He is hoping that the desire for the Arkenstone would force Thorin to negotiate with the Men and Elves instead of fighting with them.
Bilbo believes that if the men and elves have the Arkenstone, which Thorin values even more than gold, they may be able to convince the dwarves to bargain with them. If an agreement can be reached, it will spare a good deal of bloodshed.
Also, Bilbo is getting quite tired of being trapped in the mountain, and wants to be somewhere with green grass underfoot and better meals.
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Bilbo knows that Thorin will never give into the demands of the Men and Elves. He also see that it makes sense that the Dwarves compensate the men of the lake for the trouble the dragon caused, and probably felt a bit responsible since he was the reason the dragon stirred in the first place.
Bilbo himself wasn't sure at first why he kept the Arkenstone. Later he used it to eventually make peace with Bard and the other men.
He gave the jewel to Bard so that Bard could use it to force Thorin to give over a fair share of the treasure hidden in the mountain.
To make a truce with the Elvenking and the other man so that the fighting will stop.
He wants to kill him.
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He wants a truce between thorin and co. and the elves, because the goblins and the Wargs will be arriving to declare war.
Once Bilbo saw how greedy the Dwarves were, he knew the Thorin would do anything for the Arkenstone, even give up some of the treasure.
The Arkenstone of Thrain.
The dwarves are Bilbo's friends for most of the Hobbit.
He had possession of the Arkenstone.
The Arkenstone is the jewel of main focus in The Hobbit, and it's a prized treasure among the dwarves.
Bilbo hides the Arkenstone from the dwarves because he knew each one was greedy for the Arkenstone and they might kill each other or harm others for it.