An amusing anecdote which illustrates why independence of irrelevant alternatives is desirable:
After finishing dinner, Sidney Morgenbesser decides to order dessert.
The waitress tells him he has two choices: apple pie and blueberry pie.
Sidney orders the apple pie.
After a few minutes the waitress returns and says that they also have cherry pie at which point Morgenbesser says "In that case I'll have the blueberry pie."
I hate to be that guy but there are plenty of reasons why this might be logical behavior. (I say "that guy" because I'm not prepared to argue for their relevance to voting.)
The most trivial example is that he simply changed his mind—but obviously this has nothing to do with IIA.
Another reason is that he might be dining with someone (who is, perhaps, away from the table). Say that he knows his partner's preferences are C>B>A. Say his own preferences are B>A>C, but that he also has a preference for being able to sample two distinct choices. Initially, he believes that his partner will order B, so he orders A to maximize his range of desserts. Upon learning that the "irrelevant" choice C is available, he will have to order his top choice, B, himself.
The same scenario could play out with himself as the second diner, in the sense that he knows he will have two opportunities to visit this restaurant (and that their dessert choices will remain constant). If his preferences are C>B>A, and for whatever reason he decides to order his second-most-preferred dessert first and his most-preferred dessert on the next visit, then his choice is logical.
Perhaps he's of the opinion that a kitchen which prepares cherry pie cannot prepare an adequate apple pie.
The point is, alternatives are rarely "irrelevant". In fact, including the word "irrelevant" in IIA is begging the question: we are tasked with determining whether dominated alternatives (for example) really are irrelevant, and our answer might be "no".
No, you're combining a few different things that aren't related to IIA. For your first dining example, you've changed the relevant set of options to (A,A), (A,B), (A,C), (B,A), etc... where the first element is the diner's desert and the second is the partner's desert. IIA means that if (B,A) ≿ (A,B) in the original choice set, then it continues to hold when we add (Z,Z) as an option.
For the second case, you've made "good apple pie" and "bad apple pie" different elements of the choice set. And again, IIA implies that if "good apple pie" ≿ "cherry pie" that continues to hold when "bad apple pie" is an option. (And as an important aside, all cherry pie is of equal quality in this hypothetical.) Also note that it has nothing to do with the relative likelihood of different choices being delivered, so choosing to order something because you think another order is likely to be messed up is perfectly consistent with the IIA.
Arrow's impossibility theorem is a mathematical result, and IIA is a property of a mathematical representation of preferences. They have practical implications, but certainly don't imply that people in real life make transparently consistent choices.
"The most trivial example is that he simply changed his mind—but obviously this has nothing to do with IIA."
Delving further down this tangent... If that's what happened, he would be unlikely to have said "In that case, ...". That usually means "I am incorporating this new information, and it has changed my decision." More likely he would have said, "On second thought..." or "Actually..." or "Y'know..."
Not, of course, that this has much relevance outside of considering English pragmatics.
When you say it like that, IIA does sound pretty obvious. But if you change the terms a little bit, you can see why the IIA doesn't match up with how people actually vote:
Say there's an election between a moderate democrat "blueberry pie" and a third party liberal "apple pie". As a liberal, Sidney would rather vote for the third party ("Sidney orders the apple pie"). However, if you introduce a republican candidate "cherry pie", Sidney will probably vote for the democrat (blueberry pie) instead of the third party candidate, because he'd be worried about his vote costing the more moderate candidate the election.
IIA means that you won't vote differently than your preferences—but people do that all the time. And sure, a voting system where that wasn't necessary would be nice, but losing that condition isn't as nonsensical as it seems at first.
Arrow's impossibility theorem, and the IIA criterion, is about preferences not about uncertain actions. In particular, IIA doesn't mean that you'll vote differently than your preferences in some sort of election, it means that your preferences themselves don't change when you introduce other irrelevant options. In your example, it wouldn't be about how Sidney would vote in an election given the different menu of candidates, it's about who Sidney would prefer win the election.
(with the caveat that it has been a long time since I've thought about these results.)
I recall that Bordes and Tideman were a good source on this issue; I believe the (relevant) paper is "Independence of Irrelevant Alternatives in the Theory of Voting." -- The upshot is that the condition known as IIA (I think it is sometimes known as Sen's condition-alpha, and the condition that many people harp on, including Michael Dummett) is in fact a stronger condition than is needed for the result; roughly, that instead of needed a condition about consistency among selections over possible ballots (like: if y were selected in ballot [xyz], y should be selected among yz on ballot [yz]..) we simply need a requirement that the selection only involves information on the ballots (and that the relative rankings of candidates not on the ballot are irrelevant to the selection).
Ah, I think your phrasing was confusing. "IIA" is considering an attribute of the ballots, but including IIA is stronger than necessary for the theorem to hold - it can be replaced with the much weaker "nothing but the ballots".
While amusing, the anecdote is off in a very important way: it is not a single person ordering dessert, but an entire party of people who represent different conflicting majorities. In such a real situation you can get whatever "absurd" result you want by just asking questions in the right order.
Ask your dinner party "Do we prefer Apple or Blueberry?", and a majority might reasonably answer Apple. Ask them "Blueberry or Cherry?", and a slightly different majority might reasonably answer Blueberry. Ask them "Cherry or Apple?" and a different majority would answer Cherry. You would get this situation, for instance, with the following voters:
2x A > B > C
2x B > C > A
1x C > A > B
A beats B by 3:2, B beats C by 4:1, and C beats A by 3:2.
This is a rock paper scissors situation -- or a "condorcet cycle".
Now suppose instead of simple Sidney there were actually these 5 people ordering dessert. Is it really that unreasonable that the group as a whole might pick blueberry once cherry is on the table even though a majority prefers apple to blueberry?
That explains why something like IIA is a reasonable intuition about how individual preferences should behave, not why IIA is a reasonable criterion for how systems of aggregating preferences should behave. (Unless you introduce the premise that "a social preference aggregation system should make society operate in a way that would match intuitions of a single actor".)
Morgenbesser is always great, but a defense against criticisms of IIA is much simpler to mount in the context of voting: it simply amounts to the requirement that our voting procedure only takes into account information from the ballots.
The anecdote illustrates why it's desirable: Morgenbesser's change of mind doesn't make any sense, and the reason why it doesn't make any sense is that it violates IIA: whether they have cherry pie shouldn't make any difference to his preference between apple pie and blueberry pie.
(Except that it might -- e.g., imagine that making cherry pie is incredibly difficult and most kitchens can't manage it, and that the special skills and equipment required are also useful for making really good blueberry pies. Then knowing that cherry pie is on offer could actually be evidence that the blueberry pie will be good. This is maybe just a tiny bit similar to, e.g., changing your vote from A to B when you learn that C is standing, because C is a terrible candidate but might win, and in scenarios where C is close to winning B is C's main rival.)
> This is maybe just a tiny bit similar to, e.g., changing your vote from A to B when you learn that C is standing, because C is a terrible candidate but might win, and in scenarios where C is close to winning B is C's main rival.
That's just strategic voting, not an actual change in your preferences. How you go about voting strategically depends on the voting system in use. In your hypothetical scenario you make the unstated assumption that the voting system in use would not allow for you to express a preference for A over B without hurting B's ability to beat C. The inability of that voting system to fully capture your preferences is forcing you to vote misleadingly based on your knowledge of how others will probably vote.
Because if that anecdote is true, Morgenbesser is clearly insane. Drill down on your intuition about why he's insane and you arrive at "he's violating IIA".
After finishing dinner, Sidney Morgenbesser decides to order dessert. The waitress tells him he has two choices: apple pie and blueberry pie. Sidney orders the apple pie. After a few minutes the waitress returns and says that they also have cherry pie at which point Morgenbesser says "In that case I'll have the blueberry pie."