[EM] IIA Theory

[Craig Carey]

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.

[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.

In [A], a candidate can't really be added unless preferences are
 added. If preferences are added, then meaning is bad and the
 definition ought be under title [B]. So [[A] can be rewritten as

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

[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).

(For All c, x, y, not (x=y or y=c or c=x)
  (For All V, U, U = alt(V,c))
    (not (xV.yV or -xV.-yV) . implies not (xV.-yV.-xU.yU))

Call the last term z. 
Use: (a impl b)=(-a or b), (a or -b)=(ba or not b)

z = (xV.yV or -xV.-yV) or not (xV.-yV.-xU.yU)
= not (xV.-yV.-xU.yU)

So the rewrite of case [B] of the "preference" Catchpole-IIA is:

(For All c, x, y, not (x=y or y=c or c=x)
  (For All V, U, U = alt(V,c))
    not (x wins V and y loses V and x loses U and y wins U)

What does the word "remove" mean?. For (P1) removal of
 preferences permits the deletion of papers but only the
 preference is the first in a particular paper.

It is not clear what a Catchpole-IIA "remove" function is.

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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