Last week’s European Union elections showed that voters were deserting the traditional two parties in droves. Labour and the Conservatives polled a combined 23.2 percent of the vote excluding Northern Ireland, the lowest share for the two in any British election in recent history. Both parties were trounced by the upstart Brexit Party, which won with 31.6 percent, and the Tories finished fifth behind the pro-E.U. Liberal Democrats and the Greens.

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English voters have traditionally been willing to back minor parties in E.U. elections only to come back to their traditional parties for British parliamentary elections. Polls show this will happen again, but only slightly. According to the polls, between 50 and 70 percent of the voters who backed Brexit, the Liberal Democrats or the Greens in the E.U. election plan to vote for the same party in the next general election. And that makes the race in Peterborough a crucial test for both major parties.

The city of Peterborough voted strongly for the Brexit Party last week (38 percent) and to leave the E.U. (61 percent) in the 2016 referendum. The Peterborough constituency itself, which includes only a portion of the city, voted to leave by even higher margins. Assuming the poll data mentioned above will apply to the by-election, Brexit Party’s candidate, Mike Greene, will likely receive between 20 and 30 percent of the vote.

That normally would not be enough to win, but there are 15 candidates in the race. Some of the minor contenders, such as the candidate of the Official Monster Raving Loony Party, won’t be a factor. But most of England’s other significant major or minor parties has a candidate, and this will split the non-Brexit Party vote so significantly that Greene could slip through.

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The Labour Party’s candidate, Lisa Forbes, is probably the most likely to emerge as Greene’s primary challenger. Labour held the seat before the incumbent’s recall triggered the by-election, and it finished second in the E.U. elections here with 17 percent of the vote. Polls show that Labour will pick up roughly 20 percent of voters who backed either the Liberal Democrats or the Greens in the E.U. vote. Those two parties received 26 percent in Peterborough, suggesting that Forbes could get between 20 and 25 percent overall. That closes the gap but does not give her the lead.

The Conservatives and Liberal Democrats also have a decent shot to win. The Tories received 11 percent of the E.U. vote, but polls show between a quarter and a third of Brexit Party voters plan to vote Conservative for parliament. That puts the Tory candidate, Paul Bristow, in roughly the same position as Forbes.

The Liberal Democrats received 15 percent last week and are likely to gain votes from Green voters that will offset the voters they will lose to Labour. The Liberal Democrats, the Greens and a small third party — all of which support remaining in the E.U. — attempted to field a single candidate together, but those negotiations broke down earlier this month. Even still, pro-Remain voters could tactically vote for the best-placed Remain candidate regardless of the presence of other candidates. Should that happen to even a small degree, the Liberal Democratic candidate, Beki Sellick, will be in the mix.

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A victory by the Brexit Party or the Liberal Democrats would shock Westminster and push both major parties to clarify their stances on Brexit. The candidates to succeed outgoing Prime Minister Theresa May as the Conservative leader would be even likelier to emphasize a hard “no deal” Brexit in an attempt to woo Brexit Party voters. Meanwhile, Labour leader Jeremy Corbyn, the lifelong Euroskeptic who heads a party that is increasingly pro-Remain, will be under even more pressure to shift unambiguously in favor of holding a second referendum if Labour fails to win. This would also undercut the Liberal Democrats.

But even the fact that this is a four-way race could itself be historic. England has not experienced serious four-party politics since the period between 1918 and 1924. That experience led to the downfall of the once powerful Liberal Party and the rise of a new party, Labour, to take its place. The Peterborough by-election is shaping up to be the first shot in a multiyear battle between the dominant parties and their suddenly powerful competitors to see if history will repeat itself. No one should be confident that it won’t.

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