Tyrion is not a secret Targaryen

[Annara Snow]

None of Joanna Lannister's children are Aerys'. They're all Tywin's. This is my 2 cents on the subject, copy-pasted (more or less) from WatchersontheWall comment thread for the latest Game of Owns podcast, where TWOAIF and this theory was mentioned:

"I disagree about the “Aerys is Tyrion’s father” theory. I never believed in it, and I think that “The World of Ice and Fire” only makes it even less likely. The book hints that Joanna may or may not have had an affair with king Aerys in his youthful pre-crazy days when she was a young lady-in-waiting to queen Rhaella (the only thing that’s 100% is that he was really, really into her), but it makes it absolutely clear that she did not have an affair with Aerys after she married Tywin. (...) Rhaella sent Joanna away after Aerys publicly humiliated Joanna and Tywin at their wedding, “joking” that it’s a shame that the “first night” custom had been abolished, and “taking liberties” at the bedding. Since the bedding consists of the men disrobing the bride and making lewd jokes about the upcoming consummation (with the women doing the same for the bridegroom), this probably means Aerys took the opportunity to grope Joanna and probably said exceptionally inappropriate things, too. Tywin was furious, as you’d expect, and Rhaella dismissed Joanna from her service (probably to protect her from Aerys, among other things) and Joanna left the court and went to live in Casterly Rock. That was in 263; in 266, she gave birth to Jaime and Cersei (the timeline makes the other popular theory, that Aerys was their father, impossible).

The incident (...) was in 172 on the occasion when Tywin and Joanna were present at the anniversary tournament of Aerys’ 10th year on the throne. (...) Aerysasked Joanna, in front of everyone, if her breasts “which used to be so high and proud” were ruined from giving a suck to her twins. Tywin tried to resign after this new insult, but Aerys wouldn’t have it. Two notable things: 1) even though he was deliberately being an asshole and trying to insult both her and Tywin, Aerys did not speak as someone who’s seen Joanna since she had given birth to Jaime and Cersei – and before that, they weren’t even in the same location since her wedding; it’s clear they hadn’t seen each other for a long time. And 2) after this incident, why would Joanna ever want to jump into bed with a man who publicly insulted her like that (twice)? Does anyone think Aerys had charmed her with his insult? It’s also not like she had anything to gain from it, either. (...)

So, the only way TWOAIF supports the theory is by having Joanna and Aerys in the same location at one point roughly in the timespan of Tyrion’s conception (which may also be a few months earlier or later). But the theory entirely rests on the possibility that Aerys raped Joanna, that Joanna hid it from Tywin, and that, for some reason, she decided not to use moon tea to abort the baby when she got pregnant and knew it could be the result of the rape. It “supports” it only if one already really wants to believe that Aerys is Tyrion’s father and is looking for anything that makes it possible. Heck, if we didn’ have Catelyn’s POV, and someone were to propose a theory that Littlefinger was Robb’s real father, you’d find more evidence for that than you’d find for Aerys being Tyrion’s father.

The TWOAIF info is interesting for other reasons. It shows the development of the complicated relationship between Tywin and Aerys, and how it contributed to Aerys’ descent into madness and paranoia and some of his weirder decisions, as well as Tywin’s growing bitterness because of Aerys’ repeated insults. They started off as best friends, and grew into bitter rivals. Aerys comes off as a very insecure man who felt threatened when he realized that everyone considered Tywin more capable and the one who really rules the kingdom. He also comes off as very jealous and resentful of Tywin because of Joanna, and his continued obsession with her was probably partly because she was “the one who got away” (whether or not she was his lover when they were young and then he couldn’t have her anymore, or she never was) but even more so because he envied Tywin in having such a happy marriage, and having something Aerys wanted. We know that Aerys’ rivalry with Tywin was why he took Jaime into the Kingsguard, so he could take Tywin’s son and heir from him. But I think Aerys’ behavior towards Jaime is still bizarre – keeping him as a hostage would make sense, except keeping an armed man who’s the best swordsman in the realm by your side (without any other Kingsguard around!) doesn’t seem like a good idea – who’s keeping who hostage in that situation? Aerys enjoyed showing that he possessed Jaime now, and even ordered him to kill Tywin for him and expected him to carry out that order. I think that Aerys may have had a Littlefinger-like “I could/should have been your father!” thing going on with Jaime (well, except for the fact that Aerys did not want to literally fuck Jaime)."