[EM] Is "Inverse Nanson" better than standard Nanson?

[Markus Schulze]

Dear Blake,

Condorcet wrote ("Essai sur l'application de l'analyse a la
probabilite des decisions rendues a la pluralite des voix,"
Imprimerie Royale, 1785, p. 125-127):

> When you want to apply the considerations we have just made
> to situations with n Candidates, you can follow the following
> rules. First: All the opinions that are possible and don't imply
> contradiction are reduced to the indication of the order of merit
> of your judgement between the Candidates. For example, the six
> opinions above are reduced to the six combinations (1) A, B, C;
> (2) A, C, B; (4) C, A, B; (5) B, A, C; (7) B, C, A; (8) C, B, A,
> which we mark here with the same numbers as the corresponding
> opinions (See page 120!) and which indicate the different orders
> in which A, B, C can be arranged. Thus, for n Candidates you have
> n*(n-1)*...*2 possible opinions. Second: Each Voter gives his
> opinion by indicating the order of value of the Candidates, when
> you compare them head to head, you have in each opinion n*(n-1)/2
> propositions to consider separately. By taking the number of
> times a given proposition is contained in the opinion of one
> of the q Voters, you have the number of votes adopting this
> proposition. Third: 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 to 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 and accept the
> resulting opinion of the remaining propositions. Fourth: In those
> cases, in which you don't have to make a choice or in which you
> can defer, you calculate the probability of the opinions that
> give the preference to A, to B, to C etc. and you pronounce the
> election only when it chooses a given Candidate with a probability
> of more than 1/2; this cannot happen when the votes lead to one of
> the 2^(n*(n-1)/2)-n*(n-1)*...*2 absurd opinions; and this cannot
> happen in the other n*(n-1)*...*2 opinions unless each of the
> n-1 propositions A > B, A > C etc. which essentially form the
> opinion in favour of A, for example, are those winning most of
> the votes; there is nevertheless a very big difference between
> this case and those with an impossible opinion. In the latter
> case, you have to pronounce a proposition that really has the
> plurality against it, this doesn't happen here: Thus, unless it
> is inconvenient to defer the election, you can pronounce the
> possible opinion that we have gotten with this method; instead
> of having a real necessity to make a choice to get the opinion
> when the propositions that form it imply contradiction.
> Fifth: ...

You wrote (8 July 2001):
> I have to agree with Mike.  Condorcet says to "create" an opinion
> of the n*(n-1)/2 propositions. I take this as a description of
> how to find the initial opinion, not a definion of what an opinion
> is.  What Condorcet means by "opinion" can only be pieced together
> in context.  So, I don't view the n*(n-1)/2 as a fixed part of the
> definition of an opinion.

I don't agree with you. To my opinion, it is clear that an
opinion is a set of n*(n-1)/2 propositions (one for each pair
of candidates).

Markus Schulze