In what way is Bilbo trying to help "everyone concerned," including himself, by giving Bard the Arkenstone?

Thorin continues to speak of the Arkenstone because it means so much to him, as it is an heirloom and he threatens to take revenge on whoever has prevented him from getting it. In spite of this warning, Bilbo decides that he will leave the mountain and offer the Arkenstone to Bard; then, Bard can offer the Arkenstone to Thorin in exchange for a fair portion of the treasure.