[EM] IIA Theory

[David Catchpole]

To paraphrase Eric Cartman, 'Ay! I know that I stuffed up in my
definitions but that's low! (and wrong!) (more further down)

On Tue, 5 Oct 1999, Craig Carey wrote:

> 
> 
> AN ANALYSIS OF THE CATCHPOLE IIA RULES IN ATTEMPT TO GET RID OF
>                ALL THEIR AMBIGUITY (PART I)
> 
> 
> Conclusion: Nothing of value was found. 5 October 1999
> 
> At 19:20 04.10.99 , David Catchpole wrote:
> 
> >Unfortunately, I miswrote the IIA statements because I got into the groove
> >of writing "removal or addition"-
> >
> >Here they are corrected-
> 
> [1] ---------------------------------------------------------------------
> >for models of preference-
> >
> >"The removal or addition of an option should not alter the preference
> >between any other pair of options"
> 
> [2] ---------------------------------------------------------------------
> >for probabilistic models of choice-
> >
> >"The removal (addition) of an option which has not been selected will
> > not reduce (increase) the chances of any outcome where that option
> > is not chosen"
> 
> 
> [1] ===================================================================
> 
> PART 1:  "For models of preference"
> 
> >"The removal or addition of an option should not alter the preference
> >between any other pair of options"
> 
> Two rewritten versions follow. Interpretation [A] seems to
>  be bad (argued for below).
> 
> [A]
> "The removal or addition of a candidate does not cause the
>  win-lose state of two other candidates that neither both win nor
>  both lose, to become swapped.

Remember that I'm talking about the whole electoral schema (e.g. who wins)
rather than "win-lose slates". However [A] is patently true for any
majoritarian single-member system- preferneces between A and C and B and C
do not influence preferences between A and B, to the extent that C is a
real additional / removed candidate and voters are rational.

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

That's nice (without the "preferences" bit- IIA is a condition on static
voter preferences and variable candidates) because it's closer to the
independence of the results at least of a Condorcet system from such
win-win, lose-lose comparisons.
Again, though, it doesn't resolve the problem of seeking for results
rather than minutiae.

> Regarding [A] under the 'removal' case, if there are N candidates
>  and 2 winners, and it is very close call between candidates
>  E and F, and preferences for were scattered throughout the voting
>  papers, and A received over 15,000 votes, then removing A could
>  cause a swap in the win lose state of E and F, with perhaps B
>  becoming the winner once A left

This is certainly the case for FPP but not for Condorcet (if a
Condorcet winner exists...).

> [B]
> "The removal or addition of preferences for a particular candidate
>  do not cause the win-lose state of two other candidates that
>  neither both win nor both lose, to become swapped.
> 
> Write the alteration as function mapping a voting system into
>  a set of voting systems, called alt(V,c).
> It is not clear what a Catchpole-IIA "remove" function is.

The removal of a candidate from the ballot paper would be the easiest way
to explain it.

Trying to read set algebra over e-mail is no fun, even more so if it's
been written as if for a TeX compliler. 

> 
> I'll stop this here. It might be easier to show the whole
>  rule oughtn't be imposed (but instead rejected).
> 
> 
> =======================================================================
> 
> 
> PART 2:  "For probabilistic models of choice"
> 
> 
> [2] ===================================================================
> 
> >"The removal (addition) of an option which has not been selected will
> > not reduce (increase) the chances of any outcome where that option
> > is not chosen"
> 
> Case 1: "option" means "preference" and "selected" means "winner":
> 
> This is a wrong interpretation as option would probably mean
>  candidate.
> 
> It can be reworded into this:
> ------------------------------------------------------------------
> "The removal of preferences for a loser will not result in the
>  candidate winning."
> ------------------------------------------------------------------
> 
> The rule is an instance/corollary of (P1). The rule is not (P1), so
>  it is weaker than (P1). ((P1) says that deleting preferences at
>  and/or after a preference for some loser never results in that
>  loser turning into a winner).
> 
> Removal of first preferences perhaps means removal of papers.
> 
> [3] ===================================================================
> 
> >"The removal (addition) of an option which has not been selected will
> > not reduce (increase) the chances of any outcome where that option
> > is not chosen"
> 
> Case 2A: "option" means "candidate", and "selected" means "preference".
> Case 2B: "option" means "candidate", and "selected" means "wins".
> 
> Both these cases are just bad. That can shown this way...
> 
> >"The removal (addition) of a candidate which has not been selected will
> > not reduce (increase) the chances of any outcome where that candidate
> > is not chosen"
> 
> [3.1a]
> >"The removal of a candidate which has not been selected will
> > not reduce the chances of any outcome where that candidate
> > is not chosen"
> 
> Once a candidate has been removed, its chances of being chosen
>  ought not be considered
> 
> [3.1b]
> >"The addition of a candidate which has not been selected will
> > not increase the chances of any outcome where that candidate
> > is not chosen"
> 
> Before a candidate has been added, i.e. while it is not being
>  voted upon, it is undefined whether the candidate wins or loses.
> All of [3] can be ignored hereafter.
> 
> 
> [z] ===================================================================
> End of response to Mr David Catchpole <s349436 at student.uq.edu.au>
> 
> _____________________________________________________________
> 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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