[EM] Duncan Black on Condorcet

[Markus Schulze]

Dear Blake,

Black wrote (Duncan Black, "The Theory of Committees
and Elections," Cambridge University Press, 1958):

>    With N candidates, the number of pairs of propositions
> of the type A:B will be equal to the number of
> combinations of 2 things chosen out of N, i.e. N*(N-1)/2.
> A set of propositions is formed by choosing one from each
> of these pairs; and the number of sets will therefore
> be 2^(N*(N-1)/2). The number of sets of consistent
> (transitive) propositions will be the number of ways of
> making an ordered selection of N objects, i.e. N!. The
> number of sets in which the propositions are inconsistent
> will be 2^(N*(N-1)/2)-N!.
>    As the number of candidates increases, the inconsistent
> sets will form a rapidly increasing proportion of the
> whole as is shown in the table of Fig. 166.
>
>     Fig. 166
>
>     No. of candidates  Consistent Inconsistent
>     2                  2          0
>     3                  6          2
>     4                  24         40
>     5                  120        904
>
>    The order in which the candidates have been ranked by
> the majorities will show which one ought to be chosen;
> and if we want to elect 2, 3, ... candidates, the top
> 2, 3, ... should be chosen.
>    Condorcet adds that if we are not forced to make an
> election, before choosing the top candidate we can insist
> that the probability that he is the best candidate should
> be greater than 1/2. But if we have no practical means of
> reckoning v and e, we cannot calculate the probability
> that he mentions.
>
>    The inconsistencies may be such that we are still able
> to pick out the best candidate, or the best and also the
> next-best, etc. For instance with the five candidates
> A, B, C, D and E, the voting may yield the set of ten
> propositions: A > B, A > C, A > D, A > E, B > C, B > D,
> B > E, C > D, E > C, D > E. Contradiction only arises
> between the last three propositions. There is no
> contradiction between the first four, which show that
> A is the best candidate; and there is no contradiction
> between the first seven propositions, which show that
> while A is the best, B is the next-best candidate.
>    But the contradictions may be such that no candidate
> is seen to be better than all of the others in this way.
> Here Condorcet gives the instruction that we should make
> out the list of propositions that result from the voting,
> then remove from it those propositions that have the
> smallest majorities in their favour, and adopt the
> decision that follows from the first consistent set of
> propositions remaining. He gives no argument for the
> rule beyond that it accords with his procedure in the
> case of three candidates. And he states that if we admit
> this method a different result may follow according as
> we begin with the set of propositions (A:B, A:C, A:D,
> etc.) or (B:A, B:C, B:D, etc.) or (C:A, C:B, C:D, etc.)
> and so on.
>    As Nanson says "The general rules for the case of any
> number of candidates as given by Condorcet are stated
> so briefly as to be hardly intelligible ... and as no
> examples are given it is quite hopeless to find out what
> Condorcet meant." The following three interpretations,
> however, might be suggested.
>    (i) We might write down the list of propositions for
> A in relation to each of the other candidates, those for
> B, etc., as shown; and then deem the largest size of
> minority to be a majority, then the next largest minority,
> and so on, until all of the propositions relating to
> one of the candidates become majorities (either true or
> deemed) and elect him.
>    (ii) It would be more consistent with Condorcet's
> words, though not with his lists of symbols, to take into
> account all of the propositions derived from the voting
> and disregard the proposition G > H, say, which has the
> smallest majority in its favour, and the corresponding
> H > G. Then if propositions of the type H > I, I > G
> existed, we would deem H > G. Proceeding to delete the
> smallest, next-smallest, ... majorities, we would select
> that candidate who was the first to secure a majority or
> deemed majority over each of the others.
>    (iii) It would be most in accordance with the spirit
> of Condorcet's previous analysis, I think, to discard
> all candidates except those with the minimum number of
> majorities against them and then to deem the largest
> size of minority to be a majority, and so on, until one
> candidate had only actual or deemed majorities against
> each of the others.
>    Contrary to what Condorcet says, however, each of these
> three interpretations would give rise to the choice of a
> definite candidate (ties in the voting apart), so that
> none of them may be what he had in mind. It is a pity
> that on this crucial question his argument should be so
> fragmentary.

Markus Schulze