How do I facilitate compromise within a relationship?

My boyfriend has this mindset that everything in the world is a test of his dominance/strength/intelligence, etc…
So he is very competitive, even with things like Jeopardy! He wants to be right all the time, even if being right isn’t important.
I see this as some what damaging to his ability to compromise, because he sees it as giving in.
That being said, he’s and amazing lover and partner. I just feel as though this could inhibit out relationships growth.