*Post by Greg Dennis*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.

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