[EM] IIA Theory

[David Catchpole]

> ===================================================================
> 
> [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.'

Non passarin. I don't get the reference to Iraq and you should already see
that Schultz-IIA per se also fails the same example.

I've already said that for many examples the rule may not possibly be satisfied.
IIA's not a general criterion- the criterion which it applies to an
electoral system is that _where_ it may be satisfied, it should be. For ye
olde paradox of voting, we can't find an IIA (neither Schultz nor
otherwise) result without completely subverting majority rule.

I don't want to get into a flame war, as surely as you don't, and I deeply
respect the fact that someone actually is in the list who is applying
something approximating mathematical rigour to other people's ideas, so
possibly before we demand retractions we should simply agree to disagree.


> (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))].

This much is true. Now, to extend this to a more general criterion, state
"For all voting schema V-
if there exists an electoral system W' such that-
	for all candidates b, for all candidates a,
	a is not b and b is an element of "-W'(V,b)" implies
	W'(V,a)=W(del(V,b),a)
-then-
for all candidates b, for all candidates a,
a is not b and b is an element of "-W(V,b)" implies W(V,a)=W(del(V,b),a)"

This _is_ a criterion which may be met always and I'll call it "Catchy-IIA
criterion," say?

> 
> 
> Here's an example which both STV & IFPP agree, and which the rule
>  fails:
> 
> A. 2
> BC 2
> CA 1

A common or garden "paradox of voting."

A vs. B- A
A vs. C- C
B vs. C- B

This will always fail to satisfy "Schultz-IIA" too. Say A won A vs. B vs.
C? "Schulze-IIA" says that because A wouldn't have won had B not stood, A
should not win. What about B winning the 3 candidate race? No, "S-IIA"
says that because B wouldn't have won had C not stood, B should not win.
Okay, how about C? Same problem.

Just because IIA as a specific rule on an election schema may not be
satisfied does not mean that it is not a useful principle where it may be.
This has been the point of my past umpteen e-mails.

> 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"

This refers to models of social preference, such as that applied by Arrow
and Sen in their work. Does the society have a transitive preference
schedule such as that possessed by an individual? An example of such a
transitive social preference schedule might be "100 greatest hits of all
time" in a ubiquitous fan magazine. 

I apologise that my nomenclature probably confused you in basically every
e-mail I've sent. It's my fault, but remember that a quick sortie to
various voting theorists' work provides grounding for understanding the
context in which we discuss stuff on this list.