ROCHESTER, N.H. >> There are few titles in American democracy as privileged as “undeclared New Hampshire voter.”

Presidential candidates obsess over them. Operatives tailor advertisements to their whims. And in an election season more volatile than any campaign is likely to have imagined, the state’s electoral free agents here are beginning to grapple with how to exercise their unusual power to control the fates of candidates in either party.

About 40 percent of the New Hampshire electorate is independent (officially called undeclared) — a greater voter share than either party can boast — and is allowed to participate in either primary.

And their choices could be decisive for two very different candidates, Donald Trump and Sen. Bernie Sanders, who are counting on independent support to win the state.

Early indications suggest that independents are being drawn to the turbulent Republican race, where the large number of candidates can give these voters an outsize role in the outcome of the Feb. 9 primary and shape the contest beyond.

Janet Doyle, an undeclared voter from Portsmouth, said she usually votes Democratic. But she suggested a sort of civic duty this time to participate in the Republican primary instead.

“It’s where New Hampshire can make the most difference this year,” she said, ticking off the names of some moderate Republicans she might support, including Gov. John R. Kasich of Ohio and Jeb Bush. “Anything to stop Trump.”

Sanders, desperate to broaden his appeal beyond the left, has taken pains to court the undeclared, particularly since independents have often provided significant support to candidates who were not the party favorite. Bill Bradley and Barack Obama won the undeclared vote in their primaries in 2000 and 2008, but lost among registered Democrats and wound up narrowly losing to Al Gore and Hillary Clinton in the state.

Tad Devine, a senior adviser to Sanders, said the campaign was courting independents with mailings and through other voter outreach efforts in part by highlighting issues like campaign finance reform and the power of special interests in Washington, which the campaign expects to resonate particularly with independents.

Neil Levesque, executive director of the New Hampshire Institute of Politics and Political Library at Saint Anselm College, noted a recent Sanders ad to hit the airwaves, titled “Rock” and narrated by a dairy farmer from Vermont, Sanders’ home state. (“Bernie cannot be bought out by big money,” the farmer says, adding, “He’s a rock.”)

“There’s not a lot of substance as far as liberal-leaning policies,” Levesque said. “It’s about the fact that he’s independent. Like the farmer.”

But many voters see the Republican primary as more competitive and intend to take that party’s ballot, according to recent interviews with two dozen independents.

In 2008, the last time the state hosted two competitive primaries, about 530,000 ballots were cast, with about 197,000 of them coming from independents.

No one has ever won a New Hampshire primary without a plurality of his or her own party’s voters. But with such a large Republican field this time, if one candidate amassed a large number of independents and a smaller but healthy share of party members, victory could be possible — especially if the candidate needed only 20 percent to 25 percent of the total vote to prevail.

“It’s more likely that you could win due to undeclared voters in a big field than if you had three people running,” said Andrew E. Smith, director of the University of New Hampshire Survey Center and an author of “The First Primary,” a new book about the New Hampshire contest.

Russ Miles, an independent from Rochester, said he agreed more with Trump than with other candidates on major issues like immigration, though he was uncomfortable with the billionaire’s temperament.

“I think about voting for him, but I wish he’d calm down a bit,” Miles said.

Many independents like Trump: In a CNN/WMUR poll this month, 34 percent of undeclared voters who plan to vote in the Republican primary said they supported Trump, compared with 16 percent for Sen. Marco Rubio of Florida and 7 percent each for Bush, Gov. Chris Christie of New Jersey and Kasich. Some political analysts attributed Trump’s high numbers to his well-known name, because these voters tend not to think about politics or their candidate options until closer to Primary Day.

Others are looking elsewhere. Barbara Novak, who voted for Obama in 2008, said she would vote Republican this time around because she wanted to help Christie’s chances to be the party’s presidential nominee. Novak said she was concerned about national security, and was not certain she would vote for Christie in the general election in November, but would like the option.

“He’s very forceful, and by next fall I think that’s exactly the kind of leader we may need,” said Novak, of Hampton Beach.

Fifteen of the last 20 winners of New Hampshire primaries went on to become their parties’ nominees. One who did not, Clinton, who won the state in 2008 but lost the nomination to Obama, is hoping to impress independents this time by focusing keenly on local issues like the heroin epidemic and mental health care needs.

“Independents are hugely important to the primary and they want to be heard, and Hillary has been listening to them closely,” said Jennifer Palmieri, a Clinton spokeswoman.

Some independents are fed up with the political establishment. A few mischief makers have pledged to vote for candidates they generally dislike — leading a conservative, for instance, to support Sanders in a bid to damage Clinton’s standing in the general election.

More often, though, undeclared moderates have begun to reckon with their responsibility as a powerful, pliable check in an unpredictable election year. And their growing numbers suggest their potential to shape the outcome in New Hampshire.

In 2000, undeclared voters cast about 29 percent of the 240,000 Republican votes and 27 percent of the 157,000 Democratic votes, according to data provided by the secretary of state’s office. But by 2008, about 42 percent of the 289,000 Democratic ballots cast came from undeclared voters — compared with about 31 percent among Republicans.

“They tend to go where the action is,” said Thomas D. Rath, a former state attorney general and a longtime Republican strategist who is advising Kasich.

But the fickle nature of many independents can make them difficult to court.

At a rally in Rochester this month, where Rubio spoke passionately about his opposition to abortion rights, he drew applause from many in the audience but alienated Christie Cole of Newmarket, an independent who had been thinking of supporting him.

“I believe it is a woman’s choice — not a government ‘issue’ for a male to decide on,” Cole, 21, said.

As Rubio posed for photographs with voters after the rally, Dave Crossan and his wife, Marilyn, watched him with mixed feelings. They believe that his Cuban roots could help attract Hispanic voters who are crucial to winning the presidency, but they dislike that he has skipped many votes in the Senate. They also admire Kasich, but are torn about his lack of traction in the polls. Trump and Carly Fiorina do not seem like winners to them.

So Crossan may vote for Sanders, if only to make Clinton sweat.

“I don’t want Bernie Sanders to be president,” he said, “but I’ll vote for him anyway.”