The Joker is a tough trickster, but I don’t think that he can really compete with Pegasus. Pegasus has a lot of monsters at his disposal including the Toon World monsters. It is very difficult to hit them with any kind of ability due to the way that they can shift their bodies and evade attacks. Sheer numbers can also help out quite a lot in a fight and Joker’s trick flowers and electronic gadgets can only get him so far. Pegasus can also shoot eye lasers if necessary. Pegasus wins.
DReager1.com 
Excellent Work, DrEager! I think even if his toon monsters didn’t get to The Joker, Pegasus would most likely collapse trying to enter the Joker’s mind for a very long period of time, as demonstrated by Martian Manhunter’s straining to turn & keep the Joker sane long enough to get a very powerful artifact away (Grant Morrison’s Justice League, Volume 3). Also, another battle request: Karolina Dean (Marvel Comics) vs Sinestro, both in the interest of giving the latter character more battles and giving Karolina a Debut on the Blog.
Absolutely, Pegasus won’t be winning a battle of minds with the Joker in any shape or form. Trying to comprehend Joker’s maze of a mind is truly an impossible feat. If MM couldn’t do it, Pegasus certainly can’t. Request accepted, I’m not familiar with Karolina Dean so it’ll be cool to learn about her.