# [EM] Condorcet cyclic drop rule

Blake Cretney bcretney at postmark.net
Wed Apr 4 10:30:42 PDT 2001

```Dear Markus,
> Dear Blake,
>
> the MinMax interpretation of Condorcet's wordings has been
> proposed by Black (Duncan Black, "The Theory of Committees
> and Elections," Cambridge University Press, 1958).

Is that really an interpretation, or a new method based on Condorcet's
principles?

Here's what Mike has to say, at
http://www.barnsdle.demon.co.uk/vote/condor2.html

> We call that method "Plain Condorcet" ("PC").

Note that originally Mike called this "Condorcet's Method", and some
people, like Rob Lanphier, still do.

> It turns out that this method was proposed in the late 18th
> century, by the founder of voting theory.
>
> The Marquis de Condorcet, in the period just after the
> French Revolution, participated in the discussion of how
> the new government should be set up. This included voting
> system proposals.
>
> He proposed that if there's a candidate such as I've called
> BeatsAll winner, that candidate should win. And he proposed
> a natural & obvious solution for when there's no BeatsAll
> winner.
>
> Here's what he suggested. What I call pairwise defeats,
> he called "propositions":
>
> If the propositions can't all exist together [because
> there's a cycle instead of a BeatsAll winner], then, one
> at a time, drop the proposition with the smallest majority,
> until there's an unbeaten candidate.

The square brackets, and context seem to imply this is a quote, but it
isn't offset or in quotation marks.  Mike, is this intended as a
quote?  Do you have a reference?

Indeed this statement is equivalent to MinMax, although an odd way of
describing it.

> In other words, drop the weakest defeat. Repeat till there's
> an unbeaten candidate.
>
> (The strength of B's defeat by A is measured by how many
> people ranked A over B).

Here's another quote from Condorcet as a comparison.

Iain Mclean, Fiona Hewitt, 1994
"Condorcet:  Foundations of Social Choice and Political Theory"
Edward Elgar Publishing Limited

p 238 (of the translation) from "On Elections" 1793
> 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.

That sounds a lot like Ranked Pairs.  Reject smaller majorities, but
only in favour of larger ones in the complete order.  On the other
hand it seems oddly vague.  I suspect that Condorcet thought that if
you kept successively dropping majorities (from smallest up), you
would have a valid ranking as soon as you eliminated all the cycles.
If this were true, his description would not be ambiguous.
Unfortunately, this isn't the case.  So, you can take Condorcet's
statement as vague, but it's probably better seen as incorrect.

It certainly doesn't look like MinMax.  There's no "until there's an
unbeaten candidate."  Condorcet implies that you will drop majorities
until all the judgements can exist together.    In any case, I think
it's clear now that if this could be "interpreted" as specifying
MinMax, it would be the marginal form of MinMax, since Condorcet
specified margins.

---
Blake Cretney

```