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“You don’t say you have a mandate, you don’t threaten people,” said Chris LeHane, a longtime Democratic strategist who worked for former President Bill Clinton. “As soon as you say you have political capital, you start losing it. Reagan didn’t say he had political capital, and George W. Bush said he had political capital and he lost it.”

Mark Mellman, a longtime Democratic pollster and strategist, said Mr. Obama is smart to keep a low profile and reach out to congressional leaders rather than immediately using his bully pulpit to push his weight around.

“I’m always wary of calling anything a mandate,” he said.

At the same time, he said, “there’s no question from every poll that’s been done that people want to see taxes on people who are the wealthiest Americans go up to help prevent cuts on programs that really matter and help reduce the deficit.”

“I think it would be a mistake for the Republicans to stand in the way of that,” he added.