[EM] Fw: IBCM, Tideman, Schulze

[Markus Schulze]

Dear Mike,

you wrote (4 June 2000):
> But Schulze's method does have one thing that counts in its favor:
>
> In public elections it chooses the same winner as SD & SSD, which
> , when they undefeat a candidate, are undefeating the candidate
> who can be elected while overruling as few opposing votes as
> possible, consistent with BC compliance. But Schulze's method,
> as I was saying before, doesn't have the obvious & natural
> motivation & justification that SD, SSD, & Tideman have--something
> that's important for a public proposal.
>
> Besides, if the standard of overruling as few voters as possible
> were more important than all other criteria & standards, then
> we'd be advocating PC. And the people's expressed choice between
> the Tideman winner & the Schulze winner may be more compelling
> than a general reckoning of which winner has more voters
> voting against him--for the purposes of choosing between Tideman
> & Schulze.

It seems to me that you consider iterative methods to be more
intuitive than non-iterative methods.

Actually the Schulze method can also be explained as an iterative
method:

Step 1: If there are still pairwise comparisons that haven't
  yet been dropped between candidates that haven't yet been
  eliminated then drop the weakest pairwise comparison between
  candidates that haven't yet been eliminated. If there is more
  than one weakest comparison then all the weakest comparisons
  are eliminated simultaneously. Eliminate all those candidates A
  for which there is a candidate B such that there is a path of
  non-dropped pairwise comparisons from candidate B to candidate A
  but no path of non-dropped pairwise comparisons from candidate B
  to candidate A.

Step 2: Repeat Step 1 until there are no pairwise comparisons
  that haven't yet been dropped between candidates that haven't
  yet been eliminated.

The difference between Schulze and SSD is: SSD cnsiders only
those pairwise comparisons that are pairwise wins while Schulze
considers all pairwise comparisons.

Example:

  A:B=50:50
  B:C=43:48
  C:A=35:44

  SD chooses candidate A decisively because candidate A is
  the unique Schwartz winner.

  The Schulze method drops the "35" in the pairwise matrix.
  Then it drops the "43" in the pairwise matrix. Then it
  drops the "44" in the pairwise matrix. Then it chooses
  candidate C because there is a path of non-dropped pairwise
  comparisons from candidate C to candidate A but no path
  of non-dropped pairwise comparisons from candidate A to
  candidate C and because there is a path of non-dropped
  pairwise comparisons from candidate C to candidate B but no
  path of non-dropped pairwise comparisons from candidate B
  to candidate C.

Remarks:

(1) To my opinion, Schulze is more similar to Condorcet's bottom-up
proposal than SD. Condorcet wrote that _the weakest pairwise
comparisons_ should be dropped successively. He didn't write that
_the weakest pairwise comparisons that are in a directed cycle_
should be dropped successively.

On page 126 of his "Essai sur l'application de l'analyse a la
probabilite des decisions rendues a la pluralite des voix"
(Imprimerie Royale, Paris, 1785), Condorcet wrote:
> Create an opinion of those n*(n-1)/2 propositions, which win
> most of the votes. If this opinion is one of the n*(n-1)*...*2
> possible, then consider as elected that subject, with which this
> opinion agrees with its preference. If this opinion is one of the
> (2^(n*(n-1)/2))-n*(n-1)*...*2 impossible opinions, then eliminate
> of this impossible opinion successively those propositions, that
> have a smaller plurality, & accept the resulting opinion of the
> remaining propositions.

In "Sur les Elections," Condorcet wrote due to McLean's
translation (Iain McLean, Fiona Hewitt, "Condorcet," Edward
Elgar Publishing, 1994):
> A table of majority judgements between the candidates
> taken two by two would then be formed and the result - the
> order of merit in which they are placed by the majority -
> extracted from it. If these judgements could not all exist
> together, then those with the smallest majority would be
> rejected.

(2) To my opinion, Schulze is more intuitive than Tideman
because why should I proceed overruling voters until there
is a complete ranking when I am only interested in the
winner?

Markus Schulze
schulze at sol.physik.tu-berlin.de
schulze at math.tu-berlin.de
markusschulze at planet-interkom.de