Ok so lets actually look at the facts here and using game theory explain why the white house has boggled this game of brinkmanship, why they must now negotiate or let this thing go sideways.
The demographic changes across the nation have destabilized equilibrium, Republicans realize that the game must change for them to survive. They are willing to elevate to a game of chance.
Democrats pushed through the Healthcare legislation using a supermajority in the senate and ignoring republicans in the house.
Republicans voted 42 times to change the bill which is the effect of raising your hand and saying this doesn’t work… you need to negotiate something for us or we will launch a nuclear weapon.
They were ignored, so they shut down the government, call it a warning shot.
The white house responds with a firmer non-negotiation move, they can not retaliate at this point. So their move is to effectively pass… republicans cant move because its not their move to make
Republicans have now gone all in. They cannot walk away with a loss if they do they will never regain political capital.
Allowing this to bog on will result in the blame going down 50/50 split, everyone losses is better for the Republicans than the republicans losing alone.
Republicans can’t win and have given several soft outs to the white house which they have refused those outs are not likely to disappear soon.
Here is the end state…
Republicans have no reason to give up without getting something in return, actually they can’t they are as they say pot committed. They gain nothing by giving up, the democrats on the other hand risk being blamed for this crisis if they let it trudge along. All of which creates room for a third party.
Republicans have no self interest in simply rolling over and Obama could look a hero if he negotiates, but that isn’t likely. Additionally republicans have tried to soften the situation almost 100 times, and they have been blocked, this entrenches them more. The only path forward is to open a dialog but all sides are entrenched.
A default is very likely, months and months and months of a shutdown and nothing happening is also very likely. It’s game theory, once you go over a brink and nothing happens you inevitably go over the next brink… There is no mediator to soften the blow or force open negotiations, yet. Im not sure how this ends if each side continues to play the way they are. Democrats are not playing the strong hand giving some soft out to the republicans to claim the win, they are instead trying to pass all the blame. It’s not the correct move… ultimately this will get interesting as a Nash equilibrium is pretty far off…a 3rd party.
In short, this is not a dictatorship and Obama must negotiate or this stalemate may never end. He and only he can end this.