In January, I wrote a post diagramming the negotiations between President Obama and the House Republicans in game-theory terms. I thought the result that actually occurred was highly predictable if one assumed that the situation was a one-off Prisoner’s Dilemma with two players (Obama and Boehner). But I raised two complications: the game might be “Chicken” rather than a Prisoner’s Dilemma; and it would be repeated, which changes the strategic calculations.

Now it is being repeated. The President sees it as “Chicken” this time and has made the recommended play. To win Chicken, you should throw your steering wheel out of the window to show that you are not going to swerve. Then the other player has the choice of dying in a draw or living by losing. Obama tried to create that situation for John Boehner by calling the Speaker privately to say that he will not negotiate over the debt limit. I say Obama “tried” to create that situation because, in fact, the president still holds the steering wheel, and Republicans know he can negotiate if he wants to. That is why they are still driving straight.

In the Economist, “M.S.” suggests a play for Obama: promise to veto any debt-ceiling increase unless Congress passes immigration reform. This would be symmetrical to the Republicans’ threat to undermine or repeal Obamacare in return for a debt-ceiling increase. M.S. thinks this gambit would reveal the reckless behavior of the House Republicans. Maybe, but I think the public would draw the conclusion that both sides are maniacs. The president should simply point out that he is not making that kind of play.

In any case, the situation is a lot more complicated than a two-player Chicken Game. For one thing, the Republicans get to choose whether to play the “shut-the-government” game or the “default-on-the-debt” game, and in which order. Besides, there are many players. To name just the obvious ones: Tea Party House Republicans (who are showing that they understand the situation, they don’t care about Boehner, and they do not want to lose); Senate Republicans of various stripes; Wall Street and other big donors; Congressional Democrats; reporters; and the American people, most of whom have not paid much attention yet–but they will.

If the government shuts down or defaults (or both), then clearly the next stage in the “game” will be an effort to pin the blame on the other side. Game theory is not especially helpful for understanding this stage, which will be a struggle for rhetorical framing rather than a negotiation. Probably the best guess is that the Republicans will take the blame– which I think they deserve–but that is uncertain because the country and media are polarized and the Republicans have their talking points ready and honed. Pew finds that Republicans will start the blame-allocation phase about on par with the President, and that may embolden them to take us into default.

The Democrats will win the finger-pointing phase if people blame Republicans and decide to vote against them in 2014, but not if people blame Congress, which you cannot vote against. This is a sticky reality for the President, because attacking Republicans makes him look partisan (when people see partisanship as the problem). But I think he has to do it anyway. He will have two free cards to play: 1) Congress may not even succeed in sending him a bill to veto, in which case it will be much easier to pin the full responsibility on the House leadership, and 2) the House has taken about 124 recess days in 2013, by my unofficial calculation.

Congressional procedure is always a confusing and boring topic, but if I were the president, I would say: “The government has shut down because the House Republicans could not even send me a budget bill to consider, even though it is their constitutional responsibility to pass a budget by the end of every fiscal year. They took 124 days off this year. They can reopen the government in fifteen minutes by sending me a clean budget bill that preserves the status quo.” (Of course, the status quo is a unacceptable, but it beats a disaster.)