This recent article in the New Yorker offers hope of a shutdown solution by invoking the Soviet "dead hand "strategy during the nuclear cold war standoff.
I like the summary of the use of game theory to explain previous political events. I do not believe the suggestions in the article to solve the shutdown solution hold water.
How Cold War Game Theory Can Resolve the Shutdown [outside link]
And so there’s one last question for Washington this week: What can cool down the right-wing hotheads and extremists? Perhaps a rigid agreement that Obamacare could be defunded if an independent panel of economists agreed at some point that it had slowed economic growth by two per cent or more? Or perhaps a strengthened version of the sequester promise that Washington accepted last time (a supposed mini-doomsday machine that was detonated): if debt passes a certain level, massive cuts to Social Security and welfare benefits could go into effect.
Perhaps, in fact, there is an asymmetry that Obama can take advantage of after all. Ted Cruz and the other members of the suicide caucus say they honestly believe that the Affordable Care Act will destroy the country. So promise to give them whatever they want if it really does. If Obamacare ends up not having much of an effect, then nothing will change—we’ll just have lived for a little while with a little more risk, which is something that, during the Cold War, worked out just fine.