[EM] "No splitting rule"

[Craig Carey]

Abstract: the wording of the extended multiwinner version of Mr
 Catchpole's (approx.) 18 November 1999 rule 2 is analysed and it is
 shown to have probable vagueness in its definition (more down below).


At 13:06 18.11.99 , David Catchpole wrote:
>Hey guys- here's two types of single-winner no-splitting rule. They
>attempt to disallow additional candidates splitting the vote and
>disadvantaging their sides- however, they do allow additional candidates
>helping the chances of other candidates "on their side"
>
...
>Rule 2 is slightly more restrictive in that it states that-
...
>An extension of rule 2 into multiple winner elections (with respect to
>removal)
>
>With the removal of a candidate A, either-
>
>(i) A was one of the old winners; or
>(ii) There's no change; or
>(iii) Of those who rank some possible combination of winning candidates
>including A over the old and new winning combinations, at least as many
>prefer the old winning combination to the new winning combination as
>prefer the new winning combination to the old winning combination (whew!)


That last rule can be reworded, on using (a|b|c) = (a|b|(-a)(-b)c) :

If candidate A loses and is removed and the removal alters the set
 of winners, then 
   For all sets of preferential voting papers [preference lists with
       weights], P say, then:
     If the group of papers P "ranks some possible combination of winning
       candidates including A over the old and new winning combinations",
     then "at least as many papers [accounting for their weights] prefer
       the old winning combination [with the loser A excluded] over the new
       winning combination [with at least one new winner] as prefer the new
       winning combination to the old winning combination [without A]". 

It can be always allowed to have candidate A not win in both the
 'before state' and the 'after state' too.

But Mr Catchpole wrote: "Of those who rank some possible combination of
 winning candidates including A over the old and new...".
 A can always be a not winning candidate in both the before and after
 states.

Also, how do groups of papers "rank" winners of possible subsets of
 the set of all winners, over alternative subsets containing a different
 number of winners and also actual different winners?.

So the definition seems to be not precise.



      A note on Condorcet's picking 0 winners when there should be 1

A note on Condorcet and pairwise comparing. This is relevant to the new
 rules.

Let's estimate F, where 10**(-F) equals the probability that the Condorcet
 method actually returns a single winner when there are 10**99 candidates.

The approximate derivation here requires the drawing of a graph, and noting
 or saying that the directions of the arrows are quite random, and that
 F = log base 10 of approximately this: 10**99 times (1/2)**(10**99 - 1).

So F = -3*(10**98) (approximately).

So Condorcet rarely picks a winner for large enough elections.
The next step in the argument is to wait for the defender of Condorcet to
 say that there are ''too many candidates'. That happens.
It is not trivial to prove that, but a statement will substitute for a proof.

At a certain number of candidates, Condorcet becomes a bad method.


                     Bounding where Condorcet goes bad

Maybe Condorcet is OK for 102 candidates but it is not OK for 103.
Or does it go bad gradually and the only way to save Condorcet is to say that
 it gets worse slowly and use ideas from probability theory to save the day.

David was intoducing new original rules using ideas of pairwise comparing.