It would seem on surface that 'J' would win a fight because J's have it together and can be 'hardcore' about their habits, their plans, and basically their actions and want to implement order and direct P's freeform energy.
HOWEVER, I think 'P' wins because as some people have alluded to, P's drive y'all CRAZY. We will drive you nuts into an early grave.
And I say this with utmost pride and respect when I say my dad is possibly the most anal person you will ever meet short of being OCD. He's not obsessive compulsive or has a PDD that I know of, he's just incredibly on top of things. I think he's INTJ.
As much as I found his insistence on finishing stuff and following rules and being logical borerline harassment growing up, I know I'm responsible for many more of his grey hairs! AHAHAHAHA. Between me and my INTP brother and my XXXP mom, I swear to god we all aged the poor man 15 years. I told my brother this too, but he's such a bastard (in this particular case, of the INTP variety) he doesn't care -- rather I don't think he quite comprehends.
So yeah. P for the win. In the most indirect, frustrating way possible.
Unless I work for you, your J'ness has no hold on me and even then... I think hardcore 'P's spend a lifetime perfecting the psychic/psychological/and interpersonal art of indirectly escaping your clutches. Or we're just so damn off in our own little grooves we don't even realize we're supposed to be listening to you...oh wait, sorry what was that? Were you talking to me?
I know, it must be frustrating as hell.