[EM] Duncan Black on Condorcet

[Markus Schulze]

Dear Blake,

it is Fishburn who introduced the term "Condorcet's function"
for the MinMax method. He also uses the term "C6". He wrote
(Peter C. Fishburn, "Condorcet Social Choice Functions,"
SIAM Journal of Applied Mathematics, vol. 33, p. 469-489,
1977):

> The designation of C6 as Condorcet's function may be arguable
> and Young, who presents the only recent analysis of C6 that
> we are aware of, prefers to call it the minimax function. My
> designation of C6 stems from Condorcet's proposal for dealing
> with four or more candidates in the absence of a majority
> candidate. Unfortunately, this proposal is not very clear
> according to Black. Black says that "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." A proposition in
> Condorcet's terms is a statement of the form "x has a simple
> majority over y." Although Black gives several interpretations
> for the above quotation, we end up with C6 if we interpret it
> as follows. First, list all ordered pairs (x,y) from AA for
> which x <> y and p(x,y) >= n/2, where n is the number of
> voters. Then, beginning with the smallest p(x,y) value,
> delete all pairs from the list that have this smallest
> value, then delete all pairs with the smallest remaining p(x,y)
> value, and so forth, until a point is reached where one or
> more candidates do not appear in the second position of any
> remaining ordered pair. The candidates absent from second
> positions at this point constitute the choice set. [If all
> pairs must be deleted to reach the indicated point then
> C(A,p) = A.] It is not hard to prove that C6(A,p) is the
> choice set obtained by this method.

"A" is the set of alternatives. "p" is a profile of the voters.
"(A,p)" is an example. "CX(A,p)" is the set of winners of the
election method "CX" in the example "(A,p)".

Markus Schulze