Washington ? In the late 1950s, the famous nuclear strategist Herman Kahn facetiously suggested building what he called a “Doomsday Machine.” A computer would be wired to detonate a vast array of nuclear bombs if the Soviet Union took an action defined as intolerable.

Kahn was joking. The point of his black humor was that official U.S. nuclear strategy was as crazy as his Doomsday Machine, in that it posited a war from which there was no rational escape. The alternative, wrote Fred Kaplan in his book “The Wizards of Armageddon,” was finding strategies short of all-out annihilation. Indeed, Kahn proposed 44 different “rungs of escalation,” including such stages as “ostensible crisis,” “barely nuclear war,” and “local nuclear war.”

These old books about nuclear strategy are oddly useful this week in considering Washington’s looming political catastrophe. House Republicans have put the country on a doomsday path toward financial default. If the two sides can’t find a way off their ladder of escalation, they will soon reach a point of “assured destruction” in global financial markets. The fact that it will be the Republicans’ fault will be little consolation to the Democrats as they try to clear the wreckage.

This impending debt-ceiling crackup finally seems to have roused the attention of both sides so that they are at least talking about negotiations. That’s a change from the first week of the shutdown, when House Republicans were still giddy with their self-destructive power and the Democrats seemed happy to let them commit political suicide.

One of my favorite books about strategy is a thin volume called “Every War Must End,” by Fred Ikle. It was published in 1971 when the U.S. was still trying to resolve the Vietnam War on acceptable terms.

Ikle’s theme was that leaders often start wars without a clear idea of how to finish them. World War I was basically a mistake. “The major European powers misjudged how their mobilization schemes would interact,” wrote Ikle. But they couldn’t escape the lockstep process of escalation, and then fought on, hoping for victories that would justify the terrible cost. All sides would’ve been better off if they could have found a settlement quickly, but they didn’t know how.

Japan’s attack on America in 1941 was folly, yet it proceeded to the grimmest result: Three months before Pearl Harbor, the Japanese emperor asked how long it would take to defeat the United States. The army chief told him that the war would be over in three months. Other Japanese knew better. The navy chief warned: “Even if our Empire should win a decisive naval victory, we will not thereby be able to bring the war to a conclusion.”

Unwilling to admit their colossal error and sue for peace, the Japanese military staggered on. The United States decided that the Japanese political system couldn’t surrender, and that the only way end the war was to use nuclear weapons.

Let’s return to the current Washington political war. It seems fair to say that House Speaker John Boehner embarked on an unwinnable campaign without a clear conception of the endgame. This has left President Obama and the Democrats with the dilemma of whether to let the tea party Republicans take the country over the cliff to default — and thereby discredit themselves. That’s a strategy for total victory — a bet that the tea party would be so damaged by forcing default that its hold on the GOP would be broken — but such epochal triumphs rarely happen.

A more modest Democratic strategy would be to allow the House Republicans to save face enough to reach a compromise. Imagine a Versailles peace treaty in 1915, instead of 1919. Obama could offer confidence-building measures, such as pledging deficit-reduction measures, to convince Boehner to stop his threats. Next could come a “grand bargain” reforming entitlement programs. Such a deal might actually leave the country better off.

Traditional societies where wars are frequent, such as the Pashtun-speaking tribal areas of Pakistan, have elaborate rituals for the face-saving process that ends wars. A losing fighter will sometimes put grass in his mouth, humbling himself like a field animal, and come into the house of his adversary — where the victor is obliged to offer hospitality and respect that make reconciliation possible.

Obama seems to have won the political argument; the whole country can see that the tea party is behaving irrationally. But now Obama needs to help Boehner turn off the Doomsday Machine. Better a little grass in the mouth for both sides than financial default.