In Scott Christensen's book "What about free will?" on page 119 is "Pharaoh is held responsible for his actions". The reason given for God attributing culpability is "You are exalting yourself...".
If Pharaoh was made by God in such a way that he could not do anything but exalt himself then Pharaoh would not be ultimately responsible for hardening his heart. If Pharaoh was the instrument in God's hand of Pharaoh's hardening then he would be instrumentally responsible.
By not separating out instrumental from ultimate responsibility but by sliding seamlessly from instrumental to ultimate, Pharaoh becomes ultimately responsible for his actions. I think A.W. Pink's book is another in this compatibilist way of using "responsible" without differentiating between instrumental and ultimate. So, do Bible believing compatibilists use the word "responsible" with one meaning or two?