Even if they are ignoring you because they never want to speak to you ever again, you ignoring them may still get their attention, and if you have rattled them enough, it could even escalate into a row because now issues of one-upmanship could come into play.
If the reason that they are ignoring you is because they like you (reverse psychology and all that) then you already have this person’s attention, so it really doesn’t matter how you react because you are going to get noticed. On the other hand, they may be playing hard to get, so if you ignore them, you may get attention but it could backfire because you are not playing the game according to their rules (even though you have no idea what those rules may be).
The whole thing about this scenario is that it is playing games. Nobody knows what one person is thinking, or what the other person wants. It would appear to be some obscure mating ritual, that you often see birds and other animals performing, though in these instances, the birds and the animals know exactly what the outcome will be. In this case, you have no idea and so it has the potential to go horribly wrong. An alternative would be just to speak to the person involved, and see find out what’s what. Unless you like playing charades, of course.