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With all of Tywin Lannister's talk of protecting the family legacy why did he never remarry after the death of his wife? He was still relatively young at the time and there are examples of older lords taking new brides.

It seems odd that he wouldn't try for more children seeing as he believes only Jaime is a suitable heir to Casterly Rock. It may be that he didn't want to create alternate claimants to the Rock but seems like weak reasoning. Especially considering how he intended to wed Cersei to Loras with intent, in part, for her to make more Lannisters.

George R.R. Martin answered this question in correspondence with a fan.

Q: And lastly, why didn't Lord Tywin ever remarry after his wife died?
Surely he would have had ample opportunities to do so? Thanks.

GRRM: Maybe he didn't want to.
_{Citadel: SSM - POVS AND THE RED VIPER}

You have to know the background of Tywin's marital life to understand why he may not have wanted to remarry.

Tywin married his cousin Joanna Lannister even though such a marriage brought no political advantages to House Lannister. Why? Because he loved her. He never loved anything more than her. Given his childhood and formative years, Tywin had grown to despise laughter and viewed it with suspicion. Yet he laughed with Lady Joanna, her and her alone. Kevan Lannister, who best knew Tywin, thinks that the last joy went out of Tywin with Joanna's last breath and he never so much as smiled again.

Tywin already had three children so he didn't need to remarry to sire heirs. Even if he hadn't had children, House Lannister survived through his brothers. So getting more children was probably not a huge concern for him, especially since childbirth had cost him the love of his life. So in short, he never moved on from Joanna and he simply had no wish to remarry at all.

Of course he was no stranger to whores after Joanna's death (Shae's one example). There are theories among the fandom that the secret tunnel from Tower of the Hand to Chataya's brothel may have been dug on Tywin's orders, allowing him to visit whores in secrecy. Given how he despised whores and those who frequented them, publicly, he could hardly become a patron openly. But it seems all that was limited to physical base needs, not because he ever wanted to have a woman again in his life.

Whilst you have only tagged this with the show tag, the show is going to lack on evidence around this whereas the books have plenty. If, however, you decide not to accept books answers I will delete this answer.

There are a few reasons here but I believe it comes down to mainly that his love for his wife was so deep he didn’t want to remarry. He then of course already had heirs who at that point weren’t going anywhere and would be inheriting the Rock. Lastly, he might not have wanted to create two “rival factions” in his children as he believes a strong family is the best type of family.

His love for Joanna

He had an extremely close love for Joanna and so probably didn’t want to remarry. This is mentioned multiple times in the text but to pick a particularly telling example:

In 263 AC, after a year as the King's Hand, Ser Tywin married his beautiful young cousin Joanna Lannister, who had come to King's Landing in 259 AC for the coronation of King Jaehaerys II and remained thereafter as a ladyin-waiting to Princess (later Queen) Rhaella. The bride and groom had known each other since they were children together at Casterly Rock. Though Tywin Lannister was not a man given to public display, it is said that his love for his lady wife was deep and long-abiding. "Only Lady Joanna truly knows the man beneath the armor," Grand Maester Pycelle wrote the Citadel, "and all his smiles belong to her and her alone. I do avow that I have even observed her make him laugh, not once, but upon three separate occasions!"

_{The World of Ice and Fire, The Targaryen Kings: Aerys II}

In fact it would not be amiss to say that a part of him died that day and he was probably in deep mourning for some time and so probably did not want to.

A queer time to come visiting. His mother had died giving him birth, so the Martells would have found the Rock deep in mourning. His father especially. Lord Tywin seldom spoke of his wife, but Tyrion had heard his uncles talk of the love between them. In those days, his father had been Aerys's Hand, and many people said that Lord Tywin Lannister ruled the Seven Kingdoms, but Lady Joanna ruled Lord Tywin. "He was not the same man after she died, Imp," his Uncle Gery told him once. "The best part of him died with her." Gerion had been the youngest of Lord Tytos Lannister's four sons, and the uncle Tyrion liked best.

_{A Storm of Swords, Tyrion V}

There was a worm inside the apple, though, for the growing madness of King Aerys II Targaryen soon imperiled all that Tywin Lannister sought to build. His lordship suffered great personal loss as well, for his beloved wife, Lady Joanna, died in 273 AC whilst giving birth to a hideously deformed child. With her death, Grand Maester Pycelle observes, the joy went out of Tywin Lannister, yet still he persisted in his duty.

_{The World of Ice and Fire, The Westerlands: House Lannister Under the Dragons}

His heirs

He already had heirs at this point and Jaime at 8 was already showing signs of becoming a strong warrior. As such he wasn’t going anywhere at that point in time so in Tywin’s mind he probably had no worries about Jaime inheriting the Rock. Of course it was only when Aerys slighted him that any worry would have shown.

Day by day and year by year, Aerys II turned ever more against his own Hand, the friend of his childhood, subjecting him to a succession of reproofs, reverses, and humiliations. All this Lord Tywin endured, but when the king made his son and heir, Ser Jaime, a knight of the Kingsguard, he could abide it no longer. Lord Tywin at last resigned the Handship in 281 AC.

_{The World of Ice and Fire, The Westerlands: House Lannister Under the Dragons}

At this point Jaime and Cersei were 17ish, if I’ve worked that out correctly, and so perhaps Tywin didn’t want to create two rival factions in his children as he believes in making the Lannister’s the most powerful and strongest House.

It’s also worth noting that at this point if he did remarry there would be a strong possibility that any new heir inheriting the Rock could have been a young Lord and it seems highly likely that Tywin would share Roose’s view on young Lords. He almost certainly wouldn’t have wanted it despite being quite a young Lord and the youngest Hand himself.

"And won't my bastard love that? Lady Walda is a Frey, and she has a fertile feel to her. I have become oddly fond of my fat little wife. The two before her never made a sound in bed, but this one squeals and shudders. I find that quite endearing. If she pops out sons the way she pops in tarts, the Dreadfort will soon be overrun with Boltons. Ramsay will kill them all, of course. That's for the best. I will not live long enough to see new sons to manhood, and boy lords are the bane of any House. Walda will grieve to see them die, though."

_{A Dance with Dragons, Reek III}

Of course if you want the Word of God answer over my analysis here George R. R. Martin has been asked this before and this was his response:

And lastly, why didn't Lord Tywin ever remarry after his wife died? Surely he would have had ample opportunities to do so? Thanks.

Maybe he didn't want to.

_{Westeros, So Spake Martin, POVs and the Red Viper}