What causes the change in Bilbo's relationship to Thorin during the siege of the Mountain in The Hobbit?
The relationship between Bilbo and Thorin changes when Thorin realizes that Bilbo has been hiding the Arkenstone from him.
Bilbo is having a terrible time. None of the other dwarves dare to contradict Thorin, so the siege of the mountain continues. This does not please Bilbo at all.
Bilbo, of course, disapproved of the whole turn of affairs. He had by now had more than enough of the Mountain, and being besieged inside it was not at all to his taste. (Ch. 15)
Biblo is also worried that Thorin is looking for the Arkenstone. He has the dwarves search around for it, and Bilbo does not want to think of what would happen if they discover he has it hidden under his pillow. The Arkenstone is very valuable, and it also holds sentimental value to Thorin. Biblo will be in big trouble if Thorin realizes he has it.
Biblo believes the Arkenstone is more valuable as a bargaining chip. He offers it up to the Elvenking, who is definitely surprised that he has it.
"But how is it yours to give?" he asked at last with an effort.
"O well!" said the hobbit uncomfortably. "It isn't exactly; but, well, I am willing to let it stand against all my claim, don't you know. … (Ch. 16)
Gandalf is impressed, but Bilbo is still worried. Thorin is “stricken dumb with amazement and confusion” when Bard offers up the stone, and accuses the men of thievery. In face of Thorin’s rage, Bilbo, despite his “dreadful fright,” gathers up the courage to admit he is the one that gave it to them.
Bilbo knows that Thorin will see this as the ultimate treachery. Thorin was just beginning to trust and appreciate him, and now this happens. Unfortunately, Bilbo does not think Thorin will act rationally. He is used to being in charge, and getting his way. No one really stands up to him except for Gandalf.
When Gandalf comes to Bilbo’s defense, Thorin accuses them both of betraying him. He tells Gandalf to come and get Bilbo or he will throw him down. The other dwarves are sad to see Bilbo go, but Thorin is just angry. Bilbo knows that their relationship has taken a serious blow, but he did what he had to do to avoid a war.