Is a fixed welfare function that outputs the same answer regardless of the inputs independence of irrelevant alternative?

Question

I'm taking this course to learn game theory and I'm confused about a question in Unit 1.5.

Background. In game theory, independence of irrelevant alternatives (IIA) says the social welfare function $W$ is IIA if the selected ordering between two outcomes depends only on their relevant orderings given by the agents.

The question is

Consider a trivial welfare function in which nature has already determined that the correct welfare ordering is (A,B,C). So W is a function which, on any input of preferences, always returns the ordering (A,B,C). Is W independent of irrelevant alternatives (IIA)?

I think the answer is no because the outcome stays the same even if all agents swap their preferences of A and B.

However, the official answer is yes. The explanation says "$W$ never changes its output, so it trivially satisfies IIA". I don't know how to make sense of this explanation.

I'm aware that this question has been asked in the forum of the course. But no one answers it so far.

Answer 1

Another framing of IIA is to say "To know the ordering of the outcomes of $a$ and $b$, you don't need to know anything about the presence or absence of other alternatives".

So this "predetermined" outcome satisfies that - when working out the ranking of $A$ and $B$, you don't need to know anything about $C$, you just need to know how the agents rank $A$ and $B$. In fact, you don't even need that, but that's kind of besides the point.