Meeting someone half way is a commonly used term to describe a willingness to compromise. Yet, compromises are rarely 50/50. More often, they are 80/20 or 60/40. If you are ever in a situation in which the opposing individual refuses to negotiate in good faith, one way of moving forward is to ask them if they would be willing to switch roles, and to argue in favor of each other's position.
You may not wish to actually vocalize the arguments you would make, but at the very least, take a few moments to write them down. Write down at least five points against your position and five points in favor of the opposing side, and encourage the opposing individual to do the same. When you resume your positions, you should be able to see each side from a new perspective that facilitates a fair and just resolution.
The Lord God Almighty El Shaddai