Definition of Condorcet's method
Bruce Anderson
landerso at ida.org
Fri Apr 19 21:47:17 PDT 1996
Mike (and anyone else who might be interested):
It seems to me that, roughly speaking, your objections to my definition and use
of the term "Condorcet voting method" fall into one of four categories. First,
you object to my defining any voting method you don't approve of using the name
Condorcet. Second, you think I am making some technically erroneous statements
concerning the relationships between your definitions and mine. Third, you
think that some of my statements are true, but that they wouldn't be true if I
defined my terms differently. Fourth, you think that I might say things in the
future that might not be true.
Concerning the first category, I suspect that neither you nor I have or can have
exclusive rights to use the name Condorcet. Obviously, I like my use. I define
three different versions, Condorcet(1), Condorcet(1/2), and Condorcet(0), which
are exactly the same as each other when none of the voters have any ties or
truncations on their ballots. Thus, they share many properties in common.
Further, when they differ, this notation not only allows a particular one to be
specified, it also (in my opinion) enhances understanding by allowing each of
them to be individually specified, and then compared and contrasted with each
other. If anything, my experience on this list concerning confusion with the
definition of Condorcet's method has strengthened my belief in this approach.
In addition, the naming convention M(1), M(1/2), and M(0) is efficient,
economical, and can be extended to other voting methods. I also think that it
is a good idea to define "elemental" voting methods (i.e., ones that cannot
easily be decomposed into M1//M2) for some ranked-ballot voting methods M1 and
M2, and then to use the M1//M2 and the [M3] structures to build new voting
methods. (I'll define define "[M3]" in a future message.) Of course, one might
want to choose a few favorite voting methods and give them unique "cute" names.
For example,
"Regular-Champion" is [Copeland]//[Plurality-ext]//Random,
and one could say if one wanted to, that
"Marquis" is Beats-all//Condorcet(1)//Plurality//Random.
But I can't make you use my definitions exclusively in your writing, any more
than you can make me use yours exclusively in my writing. What we can and
should do is to establish a rigorous correspondence between our definitions;
which brings us to the second category of your objections.
Concerning the second category, I think you said that I gave an incorrect
correspondence between our definitions. First, 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)?
Any objection that fits into the third category is obviously silly.
Finally, any objection that fits into the fourth category can be considered to
be a helpful warning, but certainly not a valid objection.
Bruce
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