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[GoT show] How would things have worked out if Tyrion had after killing Shae not gone on to face Tywin but instead had gone straight onto the boat leaving Tywin alive?
gpt
Tywin quashes the Sparrows' movement before it gains traction. He forces Cersei to marry Loras and sends both of them away to Highgarden so he is free to educate Tommen. He limits Margaery's control over Tommen, but is careful to keep the Tyrells in the fold.

Before he can turn his attention to Stannis in the North, he has to resolve issues with Dorne. Qyburn is Cersei's pet, without her, Tywin puts an end to his research. Gregor is left to die and his skull is sent to Dorne as an apology. Tywin offers Oberyn's seat on the Small Council to another Dornish representative. Doran keeps Ellaria and the Sand Snakes in line while sending Trystane to take Oberyn's seat, while Myrcella accompanies. Tywin encourages King Tommen to spend time with his sister and with Trystane to foster good ties with Dorne and to limit Margaery and the Tyrells' influence over him.

In the Riverlands, House Frey quashes revolts with Tywin's support. Jaime is probably still sent to retake Riverrun from the Blackfish, this arc doesn't change much. Not sure what Jaime and Tywin's relationship is like now, but Tywin probably forces Jaime to take on more responsibility to make up for freeing Tyrion. Jaime is tasked with assembling a force to oppose Stannis in the North.

Littlefinger would be stupid to continue his treacherous alliance with the Boltons with Tywin still alive. He keeps Sansa in the Vale and focuses on consolidating his power in the region.

With Lannister and Frey support, Roose easily crushes Stannis. The Lannister army continues to the Wall and demands the removal of the Wildlings, ignoring Jon Snow's pleas and warnings about the White Walkers. The Watch is helpless to stop the massacre of the Free Folk, many of brothers are even pleased with this turn of events. Jon is shattered and resigns his command to Thorne. 

Ramsay is unable to betray his father, who has powerful allies in Tywin and Walder Frey. He is forced to watch his trueborn brother grow to become a rival despite his father's reassurances. Theon remains Reek. The Umbers give up Rickon to Tywin to earn his favor. Tywin keeps Rickon as a hostage to keep the Boltons in line and to make up for the loss of Sansa.

The Brotherhood without Banners and any remaining dissidents and insurrectionists on the mainland are hunted down. In a scandalous, unprecedented appoinyment, Tywin has Olenna Tyrell appointed Mistress of Whisperers since he desperately needs to replace Varys. 

Mace Tyrell and Meryn Trant are never sent to Braavos to treat with the Iron Bank, so Arya continues to train with the Faceless Men.

As he rebuilds the royal fleet in preparation for finally dealing with the Greyjoys, Tywin meets with Littlefinger to return the Vale to the fold. Littlefinger stalls, quietly keeping Sansa in his custody and denying any knowledge of her whereabouts. He is forced to reach out to Varys in an attempt to coordinate, since they share a common enemy in House Lannister and House Targaryen is the only piece left in the field. Varys rebuffs him for now. 

Tywin personally commands a grand coalition to attack the Iron Islands once again. Jaime and Kevan serve under his command, this force is composed of Lannister, Tyrell, Martell, Frey, and Bolton bannermen. The Greyjoys are crushed and their castles torn down. Tywin puts many thousands to the sword in order to ensure the Ironborn will never be a threat again.

A portion of the Iron Fleet under Yara manages to escape to Mereen to join House Targaryen.

Tywim begins a vast rebuilding program to restore the war-torn country, while preparing for the inevitable threat of Daenaerys. He makes common cause with Qarth and Yunkai and funds the Sons of the Harpy, hoping to keep Dany tied down in Mereen for as long as possible.

The Night King cannot breach the Wall without Viserion, so he sits around waiting.