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Snap polls suggested Mr. Cameron had the better performance on Thursday during a live television audience question and answer session.WPA Pool/Getty Images

These are desperate times for David Cameron and Ed Miliband, the two men who would be Britain's Prime Minister, and it's showing. The Conservative leader sweated profusely from his upper lip as he answered questions from a live audience here in northern England on Thursday. The Labour leader stumbled off stage after his own performance, nearly taking a career-defining fall after 30 tough minutes of questioning.

The United Kingdom may have discovered an effective new format for testing political leaders – the half hour of grilling each man faced from a live television audience here in Leeds were far more revealing than the seven-way leaders' debate earlier in the campaign – but the country is no closer to finding out who will be Prime Minister after the May 7 election.

In fact, with polls showing the two main parties deadlocked ahead of next week's general election, the path to forming a government became (if possible) even more complicated after Thursday's special edition of BBC Question Time.

The conventional wisdom for some time now has been that – if the polls are right and the Conservatives and Labour each end up with about a third of the popular vote and around 270 seats in the 650-seat House of Commons – the quickest way for either man to move into No. 10 Downing Street would see Mr. Miliband strike some kind of formal or informal deal with the Scottish National Party. The SNP is expected to emerge as the third largest party in Westminster following the election by taking at least 50 of the 59 seats up for grabs in Scotland.

A Labour-SNP arrangement would be awkward, to be sure. While both parties are left-of-centre, the SNP is dedicated to pulling Scotland out of the United Kingdom, making them a uncomfortable ally for anyone claiming to have the union's best interests at heart. But if Mr. Miliband's post-May 7 options are cooperation with the SNP or five more years of Conservative-led government, most expect the Labour leader to choose the former.

But pressed by a Leeds audience that appeared to have equal contempt for all three politicians thrown to it (Liberal Democrat leader Nick Clegg took the stage after Mr. Cameron and Mr. Miliband), Mr. Miliband used surprisingly strong language in ruling out the possibility of any kind of pact with the SNP.

"I am not going to have a Labour government if it means deals or coalitions with the Scottish National Party," Mr. Miliband told a questioner. The crowd didn't quite seem to believe him, so he repeated the pledge. "If the price of having a Labour government was coalition or a deal with the Scottish National Party, it's not going to happen."

While it's still possible the nationalists could decide to support a Labour minority government on a vote-by-vote basis, without any formal deal, Mr. Miliband's statement drew anger from SNP leader Nicola Sturgeon, who said she was "appalled" by what Mr. Miliband had said. The SNP have openly delighted in the clout they would wield over a possible Miliband government, with ex-leader Alex Salmond joking recently that he would be the one writing a Labour government's budgets.

"He sounded as if he was saying that he would rather see David Cameron and the Conservatives back in government than actually work with the SNP," Ms. Sturgeon told a separate BBC question-and-answer session shown Thursday in Scotland. "Now, if [Mr. Miliband] means that, then I don't think people in Scotland will ever forgive Labour for allowing the Conservatives back into office."

Nigel Farage, leader of the radical right-wing U.K. Independence Party – which is running third in nationwide polls with about 14 per cent support – said the promise meant Mr. Miliband no longer had a chance of becoming Prime Minister. He called for Labour supporters to throw their support to UKIP in close races where UKIP had a chance of beating the Conservative candidate.

But while snap polls suggested Mr. Cameron (despite his glistening upper lip) had the better performance on Thursday, the Conservative leader also seemed to complicate his chances of becoming Prime Minister during his own half hour on stage.

Pressed by the audience to explain which Conservative policies he would compromise in order to make a deal with other parties that would allow him to remain Prime Minister, Mr. Cameron said he would not lead a government that wouldn't deliver on his promise of holding an in-or-out referendum on Britain's membership in the European Union in 2017.

"Absolutely, that is a red line," he said. "I think the British people really do deserve a referendum on whether to stay in a reformed European Union, or leave. I've been very clear that I will not lead a government that does not deliver that pledge."

Like Mr. Miliband's promise not to cooperate with the SNP, Mr. Cameron's "red line" could complicate any post-election talks with the Conservatives' most obvious partner, the Liberal Democrats. The Conservatives and Liberal Democrats have worked together in a coalition government since 2010, with Mr. Cameron as Prime Minister and Mr. Clegg as Deputy Prime Minister.

Mr. Clegg seemed surprised by Mr. Cameron's referendum red line. He said that while he and Mr. Cameron agreed while in government that an EU referendum should be held if Brussels sought to take any more powers from national governments, Mr. Clegg said that threshold had not been met. Mr. Cameron, he suggested, was trying to pander to UKIP and the right-wing of the Conservative party.

The Liberal Democrats, who disappointed many of their supporters five years ago by entering into the coalition with the Conservatives, are expected to lose about half the 57 seats they currently hold. The party is nonetheless expected to be the fourth-largest in parliament.

UKIP, despite its surge in the polls, will likely win only a handful of seats under Britain's first-past-the-post electoral system. The centrist Liberal Democrats have said they won't enter a government that includes either UKIP or the SNP.

Despite the fact every poll taken so far predicts a hung parliament following the May 7 vote, both Mr. Miliband and Mr. Cameron insisted Thursday they still believed they could win an outright majority, leading to audible snickers from an incredulous audience.

"Unlike David Cameron or Ed Miliband, alas, I'm not pretending I'm going to be Prime Minister next Thursday," Mr. Clegg said to laughter when it was his turn to face the crowd. He said his own party's "red line" in any coalition-making would be increased education spending to cope with the skyrocketing costs associated with a rising number of new students.

"I think they know they're not going to be Prime Minister [without help from other parties]. I think they're not coming clean with you that they're going to have to make compromises."