Why didn't Tyrion confront Littlefinger for telling Catelyn that the knife belongs to him?

Question

In season 1 of Game of Thrones, Littlefinger tells Catelyn and Ned that the knife used for trying to murder Bran belongs to Tyrion ("the Imp"). He told them that he lost it to him in a wager. On that basis she accused Tyrion of attempting to murder her son, abducted him and took him to the Eyrie. He could have lost his life there, if not for Bronn.

So why didn't Tyrion confront Littlefinger for telling Catelyn that the knife belongs to him? For me, this will be the first instinct for anyone once they reach King's Landing. He didn't even need to wait to reach King's Landing, he could have sent word to his brother Jaime.

Is he not aware who told Lady Stark? Or does he have some other reason?

Answer 1

According to the books wiki he didn't know:

Unknown to Tyrion, Catelyn was told by Petyr Baelish that the dagger used in Bran’s failed assassination attempt belonged to Tyrion.

I don't think it was different in the show (for sure she didn't mention it in the capture scene) and I don't remember him finding out later. In the TV show this plot dies in season 1, while in the books we find out more about it, as far as finding out who sent the assassin.

It is unknown why the TV series never reveals that it was Joffrey who sent the assassin; Joffrey receives the Valyrian steel sword Widow's Wail as a wedding gift, as he did in the books, but no mention is made of the assassination attempt on Bran back in Season 1. It is possible that the reason is the same for not revealing the truth about Tysha in "The Children": the producers grew afraid that casual viewers wouldn't remember that this happened in early Season 1 - even though this was one of the escalating events which sparked the entire Stark-Lannister conflict.

Interestingly, in the books Tyrion did find out about Littlefinger's deed:

During a casual conversation at King's Landing, Tyrion made a comment about Littlefinger's elegant suit and handsome knife. Littlefinger drew his knife, glanced at it casually, as if he had never seen it before and said “Valyrian steel, and a dragonbone hilt. A trifle plain, though. It’s yours, if you would like it”. Noticing the mischief in his eyes, Tyrion realized that Littlefinger was teasing him insolently about the assassin's dagger, without saying anything explicit.

Tyrion suffered a lot as a result of Littlefinger's lie: he was humiliated, kidnapped, beaten, locked in a sky cell and nearly killed three times (on the way to the Eyrie, at the Eyrie and on the way back) - and yet he never attempted to settle score with Littlefinger, either openly or secretly (as he did with a singer who tried to blackmail him), even after being taunted about it. It is unclear why Tyrion let Littlefinger get away with that, in contrast to the unofficial motto of his house.

Answer 2

He wants to:

But do I dare touch him? Tyrion wondered. Even if he is a traitor? He was not at all certain he could, least of all now, while the war raged.

Granted, that's from the books, but the issue is the same. The problem is that Littlefinger had just performed a great service to King Joffrey (and Cersei). He revealed and helped destroy Ned's "treasonous" attempt to steal Joffrey's throne for Stannis. Do you really think Joffrey is going to let Tyrion do anything to Littlefinger on the evidence given to him by a traitor's wife? Tyrion does trying setting a trap, but it ends up being Maester Pycelle who falls for it, which disappoints Tyrion in the book. Attempts to do it more subversively could backfire and put Tyrion in danger, and even if they work who knows if Littlefinger has arranged for something if he is mysteriously killed? Littlefinger has also proven useful both as a master of coin and as an emissary during the war. For the time he's useful and Tyrion is patient. Shortly after the attempted trap Littlefinger leaves the capital anyways, and after that Tyrion has other issues to deal with. But you can bet he hasn't forgotten.