So far, the battle of the budget in Washington is playing out roughly as expected. While a government shutdown has theoretically been ordered, nothing much has really happened, all the functions of government deemed essential have continued and financial markets have simply yawned. The only real difference between the tragicomedy now unfolding on Capitol Hill and the scenario outlined here last week has been in timing. I had suggested that the House Republicans would give way almost immediately on the budget, if only to keep some of their p owder dry for a second, though equally hopeless, battle over the Treasury debt limit. Instead, it now looks like President Obama may succeed in rolling the two issues into one and forcing the Republicans to capitulate on both simultaneously.

The ultimate outcome of these battles is now clearer than ever. As explained here last week, the Tea Party’s campaign either to defund Obamacare or to sabotage the U.S. economy was doomed by the transformation in political dynamics that resulted from November’s election — above all by the fact that the president never again has to face the voters, while nearly every member of Congress must. This shift in the balance of power made the Republicans’ decision to mount a last stand on Obamacare, instead of attacking the White House on genuine budgetary issues, politically suicidal as well as quixotic. But while the outcome now looks inevitable, the timing of the decisive battle is important. Financial markets and businesses have responded with a tolerance bordering on complacency to the shenanigans in Washington, but this attitude could change abruptly if the House Republicans’ capitulation is delayed too long. As they say in the theater, the only difference between comedy and tragedy is timing.

The risk, as everyone now realizes, is that the battle of the budget — which turns out really to be just a minor tussle over the funding of a limited range of worthy but nonessential government services — remains in a stalemate right up to October 17, when U.S. Treasury is expected to hit its debt limit. At that point, an immediate settlement will be needed or all hell could break loose. The key question for businesses and investors around the world, therefore, is whether the Republicans’ impossible demands to defund Obamacare are removed from the budgetary bills comfortably before the October 17 deadline, or whether this capitulation is triggered by a financial crisis once the deadline draws too close.

Until recently, the Republicans believed October 17 would be their moment of maximum leverage, since President Obama would have no choice but to make concessions or face an economic meltdown. But now it is becoming clear that Republicans will be the ones facing maximum pressure as the debt limit draws near.

Game theory teaches that in such confrontations we must assess the costs to the players of fighting and also of backing down. Republican Congressmen, if they continue fighting right up to the debt limit, will have to answer to voters for the economic mayhem that a Treasury default could cause. Obama, on the other hand, has nothing to fear from elections since he will never run again. The costs of retreat also favor the White House. For the president to back down and gut Obamacare would destroy what he sees as his greatest achievement and would confirm his lame duck status. For the Republicans, by contrast, allowing the budget and debt ceiling to pass would leave them exactly where they were a few weeks ago.

In logic, such calculations should force the Republicans to back down quickly, extracting whatever small face-saving concessions the White House might offer them to avert any further political embarrassment and economic harm. Such concessions might include agreement to go ahead with the Keystone oil pipeline, or to reduce the unpopular excise tax on medical equipment, or to launch a formal process of discussions on revenue-neutral tax reforms.

Many Republicans seem to be making such calculations. Within 48 hours of the government shutdown, the Washington Post had identified 18 Republican Congressmen who had publicly expressed willingness to vote with the Democrats on a “clean” budget resolution, plus four leaning that way — and the numbers were growing every hour. Since only 17 Republican defectors would be needed to pass the budget, the deadlock seemed almost over — and this whole manufactured crisis may indeed be resolved by the weekend.

Unfortunately, however, such a happy outcome is far from certain. The reason, ironically, is that the Republicans have backed themselves into a position that is now too weak, thereby tempting the Democrats to overplay their hand.

By any rational standards, Obama should now offer the Republicans some minor concessions in exchange for a long-term deal on the debt limit that would permanently cure this headache blighting his presidency. Such a deal would save Obama from lame duck status and maybe even boost him into the ranks of transformational presidents if his health policy gains the popularity he expects. Long-term resolution of the debt limit would also be a powerful economic tonic, catapulting Wall Street to new records, underpinning the dollar and perhaps inspiring the upsurge in business confidence, investment and hiring that has remained so frustratingly elusive since the Lehman crisis.

The danger, however, is that Obama is now so confident that he will refuse any small concessions that might allow the Republicans to surrender with dignity. By trying to drive too hard a bargain, the Democrats could push the rational Republicans who are almost ready to surrender into the arms of their suicidal colleagues. The suicide mission to defund Obamacare, even if it continues, will probably be abandoned well before the Treasury debt limit is breached on October 17. But the risk of financial panic and serious economic damage will mount every day as the deadline approaches. Why take this risk? Game theory says that skillful negotiators should offer their opponents a relatively painless way to back down. Obama should look for a way to help moderate Republicans save face.

PHOTO: U.S. House Speaker John Boehner (R-OH) walks from a meeting with U.S. President Barack Obama, House Minority Leader Nancy Pelosi (D-CA), Senate Majority Leader Harry Reid (D-NV) and Senate Minority Leader Mitch McConnell (R-KY) outside the West Wing of the White House in Washington, October 2, 2013. REUTERS/Jason Reed