[EM] IIA Theory

[Craig Carey]

Mr Catchpole's "Catchy-IIA (Deterministic Independence of Removal of
 Irrelevant Alternatives)" should be rejected.

Mr Schulze's "Deterministic Independence from [removal of] Irrelevant
 Alternatives" rule is a very different rule.


Rule A: Markus Schulze, 6 October 1999:

>   Suppose, that candidate A would have not been elected
>   if candidate B hadn't run. Then if candidate B does run,
>   candidate A must not be elected.

Translate that English version into a mathematical form:

Let V be the larger voting system and U be the smaller, then the rule is

(All a,b,a<>b)(All U,V, U=del(V,b)).[-W(U,a) => -W(V,a)]
= (All a,b)(All U,V).[(a<>b)(U=del(V,b)) .=> (-W(U,a) => -W(V,a))]
= (All a,b)(All U,V).[(a<>b)(U=del(V,b)) .=> (W(V,a) => W(U,a))]

W(V,a) = (a wins V), (-x.=>.-y)=(-y or x)=(y=>x).
U is a point on the face of the simplex that contains point V.


The rule doesn't seem that important to me, because one point of
 the 2 that have their winner sets compared, is right at surface.

Possible line of research: Couldn't the point to point 'del'
 function be made to be a function that returns a set, and then
 there would be a rule saying approximately, if candidate A wins
 at point V, then for any point inside a simplex that has a
 tip at point V and that projects out towards a particular face,
 that other point that is near the face also wins.
(Irrespective of the number of winners of course). IFFP with 3
 candidates passes the rule. 

Commentary 
In (P1), when preferences are discarded and the point moves towards
 a face, the rule specifies that a loser continues to lose.
This rule of Mr Schulze, looks as if it might be an instance of a
 rule that says a winner continues to win as the point moves
 towards a face. I suppose it just a matter of dropping a line
 segment across those two points and "writing down" the term to the
 right of the "For All"s. IFPP 3 candidate needs to passed obviously
 and I suppose STV could be checked (although hopes shouldn't be
 too high for that method since it near mathematical rubbish with
 far too many internal vertices and corners, all of which will tend
 to make end up being failed by this rule.


===================================================================

[Retraction (on behalf of Mr Catchpole, who will be commenting when
 able)]

The Catchy-IIA rule.
'Shot down in mid flight in the winds above a desert nation.'


At 20:51 07.Oct.99 , David Catchpole wrote:
...
>Catchy-IIA (Deterministic Independence of Removal of Irrelevant
>Alternatives)
>
>"The removal (ceteris paribus) of any combination of candidates who fail
>to be elected does not alter the outcome"

A rewording would be this:

"The removal of a loser does not alter the set of winners"

Here's the rule in a mathematical form:

(All V)(All b, -W(V,b))(All U, U=del(V,b)).(All a,a<>b).[W(U,a) = W(V,a)]
= (All U,V)(All a,b).[-W(V,b).(a<>b)(U=del(V,b)) .=> (W(U,a) = W(V,a))].


Here's an example which both STV & IFPP agree, and which the rule
 fails:

A. 2
BC 2
CA 1

Both STV & IFPP: A wins.  (IFPP given that 2>(2+2+1)/3=5.666..)
FPTP: A or B

Apply the rule: since B is a loser the removal of all preferences
 for B (i.e. removal of candidate B) should not cause A to flip from
 being a winner to a loser. Unfortunately for the rule, it does.

A. 2
C. 3

About all methods: C wins.

In an aside to Mr Catchpole, your started with "For models of
 preference: 'The removal or addition of an option should not
 alter the preference between any other pair of options"


----------------
Some corrections to my last message: Mr Schulze didn't use the word
 "chance". I was writing on the technique of placing and upper
 bound or estimate of probability (whatever) that the formula
 gets the wrong result. An error was made in a particular line
 that should have read:

(The progression [of the number of vertices of the election
 simplexes follows in this way: 0,2,9,40,205,1236,8659,...
 following M(n)=n.(M(n-1)+1)).

If votes can be spread across K papers where K is the factorial
 of the number of missing preferences, then the number of vertices 
 would be the factorial of the number of candidates. (Bart Ingles
 was writing about vote smudging in 27+ December 1998).

8 October 1999

_____________________________________________________________
Mr G. A. Craig Carey
E-mail: research at ijs.co.nz
Auckland, Nth Island, New Zealand
Pages: Snooz Metasearch: http://www.ijs.co.nz/info/snooz.htm
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