IIA, spoiling & clone independence (was Re: [EM] New Condorcet/RP variant)
Steve Eppley
seppley at alumni.caltech.edu
Thu Nov 11 10:42:59 PST 2004
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
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