[EM] Yet another weaker IIA criterion

[Kristofer Munsterhjelm]

On 01/08/2018 10:49 PM, Greg Dennis wrote:
> To anyone's knowledge, has the following weaker IIA variant ever been
> named/defined before?
>
> If W is elected, then if a candidate is added that finishes behind W
> (lower in the social ordering than W), the winner is still W.
>
> The idea is a new candidate can't "drag the winner down," so to speak,
> but can overtake them and cause neither to win. If it hasn't already
> been named, I'm tempted to call this "Independence of Weaker
> Alternatives" but open to other suggestions.

I'm not aware of any previous mention of this variant of IIA. I'm also 
unaware of any method that passes it -- short of perhaps 
Approval/Range/MJ under the independent evaluation assumptions that are 
required to have them pass ordinary IIA.

At first one would think that it would be met by any method that works 
by sequentially eliminating losers, and by any method that's equivalent 
to its loser-elimination modification. The tempting proof is something 
along the lines of "if W is the original winner and is ranked ahead of X 
when X is admitted, then X must be eliminated before W, thus X can't 
affect whether or not W wins".

However, that is false: X can affect the order of elimination, either 
protecting someone who would lose right away, or exposing someone who 
wouldn't were X not present. See e.g. this IRV example:

3: A>B>C
3: B>C>A
2: C>A>B

First C is eliminated, then B is eliminated, so the outcome is A>B>C.

Now add candidate D to get:

3:A>B>C>D
1:B>C>A>D
2:C>A>B>D
2:D>B>C>A

First B is eliminated (because D is hiding some B first preferences), 
then D is eliminated, then A is eliminated, and C wins. So the outcome 
is C>A>D>B.

D, who was added, finishes behind A (the original winner), yet changed 
the winner from A to C; so IRV fails this IIA variant. And because IRV 
is a loser elimination method, that means that the class of candidate 
elimination methods can't all pass this IIA variant.

For an elimination method to pass the variant, it would suffice to have 
an additional property that adding some candidate X can't affect the 
elimination order of candidates who are eliminated before X. But that 
seems to be a very hard criterion to satisfy.

For the same reason, methods that pass LIIA may not necessarily pass 
this variant. If such a method also meets the property that adding some 
candidate X can't alter the social order after X, then it also passes 
variant IIA if it passes LIIA. But I suspect that the "can't alter 
social order after X" property plus LIIA is too tall an order for a 
ranked method, because of the Condorcet paradox.