The reason you’re seeing these multiple definitions of IIA is because with the appropriate setup, they’re equivalent. To be specific:
Let’s say we have a finite set of candidates C and a set V of voters. (The set of voters will remain fixed throughout.) Then we can define a (type 1) voting system for C to be a function taking (maps from V to linear orders on C) and returning a linear order on C. With this definition, the natural notion of IIA is the one you describe above.
(Note: Obviously, “type 1”, “type 2″, “type 3” are all ad-hoc terminology.)
Now we could define a type 2 voting system for a (potentially infinite) set of candidates C to be a family of type 1 voting systems, one for each finite subset of C. Then you have a natural notion of IIA for type 2 voting systems, namely, restricting from one subset of C to a smaller subset shouldn’t change the induced order on this smaller subset.
Finally we can consider type 3 voting systems for C, which we will define to be a family of voting systems, one for each finite subset of C, but these all just return single winners, not whole linear orders. The natural notion of IIA for this sort of voting system is the one Khoth describes.
It’s then trivial that every type 2 voting system induces a type 1 and a type 3 voting system, and if the type 2 system satisfies its IIA then so do the induced systems. What is less obvious but still not too hard is that if a type 1 or type 3 system satisfies its appropriate notion of IIA, then in fact it must come from an IIA-satisfying type 2 system. (Going from type 1 to type 2 is obvious; to go from type 3 to one of the other types, you run the election, put the winner in 1st, remove him, run again, put the new winner in 2nd, etc.)
So on a finite set of candidates all 3 notions are equivalent and on infinite set of candidates the 2 notions that make sense are equivalent.
I don’t have a reference on hand, which is why I don’t know if there’s standard terminology for this; this is just something I worked out some time ago when trying to figure out why I’d seen IIA defined differently in different places. :)
The reason you’re seeing these multiple definitions of IIA is because with the appropriate setup, they’re equivalent. To be specific:
Let’s say we have a finite set of candidates C and a set V of voters. (The set of voters will remain fixed throughout.) Then we can define a (type 1) voting system for C to be a function taking (maps from V to linear orders on C) and returning a linear order on C. With this definition, the natural notion of IIA is the one you describe above.
(Note: Obviously, “type 1”, “type 2″, “type 3” are all ad-hoc terminology.)
Now we could define a type 2 voting system for a (potentially infinite) set of candidates C to be a family of type 1 voting systems, one for each finite subset of C. Then you have a natural notion of IIA for type 2 voting systems, namely, restricting from one subset of C to a smaller subset shouldn’t change the induced order on this smaller subset.
Finally we can consider type 3 voting systems for C, which we will define to be a family of voting systems, one for each finite subset of C, but these all just return single winners, not whole linear orders. The natural notion of IIA for this sort of voting system is the one Khoth describes.
It’s then trivial that every type 2 voting system induces a type 1 and a type 3 voting system, and if the type 2 system satisfies its IIA then so do the induced systems. What is less obvious but still not too hard is that if a type 1 or type 3 system satisfies its appropriate notion of IIA, then in fact it must come from an IIA-satisfying type 2 system. (Going from type 1 to type 2 is obvious; to go from type 3 to one of the other types, you run the election, put the winner in 1st, remove him, run again, put the new winner in 2nd, etc.)
So on a finite set of candidates all 3 notions are equivalent and on infinite set of candidates the 2 notions that make sense are equivalent.
I don’t have a reference on hand, which is why I don’t know if there’s standard terminology for this; this is just something I worked out some time ago when trying to figure out why I’d seen IIA defined differently in different places. :)