In a wildly chaotic presidential election starring a tycoon-turned-reality-television-star from New York and a socialist from Vermont, is there room for another billionaire from Manhattan’s East Side?

That is a central riddle facing former New York mayor Michael Bloomberg as he contemplates a third-party bid for the White House.

Donald Trump’s resounding victories in South Carolina on Saturday and in Nevada’s caucuses Tuesday have solidified his status as the Republicans’ front-runner. On the Democratic side, Sen. Bernie Sanders’s unexpected surge — and landslide win in New Hampshire — have kept him in the race even as Hillary Clinton has regained some momentum following her narrow victory in Nevada last weekend.

What remains unknown is whether Bloomberg, 74, is willing to insert himself into a contest that remains unpredictable even as the field of candidates winnows.

Bloomberg went out of his way last week to bemoan a campaign he described as a “race to the extremes,” with candidates exploiting Americans’ loss of faith in a political system that is “corrupt, gridlocked and broken.”

“That’s why you see the current candidates out there doing well, and not the conventional ones,” Bloomberg told a gathering at a Manhattan book party, an apparent reference to Trump and Sanders.

Looming over the deliberations is the question of which party a Bloomberg candidacy would hurt more. With polls suggesting that Bloomberg would draw more Democratic than Republican voters, it makes little sense that Trump’s surge would prompt the former mayor to run.

That said, Bloomberg’s viability may be strengthened if Clinton is damaged by a drawn-out nomination process.

Bloomberg proved his capacity for defying naysayers 15 years ago, when he captured New York’s City Hall. But New York is not the United States, and it’s far from certain that swing-state voters would embrace a former Republican known for opposing guns, smoking and Big Gulp sodas.

Nor is it clear that a measured, sober-toned business executive could command an electorate that so far has rewarded unceasing bombast over pragmatism.

“You don’t solve problems by pointing fingers or making pie-in-the-sky promises,” Bloomberg told the gathering last week. “You solve them by bringing people together around common goals, promoting innovation, demonstrating independence and recognizing that compromise is not a bad word.”

Bloomberg’s potential as a candidate is rooted in what the Gallup organization has identified as an ever-broadening swath of American voters who define themselves as independent and express dissatisfaction with both major political parties.

For those voters, familiar names such as Clinton and Jeb Bush may seem like more of the same; meanwhile, Sanders, Trump and Sen. Ted Cruz (R-Tex.) may be too extreme, creating opportunity for Bloomberg, a former Democrat who turned Republican before declaring himself unaffiliated in 2007.

“There is a broad constituency, and there will be a broader one still, given the polarization of this election,” said Douglas Schoen, Bloomberg’s pollster, who recently wrote a Wall Street Journal op-ed article headlined, “Why Mike Bloomberg Can Win.”

“He’d be the only one talking about bipartisan consensus and bipartisan decision-making and results-oriented policies,” Schoen said. “In a particularly unstable political situation like the one we’re in now, where there’s so much anger, it’s impossible to write off any credible candidate.”

Bloomberg would face daunting challenges, not the least of which would be prevailing in enough states to reach the 270 electoral college votes needed to win the presidency.

American voters have never elected a third-party candidate to the presidency. In 1992, when he faced President George H.W. Bush and then-Gov. Bill Clinton, Ross Perot managed to draw 19 percent of the popular vote. But he did not win a single state.

On a logistical level, Bloomberg’s campaign would need to quickly assemble an army to gather more than 900,000 signatures needed to qualify for the ballots in all 50 states, with a parade of rolling deadlines beginning this spring.

“Every state has a different process for getting on the ballot, and it’s implicitly designed to make it very difficult,” said Reed Galen, a Republican strategist who advised Sen. John McCain’s 2008 presidential campaign. “Is it plausible for him to get on the ballot? Yes. Easy? No.”

Perhaps more crucially, Bloomberg would need to sell himself to a public that is largely unaware of him. A Quinnipiac poll earlier this month found that 56 percent of American voters had not heard of Bloomberg, the owner of the media company that bears his name.

And a new poll this week from the Associated Press showed even more dire numbers — that a majority of voters of both parties would not even consider voting for him.

Ed Rollins, the Republican strategist who advised Perot in 1992, recalled that the Texas billionaire’s appearances on Larry King’s CNN talk show galvanized a massive effort to get him on ballots across the country.

Unlike the enthusiastic hordes propelling Trump and Sanders, the strategist said, Bloomberg “has none of that. He has a billion dollars, but there’s not a movement for him.”

“My sense is that people telling Bloomberg to run are his New York dinner companions who are not happy with their choices,” Rollins said. “That’s not a constituency or a movement.

“Arguably, he was a good manager of New York,” Rollins said. “But the vast majority of Americans don’t care about New York.”

New York’s political establishment ridiculed Bloomberg when he ran for mayor in 2001. No matter his wealth, the argument went, a political novice running as a Republican could never win in a city awash in Democratic voters.

“He was seen as another one of these rich guys who had nothing better to do,” said Bill Cunningham, a former Bloomberg adviser who recalled that his own friends said he was “crazy” for signing on with the mogul’s campaign.

While the Democratic nominee, Mark Green, was weakened by a savage primary battle that year, he still held a formidable lead over Bloomberg as the general-election campaign began.

Bloomberg’s willingness to spend $50 million of his own fortune on the campaign allowed him to inundate New Yorkers with television ads and mailings. His proven financial acumen became ever more attractive after the 9/11 terrorist attacks decimated New York’s economy. And an 11th-hour endorsement from then-Mayor Rudolph W. Giuliani — exalted at that point for his handling of the 9/11 aftermath — was tantamount to a blessing.

But Bloomberg also closed the gap by identifying and, over the course of many months, methodically appealing to 800,000 New Yorkers who were unaffiliated with either the Democratic or the Republican parties. It was an effort that his advisers tout as key to his narrow victory.

For all that, said Kenneth Sherrill, a Hunter College political science professor: “Everyone was flabbergasted when he went over the top. No one expected it.”

Still, Bloomberg’s success in New York may have little bearing on how he would fare in a presidential race, the outcome of which is determined not by the popular vote but by the electoral college.

“What works in New York doesn’t necessarily work in the rest of the country,” said Stuart Rothenberg, the editor of a nonpartisan newsletter covering state and federal campaigns. “He’s the guy who’s known for being against guns and soft drinks. Is that going to play in Fairbanks, Alaska, or the suburbs of Indianapolis?”

Competing in a three-way race, Bloomberg’s progressive stances may not be problematic if he needs to win only a plurality against opponents such as Trump and Sanders.

But his success would depend on independents remaining estranged from the two major parties by November — and on their willingness to expend their vote on a candidate whose viability is uncertain.

Despite claims of political independence, voters’ “partisanship runs pretty deep,” Rothenberg said. “There are people who say they’re independent who aren’t really independent. They don’t want to throw their vote away.”

Then there’s the question of which party a Bloomberg candidacy would damage more. A recent USA Today-Suffolk University poll showed Sanders in a close two-way contest against Trump but losing to the reality-television star if Bloomberg were on the ballot (Bloomberg placed third).

Ultimately, Bloomberg’s challenge would be to identify those states in which he could win a plurality and put together the necessary electoral votes, an analysis Rothenberg undertook this month in a column in Roll Call.

What Rothenberg concluded is that Bloomberg would have little chance in what Gallup and he have identified as the country’s 20 most conservative states, including Mississippi, Alabama, Louisiana, Oklahoma and South Carolina.

Bloomberg would also face imposing challenges in the country’s most left-leaning states, such as Oregon, Vermont, Massachusetts, Michigan and Maryland.

There’s little question, though, that Bloomberg’s candidacy would upend the current list of key swing states in the general election — states including Ohio, Iowa, New Hampshire, Florida, Colorado and Virginia.

Where Bloomberg could find success, Rothenberg said, is in states such as California, New York, Connecticut and New Jersey — all states that typically vote Democratic. Ohio is the only swing state on that list. And with so many states probably out of reach, Bloomberg “would have to run the table” in those and other states. “That won’t be easy. My bottom line is I don’t see it.”

Trump’s and Sanders’s success may have made the art of political predictions risky, Rothenberg said, but the math isn’t adding up, “even for Michael Bloomberg with all his money.”

“When you look at the puzzle,” he said, “it’s, at the least, very, very difficult.”

Scott Clement contributed to this report.