The newest entry into the DC Versus series pits two of the richest and smartest people in the DC Universe against each other. Batman vs Lex Luthor. This is one of the battles that I've thought about many times, because as much as Luthor is put as the arch-nemesis for Superman, he actually makes a better Batman villain. And from a psychological angle, the one type of villain Batman doesn't have. The closest to Lex is Tommy Elliot / Hush.
Lex is often time written completely wrong and this video carries forth that error. He is not evil nor is he a sociopath. Lex Luthor suffers from Narcissistic Personality Disorder. He must be the best, he must be the one to save the day, he has to be loved by everyone. If Superman had never come to Metropolis, there is a better than average chance the Luthor would be considered a powerful and benevolent figure that helps those that need it. But, on the other hand, if Lex had set up shop in Gotham he would be obsessed with taking down The Batman if the citizens worshiped him. Lex has all the money, power and influence he needs. He doesn't hurt people because he enjoys it, he does it to show his superiority. It explains why he is willing to put on a super suit and try to fight Superman one-on-one. The smarter thing would be to put a better fighter in the suit and if he dies, it doesn't matter.
The fight between Batman and Lex Luthor would be a battle of wits and influence. Which sounds like a fun battle to watch.