James Gordon Jr. vs Joker

 Joker realizes James Gordon Jr. only used him as a tool. Joker doesn't think that's too funny. Meanwhile, James Gordon Jr. is trying to make a name for himself amongst Batman's enemies, as they could possibly assist him later. What better way to get attention than to take out the man who laughs? They're both in Arkham Asylum and they both want the other dead. 
 
Joker may know the layout better, but James Gordon Jr. is brilliant, and he has been there before. And a doctor/security officer could be manipulated by Gordon a heck of a lot easier than the Joker - everyone knows who he is, after all. While Joker might be able to get some of the inmates on his side, so could Gordon.
 
Characters are as they are at the end of Scott Snyder's run on Detective Comics (#881). Both are ruthless, and will do whatever they feel necessary to attain their goal. 

James Gordon Jr. hasn't got a prayer. BTW Joker is actually the better of the two at manipulating people to do what he wants.

I wouldn't count him out. Whenever Batman bests the Joker when the Joker is at his best, he does it through brains. There is nothing super special that Batman could know about the Joker that James Gordon Jr. couldn't immediately deduce, given how smart he is. James Gordon Jr. is also a master manipulator. He managed to get away with killing who knows how many people without being caught. I doubt you can do that without knowing how to lie to people and get them to believe you or leave you alone, I.E. manipulation.