try to avoid in the future.
> But, returning to objection #2, would you mind letting me be the
> one to define my proposal? If you must dump a heap of new definitions
> of new methods on us, I'll have to insist that you not falslely
> claim that any of them are the same as my proposal.
>
> ***
>
> There seems, Bruce, to be a misunderstanding about the purpose of
> the Single Winner Committee. We're not here to invent an ever-increasing
> proliferation of new definitions for Condorcet's method or other
> methods. Our purpose is to compare a few specific proposed methods
> by certain proposed standards. If you want to invent lots of new
> versions of Condorcet's method then I certainly encourage you to do
> so in journal papers, but not here.
>
> If you believe that one of your versions is better, then say why
> or don't bring it up. We're only interested in _proposed_ methods.
> What exactly is your proposal? Copeland//Plurality.ext? If so then
> answer the posted arguments about why it isn't as good.
>
> ***
>
> And your "Condorcet(1/2)", as I said, shouldn't be called "Condorcet's
> method", since its practice of inventing preferences that were never
> expressed by the voter wasn't proposed by Condorcet. You're making
> a very radical change in Condorcet's proposal there--too radical
> to use his name for the method. Likewise, of course, with "Condorcet(1)".
>
> ***
>
> Mike
>-- End of excerpt from Mike Ossipoff
Since Steve and Demorep seem to have thought that Condorcet(M) was
Condorcet(1/2), I think that I am adding to clarity here.
Here's my proof that Beats-all//Condorcet(1) and Condorcet(M) are equivalent:
Since r(i,j;1) = p(i,j) + q(i,j) = v - p(j,i),
min/j of r(i,j;1) = min/j of (v - p(j,i)) which is the same optimization problem
as -max/j of p(j,i). Thus, defining i to be a Condorcet(1) winner if and only
if min/j of r(i,j;1) >= min/j of r(k,j;1) for every candidate k on the ballot,
is the same as defining i to be a Condorcet(1) winner if and only if
max/j of p(j,i) <= max/j of p(j,k) for every candidate k on the ballot, which is
the second part (after Beats-all) of Condorcet(M).
Bruce
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