[EM] Herve Moulin's proof not really a proof

[Markus Schulze]

Dear Nicholas,

you wrote (13 June 2012):

 > Actually, on a weird second thought, wouldn't a method that refused to
 > identify a winner in a three-way tie (Condorcet paradox) be compatible
 > with both?

In Woodall's terminology, the output of an election method is
a probability distribution on the set of candidates.

He defines the participation criterion as follows:

Suppose a set of voters is added where each voter strictly prefers every
candidate of set _A_ to every other candidate. Then the probability that
the winner is chosen from set _A_ must not decrease.

In my paper (http://m-schulze.webhop.net/schulze1.pdf), the output of
an election method is a set of winners _W_ rather than a single winner.
In (4.7.16) -- (4.7.17), I define the participation criterion as follows:

Suppose a set of voters is added where each voter strictly prefers every
candidate of set _A_ to every other candidate. Suppose the intersection
of _A_ and _W_ was non-empty, then the intersection of _A_ and _W_ must
be non-empty afterwards. Suppose _W_ was a subset of _A_, then _W_ must
be a subset of _A_ afterwards.

It is easy to see that Moulin's proof also works when Woodall's or my
definition of the participation criterion is used.

Markus Schulze