[EM] IIA Theory

[David Catchpole]

(responses further down)

On Tue, 5 Oct 1999, Craig Carey wrote:

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

Again, IIA is a condition on static voter preferences and on veriable
candidates.

> [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".

2B is the bone of my gist. I apologise that perhaps selected, chosen, etc.
became confusing terms but I like to think the thing is part of social
choices other than voting where "wins" doesn't sound right. Because I
continuously think in preferences rather than some concept of marking a
single name on a list when I say "chosen" or "selected" I more or less
invariably mean "won." Sorry. English is a terrible language with too many
multiple entendres.

> 
> 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

That's not relevant to the condition (sorry- I think I do understand the
rationalisation for your argument to some extent, but I disagree). I
think you meant that you understand that because the candidate is no
longer involved, it is of no relevance what
options do and what options do not include that candidate.
However, consider a probabilistic election between say A, B and C. Say the
election system we have chosen gives A 1/3 chance of winning, B 1/3 chance
of winning, C 1/3 chance of winning. Were we to remove C, IIA for a
probabilistic regime says you can't have A 100% chance of winning, B 0%
chance of winning, because that's unfair. In such a case, if B wants to
have any chance of winning he has to invite his buddy C into the race, and
say if A and C are more or less of the same side of a spectrum, C has
managed to split the vote for that side, reducing its chances of winning
from 100% in the second case to 2/3 in the first. 

> 
> [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.

Aargh! Electoral nihilism (sorry again, Craig, I'm really trying not to
be a pompous prick, but not being one comes hard for me)! What one assumes
is that wired in every elector's brain is the way they would vote for any
combination of any candidates, and that any set of preferences which comes 
from such a "database" such a set has the same transitive
relations (e.g., to use Demorep's example, Demorep would always
vote George Washington higher than Joseph Stalin no matter what
other candidates he had on the ballot form). The implication of this is
that silly people should not say "FPP is independent of irrelevant
alternatives because voters don't express preferences beyond their first."
Our consideration is of the voters themselves and the static opinions
which reflect their response in a voting game, not of the response itself.
For instance, in the prisoner's dillemma, we don't assume Prisoners don't
want to go without prison time, do we?

"Addition" _is_ problematic because intuitively every system has the
potential for some candidate being added to break the rule. That's why I
like to play around specifically with independence of the _removal_ of
irrelevant alternatives, because it sets up a logical system in which
larger schema have smaller pups, so the system is still meaningful.