But for two hours, Romney, Santorum, Paul and Gingrich had a different campaign in mind, their own race for the Republican nomination and the right to oppose Obama in the fall.
After a brief lull, the campaign calendar calls for 13 primaries and caucuses between next Tuesday, when Arizona and Michigan have primaries, and March 6, a 10-state Super Tuesday.
Romney is campaigning confidently in Arizona, so much so that his campaign has not aired any television ads.
But the former Massachusetts governor faces an unexpectedly strong challenge in his home state of Michigan, where Santorum is hoping to spring an upset. Santorum's candidacy has rebounded in the two weeks since he won caucuses in Minnesota, Colorado and a non-binding primary in Missouri.
The result is a multimillion-dollar barrage of television commercials in Michigan in which the candidates and their allies swap accusations in hopes of tipping the race.
In all, 518 Republican National Convention delegates are at stake between Feb. 28 and March 6, three times the number awarded in the states that have voted since the beginning of the year. It takes 1,144 to win the nomination.
The dynamic of the campaign — Santorum challenging Romney — made their clashes Wednesday night inevitable.
Romney said Santorum voted five times while in Congress to raise the government's ability to borrow, supported retention of a law that favors construction unions and supported increased spending for Planned Parenthood. He said federal spending had risen 78 percent overall while the former Pennsylvania senator was in Congress.
Santorum retorted that government spending declined as a percentage of the economy when he was in the Senate, and he noted that when Romney was asked last year if he would support a then-pending debt-limit increase, "he said yes."
There was a clash over federal spending earmarks, as well, and Gingrich sought to intervene as if serving as a referee instead of a debate participant.
He said he supported the earmarks that Romney had sought for the Olympics in Salt Lake City in 2002, but then he accused Romney of observing a double standard by running television ads attacking Santorum for having voted for different earmarks.
He said it was silly for Romney to take the position that "what you got was right and what he got was wrong."
In the hours leading to Wednesday night's debate, Romney called for a 20 percent across-the-board cut in personal income taxes as part of a program he said would revitalize the economy and help create jobs. The top tax rate would drop from 35 percent to 28 percent, and some popular breaks would be scaled back for upper-income taxpayers. However, aides provided scant details.
"We've got to have more jobs, less debt and smaller government, they go together," Romney said in an appearance in nearby Chandler. "By lowering those marginal rates, we help businesses that pay at the individual tax rate to have more money so they can hire more people."
Romney's proposal sharpened his differences with Obama, who favors allowing tax cuts enacted under President Bush to expire on higher incomes.
Santorum, who has emerged as Romney's leading challenger in the Republican race, campaigned at a tea party gathering in Tucson, where he said his rival's new tax proposal largely mirrored one he had had already made.
"Welcome to the party, governor, it's great to have you along," he said.
Santorum's rise in the race has left Gingrich and Paul on the outside looking for a way in.
The former House speaker has yet to recover from a campaign nosedive that began after he won the South Carolina primary on Jan. 21, and he is pinning his hopes on his home state of Georgia to begin a comeback on March 6.
His campaign announced plans Wednesday to buy 30-minute blocks of television time in upcoming primary and caucus states for an infomercial on reducing energy prices.
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