I think of cases like this as an interplay between our choices and God's response, or even, between our choices and their natural consequences. We have free will, and we get to make our choices, but once the choices are made we do not get to choose their consequences. Pharaoh made his choices, and reaped their consequences. This harmonizes with Ezek 18:23, "Have I any pleasure in the death of the wicked, declares the Lord God, and not rather that he should turn from his way and live?", and with Lamentations 3, where Jeremiah says that God does not afflict willingly. If God was actively in the mix, I trust He had good reason to be, as His judgments are perfect.
Paul talks about the balance between God's side and man's side in Romans 1, where he shows God progressively "handing them over" to the next level of consequences the disobedient had opted for. It seems like more of a passive role for God, and you can picture Him wishing at each stage that men would repent, for in the end lies God's wrath, which is serious business indeed.