The current situation in politics dictates a very simple motivation in game theory for the Republicans:
Support Obama AND Country succeeds = bad
Support Obama AND Country fails = bad
Renounce Obama AND Country succeeds = bad
Renounce Obama AND Country fails = good
Their only reasonable strategy in the frame of game theory is to renounce Obama and hope the country fails. This is obviously a bad incentive for the country.
Even if Republicans agreed with an Obama initiative, there would be this factor of simple game theory tugging in the opposite direction.
A way I imagined to mitigate this problem is to have the opposite party take the vice presidency, but allow the president to choose from the top candidates from the other party. I make this modification because I imagine an Obama/Mccain ticket would be a disaster, but some other combos might not be as bad (eg Obama/Romney).
Ceteris paribus this would take away some of the ‘cult-think’ that seems to create an ‘us vs. them’ deadlock on open, rational debate. This is fueled by the entertainment-centered news industry. War and fear seem to be the only way to get things passed nowadays. If ‘your guy’ is right in the mix of things it provides a bit of an anchor for you.
I know Hannity/Colmes is not what we want, but that was a bit of a straw man scenario.