The problem is, this anecdote is completely misleading. IIA is about the group's preferences, not just your own.
Here's an easy example that shows the difference. The first is more like the anecdote, and evidently disproves it:
Given that there is no war to deal with, you prefer candidate A to B.
Given that there is a war to deal with, you prefer candidate B (who would stop the war quickly) to A.
Candidate C is a war-candidate; though ranked below A and B no matter what, the important thing is that he is running for election if and only if there is a war to deal with. You know this.
Therefore, your ranking of {A,B} is A > B, but your ranking of {A,B,C} is B > A > C. This "disproves" the pie anecdote. It does not, however, have anything to do with IIA.
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The second example is more like what IIA actually says:
Given that there is no war to deal with, everyone prefers candidate A to B.
Given that there is a war to deal with, everyone prefers candidate B (who would stop the war quickly) to A.
Candidate C is a war-candidate; though everyone ranks him below A and B no matter what, the important thing is that he is running for election if and only if there is a war to deal with. Everyone knows this.
Therefore, everyone's ranking of {A,B} is A > B, but everyone's ranking of {A,B,C} is B > A > C. Everyone votes accordingly.
IIA implies: the SOCIAL ranking of {A,B} should agree with the SOCIAL ranking of {A,B,C}, as long as the individual rankings for {A,B} all agree with the individual rankings of {A,B,C}.
Here, the conditional fails: the individual rankings are completely reversed. This example does not disprove IIA.
> For the second case, you've made "good apple pie" and "bad apple pie" different elements of the choice set.
I don't think so. Arrow's theorem has a set of outcomes as a given. In this restaurant, one of the two, "good apple pie", "bad apple pie" is not actually an outcome, so it does not exist in the ranking.
Here's an easy example that shows the difference. The first is more like the anecdote, and evidently disproves it:
Given that there is no war to deal with, you prefer candidate A to B.
Given that there is a war to deal with, you prefer candidate B (who would stop the war quickly) to A.
Candidate C is a war-candidate; though ranked below A and B no matter what, the important thing is that he is running for election if and only if there is a war to deal with. You know this.
Therefore, your ranking of {A,B} is A > B, but your ranking of {A,B,C} is B > A > C. This "disproves" the pie anecdote. It does not, however, have anything to do with IIA.
---
The second example is more like what IIA actually says:
Given that there is no war to deal with, everyone prefers candidate A to B.
Given that there is a war to deal with, everyone prefers candidate B (who would stop the war quickly) to A.
Candidate C is a war-candidate; though everyone ranks him below A and B no matter what, the important thing is that he is running for election if and only if there is a war to deal with. Everyone knows this.
Therefore, everyone's ranking of {A,B} is A > B, but everyone's ranking of {A,B,C} is B > A > C. Everyone votes accordingly.
IIA implies: the SOCIAL ranking of {A,B} should agree with the SOCIAL ranking of {A,B,C}, as long as the individual rankings for {A,B} all agree with the individual rankings of {A,B,C}.
Here, the conditional fails: the individual rankings are completely reversed. This example does not disprove IIA.