[EM] IIA Theory

[Craig Carey]

       AN ANALYSIS OF THE CATCHPOLE IIA RULES  (PART III)


At 15:52 06.10.99 , David Catchpole wrote:
>On Wed, 6 Oct 1999, Craig Carey wrote:
>> At 12:26 06.10.99 , David Catchpole wrote:
...
>> 
>> Mr Catchpole was seeking to say that the 'IIA' rule has this
>>  meaning.
>> 
>>          The election is invariant to the adding or
>>          removing of a candidate, provided that both
>>          before and after, there are no preferences
>>          anywhere for that candidate.
>
>This is still inaccurate. How perhaps it should read is-
>
>	The election is invariant to the adding or
>	removing of a candidate, if both
>	before and after, the candidate does not win.

(Start up the rule destroying machinery again)

CASE [A] the "Adding" interpretation:

Here is an Example

Multiwinner First Past the Post, 2 winners,
 and 3 candidates:

10   A., AB, AC
7    B., BA, BC
5    C., CA, CB

FPTP Winners = {A, B}.

Under the 6 October 1999 Cathpole-IIA rule:

    > "The election is invariant to the adding or
    >  removing of a candidate, if both
    >  before and after, the candidate does not win." ;

this election should have the exact same outcome:

10 : A, AB,ABC,ABD, AC,ACB,ACD, AD,ADB,ADC
4  : B, BA,BAC,BAD, BC,BCA,BCD, BD,BDA,BDC
3  : DB
5  : C, CA,CAB,CAD, CB,CBA,CBD, CD,CDA,CDB
0  : D, DA,DAB,DAC,    DBA,DBC, DC,DCA,DCB

A:B:C:D = 10:4:5:3
FPTP Winners = {A, C}
In three of the papers, a preference for "D" was inserted
 before the preference for "B".

So FPTP is failed by the rule. The new IIA rule does't take
 any issue with the fact that FPTP loses 100% of the vote
 during a 'transfer', so why wouldn't all preferential
 voting methods be failed by this IIA rule?.

(An comment on the side: Persons might believe that it is
 possible to constuct formulae where there is no wastage of
 a vote during 'transfers' across preferences for completely
 losing candidates. I presume a method without 'transfer'
 type vote wastage over losers cannot be devised.)

Sub-issue: Which methods pass the Catchpole-IIA method?

[This is quoted from an earlier message dated 19-20 September 1999]
: 
: > What is a 'fave' meth Dave?: is it a threat to any of the strict
: >  reasonable criteria that some theorists might write about?.
: 
: We all have our favourites on this list. Over time, this becomes
: really apparent- especially with respect to who gets involved in what
: discussion. ...

What's the method?. The latest Cratchpole-IIA rule ("adding" mode) may
 in fact fail a big number of preferential voting methods.


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

CASE [B] the "Removing" interpretation of the
   6 October Catchpole-IIA rule;


>	The election is invariant to the
>	removing of a candidate, if both
>	before and after, the candidate does not win.

V
  AB  a
  B   b
  C   c

U
  B   b+a
  C   c
  
In a quite general method (excluding Condorcet, etc.)
 the winners can be:

  aV = (b<a)(c<a)
  bV = (-aV).B(a,b,c)
  cV = (-aV).-B(a,b,c)

  bU = (c < b+a)
  cU = (b+a < c)

.....AB
..../.\
.../...\
../.....\
.B-------C

B is a divide between B & C. The divide passes through
 the centre of the triangle and also through the midpoint of
 the line between vertices b and c.

The rule fails the method when X = True, where X = bV.cU
  X = (-aV).B(a,b,c) . (b+a < c)
  X = [(a<b) or (a<c)].B(a,b,c) . (1/2 < c), if a+b+c=1

So this IIA rule prevents the B boundary moving
 outside of an inverted triangle that has its lower vertex on
 the midpoint between vertices (B.) & (C.).

I.e. this IIA rule prohibits these two election alterations:

  C wins :  B { C  
  B wins :  AB    

  B wins :  C { B
  C wins :  AB   

Method = any

The (P1) rule prohibits those two alterations anyway.
(P1) is quite different from IIA, since IIA applies to
 preceding preferences.


....
>www.math.nwu.edu/~dsaari . By going through Sen's, Gibbard's and
>Satterthwaite's work first you can see how Saari's criticism of IIA as
>being "absurd" (because, and this should already obvious, it fails to be
>satisfied in all cases) is itself problematic, especially in its
>"implications" towards Borda score systems, and at the same time how right
>Saari is in seeing that Arrow's theorem has at its heart the simple 
>problem of IIA occasionally failing to be satisfied where more than two
>voters have an impact on the outcome.

Those are old books. "Saari's criticsm of IIA"?. Is that yet another
 version of IIA?. I'd like to have them all listed. It is most possible
 that 4 or 5 candidate formulae will the IIA rule to be rejected or
 modified.

-------------------------
For reference, these are the wordings that survived in some sense:

   "The election is invariant to the removing of a candidate,
   if both before and after, the candidate does not win."

   "The removal of preferences for a particular candidate do
   not cause the win-lose state of any two other candidates
   that neither both win nor both lose, to become swapped."

   "The addition of preferences for a particular candidate do
   not cause the win-lose state of any two other candidates
   that neither both win nor both lose, to become swapped."



_____________________________________________________________
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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