IIA, spoiling & clone independence (was Re: [EM] New Condorcet/RP variant)

[Steve Eppley]

Hi,

Eric G wrote:
> On the current wikipedia page for the Spoiler Effect, it says:
> 
>   A voting system which satisfies the independence of irrelevant
>   alternatives criterion is immune to the spoiler effect,
> 
> Now, considering people use the term Spoiler Effect in
> the context of Independence of Clones Criterion, 

Really?  I consider some but not all forms of clone 
dependence to be cases of spoiling, but I don't recall 
seeing it used as a synonym for spoiling.

Here's a very strong "no spoiling" criterion:

   If x is an alternative that loses, then deleting x 
   from every ballot must not change the winner.

Here's a weaker "no spoiling" criterion, which perhaps 
corresponds with what most of the public and political 
pundits think when they think about spoiling:

   If x is an alternative that loses, then deleting x 
   from every ballot must not change the winner to
   an alternative preferred over the winner by most 
   of the voters who prefer x over the winner.

Clone alternatives can be spoilers.  For example, suppose 
the following ballots are tallied by plurality rule:

   25%   35%   40%
    X'    X     Y
    X     X'    X
    Y     Y     X'

Deleting X' (or X, take your pick) from every ballot 
changes the winner from Y to X (or to X', respectively).
So, X' is a spoiler given either definition of the 
"no spoiling" criterion.  X' is also a member of 
the {X,X'} set of clones.

Here's an interesting example:  Suppose the following 
ballots are tallied by Borda:

   55%   35%   10%
    X     Y     Y'
    Y     Y'    Y
    Y'    X     X

Borda elects Y.  Here, Y' is a clone of Y.  If Y' is 
deleted from every ballot, the winner changes to X.
So, Y' is a spoiler given the strong no spoiling 
criterion.  But Y' is not a spoiler given the weak 
no spoiling criterion, since most voters who prefer
Y' over Y do _not_ also prefer X over Y.

Spoilers might not be clones.  Tweak the first example 
just a little:

   25%   34%    1%   40%
    X'    X     X     Y
    X     X'    Y     X
    Y     Y     X'    X'

X' is still a spoiler--deleting X' changes the winner 
from Y to X given plurality rule--but {X,X'} is 
no longer a set of clones.

There are multiple versions of IIA in the literature, 
some strong and some weak.  I think it's reasonable 
to write the strong form of IIA as:

    If x is an alternative that loses, then deleting x 
    from every ballot must not change the winner.

Yes, you clever readers will note that that's identical to 
the strong "no spoiling" criterion above.  I basically 
agree with Wikipedia.  And I think it should be obvious 
to the casual observer that satisfaction of this criterion 
implies independence of clones.  From the beginning, I've 
always viewed independence of clone alternatives as weaker 
than (strong) IIA.

* * *

When IIA is written weak, there are companion criteria 
like "ordinality" and "rationality" (for which I prefer 
the term "choice consistency" since it's less loaded) 
that together function like strong IIA.  See below.  
For an example, look in my webpages at the statement and 
proof of Arrow's theorem for choice functions (not for 
social ordering functions, which is probably the more usual 
way to express Arrow's theorem).  I learned this form of 
Arrow's theorem from Matthias Hild when he was a visiting 
professor of philosophy at Caltech several years ago. 
(Hild, however, used the name "rationality" where I use 
the name "choice consistency.")

   Weak IIA for choice functions:  
   The choice function must neglect all information 
   about non-nominated alternatives.  

   Ordinality:  The choice function must neglect all
   "intensity" information.  In other words, only 
   "ordinal" information may affect the choice.  

   Choice consistency ("rationality"): 
   For all pairs of alternatives, say x and y, 
   if the votes are such that x but not y would be 
   chosen from some set of nominees that includes both, 
   then y must not be chosen from any set of nominees 
   that includes both.

To understand these three criteria, you need to also 
understand that choice functions are defined to take 
TWO parameters: the collection of votes and a set of 
"nominated" alternatives. (Note that the votes may 
also contain information about alternatives that 
aren't nominated.)  The second parameter, the set 
of nominees, makes it trivial for choice functions 
to satisfy Weak IIA.

I like, as Hild did, that decomposing Strong IIA into 
Weak IIA, ordinality & choice consistency separates 
out ordinality for scrutiny, and that the notation 
makes it easy to vary the set of nominees.  Focusing 
on social choice rather than social ordering also
seems like a good idea, since it's more general.

I recall an amusing argument between Don Saari and Richard 
McKelvey, a few years before McKelvey died.  The reason 
they were arguing was that they had different definitions 
of IIA in mind but didn't realize it for a half hour that 
entertained the bystanders.  :-)  What definition of IIA is 
used in Wikipedia, and what's their definition of spoiling?

--Steve