[quote=@AndyC]
<Snipped quote by Enarr>

If we're going with sheer raw power, Bizarro would reduce Mr. J to a fine red mist before Joker even knew he was there.

But assuming we account for DC's grand commandment of "Thou Shalt Not Ever Look Cooler Than A Gotham City Character," Joker would likely pull some plot-armor nonsense and mind-control him or something.
[/quote]

Don't even think it'd be mind control.  Just straight up manipulation. 

Which, technically speaking, I wouldn't consider a "win" so much as a no contest.