I didn't say "Sansa understood what Arya's endgame was." I said Arya never ended an interaction with a serious threat; she always ended their interactions by trying to steer Sansa toward moving on Littlefinger - handing her the dagger being the most obvious of these hints.
If she was trying to steer Sansa with hints (which makes no sense by the way since she can just tell Sansa what she knows), then why the hell would she threaten her life mid conversation? That isn't a thing you do when you want to help someone learn something because all they'll be thinking about is the threat to their life.
Why does Arya have the knife back when she is called to Littlefinger's execution, if not because Sansa recognized the meaning of Arya's last gesture toward her?
Because Sansa gave it back to her after she figured out what Littlefinger was really trying to do. Arya gave it away to throw Sansa off balance, because again, if she wanted to tell Sansa something she would have just told her. The whole point of that scene was to intimidate Sansa, not help her. Sansa appealed to Arya, not the other way around. That's why Arya was the contrite one in their final scene this season.
1) They're the same game - someone puts on a false face to kill their enemies. Littlefinger is playing it to turn Sansa against Arya; Arya is playing it to get Sansa to finally do something about Littlefinger.
This isn't a response at all to what I'm saying. Sansa referenced what Littlefinger told her, not what Arya told her. If Sansa already knew what LF was up to by the time she had that convo with him in the finale, she wouldn't have talked to him, she would have just killed him. It was only after that talk that she figured things out.
2) The fact that Arya gives Sansa the knife at the end of that scene is a really obvious signal that she's not actually a threat to Sansa, and therefore that her entire speech about the Game of Faces does not mean what it appears on the surface to mean. She is talking about the metaphorical Game of Faces, the one where you use duplicity and cunning to defeat your enemies, not the literal one.
You know what's a more obvious signal than waving a knife in someone's face and threatening their life? Telling them what you know. Arya's actions make zero sense if you think she wasn't being played by LF, but make way more sense if you go by the obvious.
3) If Arya was going to do anything to Sansa, she would have just said "I'm going to kill you" like she does with everyone else.
She didn't want to kill Sansa, she wanted to scare her and keep her in line. She wanted to protect Jon from her because she saw her as a conniving snake at that point because she was manipulated by LF.
What? The game I mentioned Sansa catching up on with is the conversation with LF. So I agree with you there. I think Sansa caught on later than Arya, but I don't buy for a second that Arya was played by LF. She stared him down the moment she saw him, knew where he was from, is probably heard of him enough to know his reputation. Her being suspicious of LF and tailing him enough to come to a conclusion that he's a threat is not a stretch at all, and it doesn't mean she's some super intelligent master of her craft.
We know she knew LF was a threat, but she underestimated him and thought he was working with Sansa to undermine Jon. Again, if she knew he was trying to pit the sisters against eachother, she would have told Sansa. Her actions are consistent with someone that was fooled, not with someone that was in on it from the beginning. This isn't hard.
And with Bran there to confirm any suspicion she would have of LF, that guarantees it. I wouldn't be surprised if she went to Bran pretty early on to get the scoop on LF. Seems obvious, really.
It isn't obvious really because instead of doing something about LF she directly played into his hands and antagonized Sansa. Sansa was the one who approached Bran for the scoop, that's what should be obvious to you.