[EM] Does the 'Independence of Irrelevant Alternatives

[Richard Moore]

Ken Johnson wrote:

> In your earlier post (Election-methods digest, Vol 1 #576, Message 7)
> you defined INI ("Independence of Non-supporting Information") as
> "If X wins and Y loses, and margin(X,Z) <= margin(Y,Z), then removing
> candidate Z from the election shall not cause Y to win and X to lose."
> This sounds like a reasonable criterion, but as stated above I don't
> think it has relevance to Arrow's IIA criterion because IIA relates to
> how the election result might change when you remove a candidate from
> the vote count, not from the election. (In the former case, the votes
> don't change; in the latter case voters may strategically change their
> voting preferences based on which alternative candidates are running.)

Good catch, because that was not my intention, as shown when I pointed 
out that lone-mark plurality passes INI and FPTP fails it. In LMP the 
ballots are not changed to promote lower-ranked candidates (since 
there's no way to know how the remaining candidates would have been 
ranked). In FPTP (with ranked ballots) the second choice is simply 
promoted to first choice on each ballot that had the removed candidate 
in first position, and by this action FPTP fails.

My intended meaning was something like this: if a candidate is 
disqualified after all ballots had been cast (and there is no revote). 
But the wording I chose could have either that meaning or the meaning 
you inferred, so there was some unintentional ambiguity. However, if 
you allow the ballots to be revised, then the results can change in 
*ANY* method, unless you are using a method that ignores the ballots 
altogether. There must be an assumption that the ballots themselves do 
not change when evaluating whether a method passes or fails a 
criterion, unless the criterion itself spells out the ways they are 
allowed to change (as in monotonicity, for example).

 > A
> more concise statement of INI (if this is what you meant) might be:
> "If X wins and Y loses, and margin(X,Z) <= margin(Y,Z), then removing
> candidate Z from the COUNT shall not cause Y to win and X to lose."

It doesn't appear to be any more concise, but it agrees with my 
intended meaning. I'm not sure how mathematicians would view the 
phrase "removing candidate Z from the count" -- there isn't one single 
number that could be referred to as "the count", and the counts that 
are there are not counts of candidates, so would this be considered 
vague? Perhaps Forest or Joe (if he's still on this list) could 
comment on that.

> In the case of CR and Approval all of Arrow's criteria, including the
> following interpretation of IIA, hold:
> "If X wins and Y loses then removing candidate Z from the count shall
> not cause Y to win and X to lose."
> - Ken

I believe Arrow wrote some of his criteria in such a way that 
non-ordinal methods automatically fail. I don't remember specifically 
which ones or how he defined them. Yes, they could all be rewritten in 
such a way that Approval passes them, but then the resulting set of 
criteria would not be the same set that Arrow's Theorem is concerned with.

  -- Richard