[EM] Condorcet cyclic drop rule
nkklrp at hotmail.com
Sat Apr 7 17:43:10 PDT 2001
> > Prof. Young goes on with a 4 candidate cycle example--
> > 25 Voters
> > a b c d
> > a -- 12 15 17
> > b 13 -- 16 11
> > c 10 9 -- 18
> > d 8 14 7 --
> > In step 2 of Condorcet's algorithm one would select the six propositions
> > having greatest majorities. In descending size of majority, these
> > are c > d [18 > 7] , a > d [17 > 8], b > c [16 > 9], a > c [15 > 10],
> > d > b [14 > 11], b > a [13 > 12]. According to a literal reading of
> > [Condorcet's] step 3, one would first delete the proposition b > a, as
> > it has the smallest majority in its favor. But this does not result
> > in an "opinion" because one cycle still remains: b > c, c > d, d > b.
> > Therefore one would delete the proposition d > b, as it has the next-
> > smallest majority in its favor. All cycles are now eliminated. But
> > there is a difficulty: in the resulting partial order both a and b are
> > undominated. Either one of them could be interpreted as the top-ranked
> > candidate, so the outcome is indeterminate.
You're sure that was posted to EM? Ok, but it never was proposed by
any member of EM. No question about it, Young interpreted Condorcet's
bottom-up proposal to mean drop weakest defeats till there are no
cycles. An indecisive, not so desirable proposal. If you want to
get rid of all cycles, then the top-down proposal is better,
Ranked Pairs (Tideman's method).
>You wrote (6 Apr 2001):
> > I don't remember the exact words, but in an article in _Journal of
> > Economic Perspective_, for Winter '95, it seems to me that they said
> > that Ranked Pairs was Tideman's interpretation of Condorcet's
> > top-down proposal.
>These are the exact words (Jonathan Levin, Barry Nalebuff, "An
>Introduction to Vote-Counting Schemes," _Journal of Economic
>Perspectives_, vol. 9, no. 1, p. 3-26):
> > Condorcet's (1785) seminal paper expresses the idea that a candidate
> > who would defeat each of the others head-to-head should win the
> > election. If no such candidate exists, then large majorities should
> > take precedence over small majorities in breaking cycles. In his
> > own words, the general rule was "to take successively all the
> > propositions that have a majority, beginning with those possessing
> > the largest. As soon as these first decisions produce a result, it
> > should be taken as a decision, without regard for the less probable
> > decisions that follow." How is this idea implemented? Consider all
> > the possible lists that order the candidates from top to bottom.
> > Find the largest margin of victory in any pairwise match - and then
> > eliminate all potential rankings that contradict this preference.
> > For example, if the largest victory is for candidate A over
> > candidate B, eliminate all potential rankings which place B above A.
> > Next, consider the second largest margin of victory, and eliminate
> > all potential rankings that disagree. Continue this process until
> > one ranking remains.
> > With only three candidates, this method is well defined and is
> > equivalent to ignoring the election with the smallest margin of
> > victory. The problem is that in elections with four or more
> > candidates, considering the largest unconsidered margin of victory
> > may, at some point, force us to eliminate all remaining potential
> > rankings (by locking in a cycle). Condorcet does not discuss this
> > possibility, an omission which has led to criticism and some
> > confusion.
> > T.N. Tideman suggests one solution to this dilemma of cycles:
...this dilemma about what is to be done in Condorcet's top-down
proposal. And so Tideman's Ranked-Pairs is an interpretation of
Condorcet's top-down proposal, the natural interpretation, it seems
> > simply skip over a head-to-head result that will lock in a cycle.
> > Tideman further notes that if ties exist, there may be more than
> > one potential ranking left even after we have considered all
> > victories. In this case, we declare a tie among the candidates
> > who have first-place ranks in the remaining potential rankings.
> > Tideman calls this scheme "ranked pairs."
> > In the pairwise-comparisons matrix below, we first lock in A > B,
> > then B > C. This implies A > C. The next largest victory is C > A,
> > but this locks in a cycle, so we ignore that head-to-head result.
> > We then lock in A > D, B > D, C > D, which leaves us which a final
> > ranking A > B > C > D.
> > A:B=61:39
> > A:C=41:59
> > A:D=51:49
> > B:C=60:40
> > B:D=51:49
> > C:D=51:49
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