[EM] IIA Theory

[David Catchpole]

On Wed, 6 Oct 1999, Craig Carey wrote:

> 
>        AN ANALYSIS OF THE CATCHPOLE IIA RULES  (PART III)
> 
> (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?.

FPP certainly does this- but how can you extend this to all preferential
systems? I assume you mean the particular class of preferential systems
which involve the transfer of successive minorities of highest preference?

> 
> (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.)

Not quite sure if that's pertinent (though I do agree that it is
true). Maybe you could be more specific to demonstrate whether it is
important to our present argument. Might be worth discussing with Demorep,
though, as an adjunct to this list's "Droop-Hare wars."

> 
> 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. ...

Well, I was talking about favourite criteria and principles of voting. I
honestly didn't see "meth" in the post.

My present favourite single-member electoral system, per se, is "FTC," an
acronym for which I no longer provide an extrapolation for fear of
incriminating myself. Basically, "FTC" is Condorcet using three-way
comparisons rather than two-way comparisons (three way comparisons using
ye olde "who's smallest? chuck him out" preferential method). It's
Condorcet for two-party systems, and where a solution exists it is
independent of the removal of irrelevant alternatives to the point where
three candidates only are standing. Yes, you may have guessed, "FTC" the
acronym involves Centrists and the act of copulation.

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

Of course adding does. In fact, it fails everything. There's the rub- the
simple removal rule bears fruit and still has meaning while adding just
demonstrates the fact that we can always introduce examples, no matter how
outrageously unlikely, that demonstrate the problems at the heart of
developing voting systems.
 
> 
> =====================================================================
> 
> 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.

Just amazing, ain't it? You'll find that IIA principles and monotonicity
principles (the more general forms of "P1") are closely related, even
though one is a condition on changing candidates and one on changing
voters.

> 
> 
> ....
> >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.

Author
                     Arrow, Kenneth J. (Kenneth Joseph) 
 Title
                     Social choice and individual values / Kenneth J.
Arrow.
 Publisher
                     New Haven : Yale University Press, 1963.
 Edition
                     2nd ed.

I apologise that my reference to Saari's URL was wrong. His articles
include references to Sen, Arrow, Gibbard and Satterthwaite.

Here is his new URL-

http://www.math.nwu.edu/faculty/homepages/donald.saari.html

IIA's failing to be satisfied most often goes by the more common
name, "the paradox of voting," the olde A>B>C>A problem. So, three
candidates is where the trubble boils up.

> 
> -------------------------
> 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."

No. 1. is spot on as the IIA criterion which may most often be applied
usefully. The ones below, while the first and second are nice, are not IIA
principles.

> 
>    "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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