President Barack Obama and Republican challenger Mitt Romney left Americans to vote Tuesday with a stark choice between their fundamentally different visions for the country's future after an aggressive and closely fought battle for the White House. The winner might not be known until the next morning.

Both sides cast the Election Day decision as one with far-reaching repercussions for a nation still recovering from the biggest economic downturn since the Great Depression and at odds over how big a role government should play in solving the country's staggering debt and high unemployment.

After months of campaigning and billions of dollars spent in the battle for leadership of the world's most powerful country, Obama and Romney were in a virtual nationwide tie, a sign of the country's vast partisan divide.

Obama appeared to have a slight edge, however, in some of the key swing states such as Ohio that do not vote reliably Democratic or Republican. That gives him an easier path to reach the 270 electoral votes needed to win the presidency.

"I feel optimistic, but only cautiously optimistic," Obama said on "The Steve Harvey Morning Show."

Later in the day, Obama eased the tension of waiting for the vote count playing basketball with friends in Chicago.

At midday, Romney and running mate Paul Ryan met in Cleveland for a campaign stop in the state that could be the most crucial battleground of all, Ohio. The Midwestern industrial state has chosen the winner of the last 12 presidential elections, and no Republican has ever won the White House without carrying it.

Reflecting the state's importance, Vice President Joe Biden made an unannounced stop in Cleveland to play defense, arriving at the airport just before Ryan's charter came in for a landing. Biden left the tarmac without comment to the surprised media traveling on his plane.

Romney, who cast his vote near his Massachusetts home Tuesday morning, still had a rally in Pennsylvania, traditionally Democratic territory where the Republican has made a surprise last-minute push — perhaps against all odds — to compensate for Obama's expected victory in Ohio.

Obama voted last month, a move intended to encourage early voting that tends to favor Democrats.

The president was spending Election Day in his hometown of Chicago, where he was met with applause and tears from volunteers as he entered a campaign office before picking up a phone to call voters. He congratulated Romney on a "spirited campaign" and told reporters he's "confident we've got the votes to win, but it's going to depend ultimately on whether those votes turn out."

Under the U.S. system, the winner of the presidential election is not determined by the nationwide popular vote but in state-by-state contests. The candidate who wins a state — with Maine and Nebraska the exceptions — is awarded all of that state's electoral votes, which are apportioned based on representation in Congress.

The close race raised the possibility of a repeat of the 2000 election, when the winner, George W. Bush, was not known for weeks after a protracted recount in Florida and a Supreme Court decision. A narrow victory for either candidate this time is sure to deepen polarization and leave the winner without a strong mandate to face mounting problems — notably, the "fiscal cliff" of higher taxes and deep automatic cuts in spending looming in January.

Sporadic voting problems were reported at polling places around the U.S.

In Pennsylvania, a voting machine that lit up for Republican Mitt Romney even when a voter pressed the button for President Barack Obama. Officials said the voting machine has been recalibrated and is back in service.

It wasn't just the presidency at stake Tuesday: All 435 seats in the House of Representatives, a third of the 100 Senate seats and 11 governorships were on the line, along with ballot proposals in some states on topics ranging from gay marriage to legalizing marijuana. Democrats were expected to maintain their majority in the Senate, with Republicans doing likewise in the House, raising the prospect of continued partisan wrangling no matter who might be president.

Obama and Romney have spent months highlighting their sharp divisions over the role of government in Americans' lives, especially in bringing down the stubbornly high unemployment rate, reducing the $1 trillion-plus federal budget deficit and reducing a national debt that has crept above $16 trillion.

Obama insists there is no way reduce the staggering debt and safeguard crucial social programs without asking the wealthy to pay their "fair share" in taxes. Romney, who claims his successful business background gives him the expertise to manage the economy, favors lowering taxes and easing regulations on businesses, saying it would spur job growth.

The economy has proven a huge drag on Obama's candidacy as he fought to turn it around after the deepest recession since the Great Depression of the 1930s, a downturn that was well under way when he replaced Bush in the White House on Jan. 20, 2009.

No U.S. president since Franklin D. Roosevelt in the 1930s has run for re-election with a national jobless rate as high as it is now — 7.9 percent in October.