Rank methods, Participation, Consistency

[Rob LeGrand]

Mike Ossipoff wrote:
> PC (Minmax) can be stated in a non-iterative way, and I once believed
> that that meant it's monotonic. But I seem to have heard that Dodgson
> is nonmonotonic, even though it isn't iterative.

It depends on how Dodgson is defined.  The definition that I prefer is
monotonic; it's equivalent to summing the columns of the margins matrix
and choosing the candidate with the smallest column sum.  The other
definition I've seen counts the number of reversals needed in the actual
ranked ballots; it's nonmonotonic.  An example (from Hannu Nurmi's book
Comparing Voting Systems):

5:A>B>C>D>E
2:B>A>D>E>C
4:B>E>D>A>C
8:D>A>C>E>B

A would require 3 preference reversals (switching with D) to become the
Condorcet winner and any other candidate would require more (D would need
to swap with B on two ballots, requiring four actual preference swaps),
so A is the winner of the nonmonotonic version of Dodgson.  If the two
B>A>D>E>C voters uprank A, the votes become

5:A>B>C>D>E
2:A>B>D>E>C
4:B>E>D>A>C
8:D>A>C>E>B

Now D becomes the winner using this version of Dodgson.  D wins both
times using the monotonic version of Dodgson (and every other Condorcet
method I've tried).  A wins both times using Borda.

--
Rob LeGrand
honky98 at aggies.org
http://www.aggies.org/honky98/

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