[EM] IIA Theory

[David Catchpole]

On Wed, 6 Oct 1999, Craig Carey wrote:

> At 12:26 06.10.99 , you wrote:
> >To paraphrase Eric Cartman, 'Ay! I know that I stuffed up in my
> >definitions but that's low! (and wrong!) (more further down)
> Thanks David. The could be gold under the IIA sandstone yet.
> 
>        AN ANALYSIS OF THE CATCHPOLE IIA RULES  (PART II)
> 
> 
> 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.

> Here is a question for Mr Catchpole:
> 
>  If a solution is put in a triangle and then a 4th quite
>  new candidate is added and allocated a vertex above the
>  triangle, then:
>  What is the significance of the Catchpole-IIA rule?.
>  What is a method that your rule eliminates.

I hate the "geometry of voting!" "Vertex" above a "triangle" makes no
sense until you direct me to the formalism you are using. Does "triangle"
impute a Saari triangle diagram?

The significance of IIA as restated above for deterministic systems is it
is the test for a voting schema of whether or not there is a strategy
involved in the creation of the agenda of candidates or options. Is the
vote split? Is a party better off having a splinter, as in STV? A
more-or-less equivalent criterion is the concept of social rationality,
that is that the choices of a society are as well behaved as those of an
individual, with the Demorep Society always preferring George Washington
over Joseph Stalin no matter who else is standing.

IIA makes Condorcet analogues preferable to others (again, this
is problematic, because of the realities of political behaviour, but this
is another story and it loses significance as the number of candidates to
be elected increases). A perfect example of a system which regularly
violates IIA when it really shouldn't is FPP, where the familiar
phenomenon of splitting occurs. It must be noted (important! important!)
that in deterministic electoral systems, there are always cases where IIA
cannot be satisfied. But- some electoral systems are more transgressional
than others!

> Can the IIA theory be transmuted by anybody into something
>  valuable?. I'd like a reference to the person or
>  mathematician who named/published the "IIA" rule. Was it
>  quoted accurately?.

A good start at the work would be in Arrow's, Sen's, Gibbard's and
Satterthwaite's early groundbreaking works on social choice. I would start
with Kenneth J. Arrow's "Social Choice and Individual Values" and end with
Donald Saari's criticism of Arrow's and Sen's theorems at
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.