Bilbo gives the Arkenstone of Thrain, 'the Heart of the Mountain', to Bard to aid him in his bargaining with Thorin. Bilbo recognizes the importance of the jewel, telling Bard that "it is also the heart of Thorin. He values it above a river of gold" (244). Bilbo does not take this action lightly, or chooses to give the stone away in some misguided effort to anger Thorin; rather he bestows the gem upon the Bard and the Elven-king in hopes that the Bard will be able to use it as a bargaining chip, enabling them to end the stand-off between the dwarves, men, and elves.
Bilbo points out that "winter is coming on fast," and all the involved members of the conflict will be dealing with "snow and what not and supplies will be difficult--even for elves" (243). To say that Bard and the Elven-king are surprised by Bilbo's actions would be an understatement, and they cannot help but question his motives.
Bilbo explains himself simply. Besides his worries about the weather, he informs them:
"Personally, I am tired of the whole affair. I wish I was back in the West in my own home, where folks are more reasonable. But I have an interest in this matter--one fourteenth share, to be precise" (243).
The hobbit firmly believes that without some outside involvement in the negotiations, that Thorin is "quite ready to sit on a heap of gold and starve" (243). Bilbo gives the Arkenstone to Bard, out of a sincere desire to help. Later, Gandalf, who secretly listened to the entire exchange, commends Bilbo's actions, saying: "There is always more about you than anyone expects!" (245)
Bilbo gives the Arkenstone to Bard to eleviate war. Thorin wants the treasure because he kills the dragon, and Bilbo knows that Bard can use this to negotiate with Thorin. I do not find this to be selfish. It made the dwarvs mad when he did this, and they were his friends and who he wanted to be with.
Obviously an unselfish gesture to limit the hostility but maximize the cooperation between the dwarven and man .