Start with two premises. One: The roots of the current standoff go to the heart of our current political culture, the radicalization of the GOP and extreme polarization of our politics. But the key proximate driver is Speaker John Boehner's weakness, not only his situational weakness but his characterological weakness as well. Two: the leadership of the House clearly doesn't want to be in this situation. They actively fought against it before agreeing to follow this approach under duress. That's led to a lot of erratic or in other cases simply incompetent decision making on the part of the House leadership.
If you have an empowered an effective counter-party on the other side of a stand-off, you can use some basic game theory reasoning and find your way at least to a prediction if not a solution. But the fact that the House leaders don't themselves know what cards they're holding makes the whole situation considerably more dangerous and unpredictable. With all that said, here's what I think Boehner is doing.
Outside of the Tea Party fire-eaters, the cataclysmic effects of a US default on its debt obligations is not lost on anyone in the Republican leadership. Boehner evidently began assuring top donors and nervous members of his caucus that he would under no circumstances allow an actual default to occur, even if that meant passing an extension mainly on Democratic votes. I have little doubt that Boehner is delivering the same message to CEOs and Wall Street leaders.
But the Times article from October 3rd made these private reassurances a bit too public. You might surmise that Boehner is judiciously leaking these reassurances to speak to multiple audiences at once. I doubt it. It's not that skilled an operation. There was some of that certanly. TPM's Sahil Kapur explained other steps Boehner was taking to defuse the situation and selectively leak reassurances. But again, I think the message that Boehner knows he'll have to fold got a bit too much play.
That's what's forced him to articulate a far harder line since the piece was released, basically saying over the weekend that if the President doesn't cave he really will force a debt default. It's what's behind his press secretary issuing increasingly open and jaunty threats like this one.
I think Boehner plans to take an increasingly hard and bellicose line right up to the day before default becomes unavoidable and then if he feels certain Obama isn't budging, he'll fold and allow a clean bill to get a vote in the House. That will pass and then pass the Senate too.
The key though, again, is that Boehner is personally weak and situationally weak. He's afraid of what House Tea Partiers will do to him if he relents. So he's hoping that if he plays for time and takes the country right up to the brink of catastrophe some other option might become available. Maybe Obama caves? Maybe his moderates revolt? Maybe the Tea Party sobers up? Anything ... And yet Boehner is weak. The Tea Party's ability to push him further and further in their direction is testament to that. So at the last moment, does he really assert his power? I think it's very questionable that he has it in him. The forces he's trying to ride are much bigger than he is. There's good chance he simply won't be able to apply the brakes. It's just not in him.