system
Given the following post, respond as if you're commenting on Reddit:
human
[Game of Thrones] Why Jaqen H'ghar killed 3 men in Harrenhall? Did he really believe that God of Death should have his 3 deaths back, or did he just want to lure Arya into his order?
gpt
Both. It was part gratitude for saving his life and the life of his companions(so they could carry out thier mission whatever that was, in the books he's after a book from the citadel),  and part following the precepts of the many faced god who is particular about how and when some people should die. 

It could have also been a test, to see what sort of person Arya was and if she could be helpful to the order.  When Arya names Jaquen in order to get more than three deaths she passed the test as it demonstrated she is clever and willing to kill.

Edit: On more research, a faceless man who matched the new face jaquen took when he left arya bribed a novice at the citadel  (who stole an archmaester's key which can open any lock) and killed the apprentice and took his place. we never seem more beyond that.   Maybe show Jaquen had a simular mission.