One of the reasons why the House is more dysfunctional than the Senate is becase of the role of Speaker. In the Senate, McConnel can simply ignore the crazy caucus, and still run the Senate. He only needs a majority of Republicans to vote for him to keep his power, so he can safely ignore the dozen or so crazies to get stuff done. He can schedule votes that Democrats want without any fear of losing control.
But Boehner? He needs a majority of the whole house to stay in power, which cannot happen without Democrats or Tea Party types. Since the house is closely divided, the crazy caucus can oust him at any time. But there's a way out and he's half way there already: resign right after a new speaker is elected, and in the meantime transfer ALL his decision making powers to the Majority Leader in the house, whom the crazy caucus cannot control. They will kick, and scream, and cry tyranny, but they will be completely powerless over the House forever more.
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This solution is fairly elegant... I know Boehner probably want to get out of that crazy town, but this kind of change would completely blunt the power of the crazy caucus forever. And fankly I think its the only thing that has a chance of winning... for these reasons:
1) The ~200 Democrats are loving this chaos, and will not help the Republicans look responsible. Doing so diminishes their chances to retake the house in 2020. So they will not cross over to elect anybody except Pelosi.
2) About ~200 Republicans don't want to face a primary challenge, so they would not vote for anybody but a Republican.
3) The ~40 Freedom Caucus members will not vote for anybody except a nihilist who would be content watching our country fall apart, so they could blame it on Obama.
4) About ~200 Republicans are NOT nihilists, and understand how dangerously stupid the Freedom Caucus is, so they will not vote for that kind of a republican.
5) There are a handful of Republicans who would get the Freedom Caucus votes initially... but as soon as the debt ceiling was raised they would instantly demand their ousting.
6) Some retiring Republicans/Democrats might be willing to do a cross over vote in exchange for some massive favors, but I dont think its enough to get to a majority.
THEREFORE: the best way through this impass is to diminish the role of the Speaker. Boehner could schedule a vote. It would be roughly 200 for Pelosi, 200 for Generic Republican, 40 for Captain Crazy Pants. No winner? FINE! I'll go golfing for a month, and come back to run another vote. Until then, the Majority Leader makes all the decisions for me.
Done. The crazies would gnash their teeth but be complete impotent. They dont have the votes to win Speaker. Nor do they have the votes to win Majority Leader. And once they are emasculated, they look weak, and their powers would diminish even further.
Whether they grow up or not is irrelevant... they will no longer have the power to hurt our country.