Despite his cool customer reputation, Obama has a distinct flair for the dramatic. Like the favorite in a prize fight, he enters the ring only after his opponent has shadow-boxed alone for a spell — hanging back until only he can finish the job.

But, in fact, the strategy all along was to enter the ring. The White House decided weeks ago to engage rather than fight Republicans over the budget. It was a tactical decision that reflected Obama’s leadership style.

“With him, it’s all about the timing,” said an Obama ally close to the talks…

He held back on becoming directly involved until just before the end, choosing instead to send Biden and a handful of key deputies into the negotiations. Along the way, he received criticism from lawmakers on the Hill — from Republicans for a lack of engagement, from Democrats for his willingness to compromise and from both sides for being vague about his bottom line in a deal.

He stepped in when his advisers felt he would have the maximum impact and made concessions that infuriated his base, such as accepting an extension of tax cuts for wealthier families and a smaller estate tax than Democrats liked. But his aides argued the trade-offs were necessary to meet his initial goal of preventing middle-class tax cuts from lapsing.