This is a rather complex question. Batman is clearly intelligent. He has been trained in multiple disciplines, from martial arts to forensics. He uses his intellect to solve crimes and protect Gotham. The Joker, though, is a different kind of intelligent. He has an almost insane creativity when it comes to devising his evil plans. He can read people very well and use their weaknesses against them. He doesn't follow the normal rules of society or logic, which in some ways gives him an edge. For example, Batman often has to rely on his moral code and the law, while the Joker has no such constraints. So, it's not really possible to simply say one is more intelligent than the other.
Well, their relationship is extremely complex. The Joker is constantly trying to break Batman's will and sanity. He creates elaborate and cruel schemes. Batman, while detesting the Joker's actions, also has a sort of begrudging respect for his intelligence. He knows that the Joker is a unique and dangerous threat that he has to keep in check. It's like a never - ending battle of good versus evil, where the Joker represents pure chaos and Batman represents order.
Lex Luthor is often an interesting character in such fanfictions. He might be the one pulling the strings behind the scenes, fueling Batman's distrust of Superman. Another could be Alfred. His wisdom and influence on Batman can add depth to the story. He might try to talk Batman out of fighting Superman or help him in a more covert way. And then there's Wonder Woman. She could be caught in the middle, trying to mediate between the two or having her own motives for getting involved in their conflict.
One popular plot could be that Batman discovers Superman has been mind - controlled by some villain. So, in order to save Superman and the world, Batman has to fight him. Another plot might involve a misunderstanding. Maybe some false information leads Batman to believe Superman is a threat to Gotham, and he takes action. There could also be a plot where Batman is pushed to his limits by some external force that makes him see Superman as an obstacle, and he uses his wits and skills to take on the Man of Steel.
At first, Batman would be shocked and a bit confused. But quickly, his detective instincts would kick in. He would view the deaged Joker as a new and unexpected threat. He'd start gathering information about how the deaging occurred, in case it was part of a larger, more sinister plan. Batman would also be very careful not to underestimate the deaged Joker. He might try to isolate the Joker in a secure location to prevent him from causing chaos in Gotham. And he'd be constantly on the lookout for any signs of the Joker's old, devious self emerging from this new, younger form.
It could be depicted as a reluctant partnership. Batman knows that the Joker is a dangerous criminal, but he has no other choice in the face of a greater threat. The Joker might pretend to be more cooperative than he really is, just to mess with Batman. There would be a lot of tense moments between them. For example, when they are planning their next move, Batman would be all serious and strategic, while the Joker would make sarcastic remarks and suggest wild and dangerous plans. And yet, there could also be a strange kind of respect between them. Batman respects the Joker's intelligence, in a way, and the Joker respects Batman's unwavering sense of justice, even if he doesn't agree with it.
The plot could involve a new villain in Gotham who has a technology that can control people's minds. This villain starts to turn the citizens of Gotham against each other. Batman realizes that he needs the Joker's unique brand of chaos to break the pattern of the mind - control. The Joker, on the other hand, sees this as an opportunity to create even more mayhem. They join forces, with Batman trying to direct the Joker's actions to some extent. As they fight the villain, the Joker may come up with some out - of - the - box ideas to disable the mind - control device, while Batman focuses on protecting the innocent. But of course, their alliance is fragile, and at any moment the Joker could turn on Batman.
A story could unfold where they work together to expose the corruption in Gotham's government. The Joker could use his chaos to draw out the corrupt officials, while Batman uses his influence and resources to gather evidence. However, as they get closer to their goal, the Joker might start to go too far with his methods, and Batman would have to decide whether to continue the partnership or stop him.
In the traditional Batman comics, the relationship between Batman and the Joker is not a love story. It's more of a complex adversarial relationship. The Joker is a villain who constantly tries to cause chaos and harm, and Batman is dedicated to stopping him.
The relationship can be explored by showing how they are two sides of the same coin. For example, in a fan fic, Batman could be faced with a situation where he starts to understand the Joker's chaotic view of the world a bit more, maybe due to some extreme circumstances. This would make him question his own motives and methods.