Random Ballot Tiebreaker

[Markus Schulze]

Dear Blake,

you wrote (25 Aug 1998):
> This method might work, maybe it's what Tideman intended:
> 1.  Pick a random ballot and use its rankings, consider
>     ties as unsorted with regard to each other.
> 2.  Continue picking ballots.  When you find one that
>     orders previously unsorted candidates, use the ballot
>     to sort them.  Do not change the order of the already
>     sorted.
> 3.  If you go through all ballots, and some candidates
>     are still not sorted, order them randomly.

If you go through all ballots, and some candidates are
still not sorted, means, that every voter ranks those
candidates identically. That means, that the voters are
not decisive about these candidates.

I have to add to my last mail, that I always suppose,
that the voters are decisive. That means, that for every
pair of two candidates A and B there is at least one
voter, who either ranks candidate A ahead of candidate B
or candidate B ahead of candidate A.

Those versions of "Complete Independence of Clones," that
I use, always suppose that the voters are decisive. Otherwise
my versions of "Complete Independence of Clones" don't make
any sense.

Example 1:

   There are 2 candidates (A and B) and every voter ranks
   both candidates identically. Then the only way to choose
   the winner is to choose him randomly. Every other
   method would violate Neutrality.

   Thus, candidate A is chosen with a probability of 50%.

Example 2:

   Suppose, that candidate A is cloned and that A1, A2, and A3
   are his clones. Suppose, that every voter ranks candidate A1,
   candidate A2, candidate A3, and candidate B identically.
   Then, the only way to choose the winner without violating
   Neutrality is to choose him randomly.

   But this would mean, that a clone of candidate A is chosen
   with a probability of 75%. Thus "Complete Independence of
   Clones" is violated.

Summary:

   You cannot ask an election method to meet "Complete
   Independence of Clones" in every possible situation.
   To get "Complete Independence of Clones," you have to
   suppose decisiveness of the voters in at least some very
   weak meaning.

Markus Schulze