[EM] Method definitions

[Markus Schulze]

Dear participants,

on page LXVIII 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 writes due to my own 
translation: 
> From the considerations, we have just made, we get the general 
> rule, that in all those situations, in which we have to choose, 
> we have to take successively all those propositions that have 
> a plurality, beginning with those that have the largest, 
> & to pronounce the result, that is created by those first 
> propositions, as soon as they create one, without considering 
> the following less probable propositions. 

On page 126, he writes due to my own translation: 
> 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. 

******

I interpret the first quotation as follows: "Minimize the maximum
pairwise defeat that has to be ignored to get a complete ranking
of the candidates."

I interpret the second quotation as follows: "Maximize the minimum
pairwise defeat that has to be used to get a complete ranking of the
candidates."

Therefore I think that every election method, that minimizes the
maximum pairwise defeat that has to be ignored and maximizes the
minimum pairwise defeat that has to be used to get a complete
ranking of the candidates, is a possible interpretation of
Condorcet's words. Therefore I think that the Tideman method,
the Schulze method and many other election methods are possible
interpretations of Condorcet's words.

******

Example (3 Feb 2000):

   26 voters vote C > A > B > D.
   20 voters vote B > D > A > C.
   18 voters vote A > D > C > B.
   14 voters vote C > B > A > D.
    08 voters vote B > D > C > A.
    07 voters vote D > A > C > B.
    07 voters vote B > D > A = C.

Then the matrix of pairwise defeats looks as follows:

   A:B=51:49
   A:C=45:48
   A:D=58:42
   B:C=35:65
   B:D=75:25
   C:D=40:60

1. The "strongest cycle" is C > B > D > C. The weakest pairwise
defeat of this directed cycle is C:D=40:60. Therefore the
ranking of the candidates should be compatible to all
pairwise defeats that are stronger than 40:60. That means
that the final ranking must include C > B and B > D.

2. If votes-against was used then A:C would be the weakest
pairwise defeat. Therefore he would ignore A:C=45:48. Then he
would ignore A:B=51:49. Then he would take A:D=58:42 into
consideration because otherwise it wouldn't be possible anymore
to get a complete ranking of the candidates. That means that
the final ranking must include A > D.

Therefore I conclude that every election method that is
compatible to Condorcet's words must result in a ranking that
includes C > B, B > D and A > D.

Markus Schulze
(this time without any virus)