Firstly, in cases where agents are unsure of their preferences, a sorting procedure is introduced. That does not guarantee preferences which conform to transitivity.

So, what does 'rational' mean in this context? It means two things:

If you show me two options (say, A and B), one of three things is true: I prefer A to B, I prefer B to A, or I don't care which one I get. If I prefer A to B, and prefer B to C, then I also prefer A to C.

What about the murky waters of A ≥ B or B ≥ A? The "≥" can imply indifference or welfare judgement based on availability, among other things. So in cases where a sorting procedure forces completeness, transitivity may be violated. This prompts the question of whether preference represents choice or welfare judgement, and what differentiates the two.

But something worth asking is how do we test for transitivity?

The von Neumann-Morgenstern model argues that independence is required. If we're selecting for sets of triples of choice sets containing the combinatorics of ABC, let's say (A, B, C) is selected. That tells us it's preferred to all combinations (A, B, A ...B, C, A). The set (A, B, C) alone can never imply transitivity as it is a set preferred at each choice set, not across choice sets, which is only guaranteed by independence.