After Arya saves Jaqen H'ghar and the other guards he was with, Jaqen promised her he would kill any three persons she named. Why didn't she name Tywin, who would was leading the war against the North, or Cersei and Joffrey, who were holding her sister captive?
In the books we get a clearer look into Arya's head. By the time Jaqen showed up, Arya had a amassed a long list of people she wants dead (her night time prayer in the TV show). Her first kill was an experimental spur of the moment thing (although the identity of the victim is different in the books). The second kill wasn't much better in terms of having deep thought behind it. Realizing this, Arya knew that she had to make the third kill count. Through a mixture of procrastination, not being able to make a decision, and being drunk on her power over life and death (her persona as the Ghost of Harrenhall) time slipped by and Tywin left. Knowing that sending Jaqen after Tywin (or other people not close by) would take a long time to get a result, Arya decided to use her third kill to escape.
In addition to the reasons System Down gave, she's only 10 years old. Granted, she's one of the most "with-it" characters in the story, but she's still just a kid and prone to reacting to what's happening without thinking about the future.
In the book, even though she realizes the first person she picks to die was a bit of a waste and is starting to think about how she should use her next two choces, the second person she picks is still mostly in response to what happened that day.
The first name given was likely just a test to see if he could do it.
The second she was forced to use to protect herself after she was caught with the letter.
I think the third is actually pretty clever.
You are forgetting that she was a child, alone and traumatized surrounded by enemies with just one friend to help her.
Jaqen's help meant a lot to her on an emotional level. She was no longer the mouse that she was forced to be, she could kill anyone who bothered her with a whisper. That made her feel braver.
But as I said, she is a child. It just did not occur to her until it was already too late.
In the Show, she had Jaqen kill:
- The Tickler for torturing others
- Ser Amory Lorch as he was trying to inform Lord Tywin or Arya reading secret papers, which could have had dire consequences for Arya.
Then she used her third name to blackmail Jaqen into helping her escape. So the first one was out of pure anger, the second one was out of fear and the third one was to escape. She just never thought to use one wish to kill someone who mattered.
In the books, she had Jaqen kill:
- Chiswyck for boasting about gang-raping a girl.
- Weese for hitting her.
She uses the third name to blackmail Jaqen into helping her free the captive Northmen who then took over Harrenhal in the name of Robb Stark.
So the first two were out of anger and third one was out of emotions. It did, however, occur to her before Weese was killed that she had been an utter fool and none of the people she named to be killed mattered. She should have killed Tywin Lannister, Kevan Lannister, Ser Addam Marbrand and other important Westerlander commanders instead. She wanted to change her wish, but by then it was already too late, Jaqen had already killed Weese by the time she got there.
A shiver crept up Arya’s spine as she watched them pass under the great iron portcullis of Harrenhal. Suddenly she knew that she had made a terrible mistake. I’m so stupid, she thought. Weese did not matter, no more than Chiswyck had. These were the men who mattered, the ones she ought to have killed. Last night she could have whispered any of them dead, if only she hadn’t been so mad at Weese for hitting her and lying about the capon. Lord Tywin, why didn’t I say Lord Tywin?
Perhaps it was not too late to change her mind. Weese was not killed yet. If she could find Jaqen, tell him...
Hurriedly, Arya ran down the twisting steps, her chores forgotten. She heard the rattle of chains as the portcullis was slowly lowered, its spikes sinking deep into the ground... and then another sound, a shriek of pain and fear.
A dozen people got there before her, though none was coming any too close. Arya squirmed between them. Weese was sprawled across the cobbles, his throat a red ruin, eyes gaping sightlessly up at a bank of grey cloud. His ugly spotted dog stood on his chest, lapping at the blood pulsing from his neck, and every so often ripping a mouthful of flesh out of the dead man’s face.
ACOK - Arya VIII
I didnt understand that either. I have seen the show and now also read the first 4 books. I think that this is just a weak spot of the story. Arya knows, that Tywin Lennister is in Harrenhal, she knows that Robb is fighting against him. Especially a 10 year old should think more black and white and would think, that the problem vanishes with killing the lord.
The theory "she had to try it out" also doesn´t convince me. Why should she? One reason could have been, that she feared that the assassination goes wrong and then Tywin could be warned in future. But that doesn´t make sense, as Jaqen is her only chance. Other reason could have been, that she feared, that once Tywin is dead, something may happen to her sister. This actually I would have believed. But then it wouldnt make sense still, that she didn´t name Gregor Clegane. So.... it´s just a weak spot of the story.