Maine Counts Votes Differently. That Could Put a House Republican in Jeopardy.

CAMBRIDGE, Mass. — Only one Republican holds a seat in the House from all of New England. Other than that single district in Maine, an area of farms and fishing villages and forests that sent an electoral vote to Donald Trump in 2016, the rest of the region is a vast swath of blue.

But the fate of New England’s lone House Republican, Representative Bruce Poliquin of Maine, grew uncertain this week as the margin between Mr. Poliquin and his closest opponent remained narrow and Maine officials went on counting votes a week after they were cast. This wasn’t a recount, but a new and complex method of voting that Maine residents chose, and one that now had become the center of Mr. Poliquin’s dilemma — and ire.

By Tuesday, Mr. Poliquin, a two-term incumbent, was suing Maine’s secretary of state, calling for a halt to the counting and, in essence, asking the state to take its new voting system and toss it. Under a regular voting system, Mr. Poliquin asserted, he would have won.

Neither Mr. Poliquin nor his Democratic opponent, Jared Golden, won a majority of the votes on Election Day. No final and official tally was released, although numbers cited by the Maine Secretary of State’s office last week showed Mr. Poliquin with a narrow lead after many of the votes were counted. Under Maine’s new system, known as ranked-choice voting, since no one had an outright majority of votes, a new tally was triggered in which the second and, possibly, third choices of some voters will be added to the count.

Maine’s election was the first time that ranked-choice voting was used for all votes in a general election for federal office. (It has been used in some states for primaries and for overseas voters’ ballots, as well as in municipal elections in at least a dozen cities.)

Under the system, voters rank their top choices, instead of just picking one candidate. If no one gets more than 50 percent of peoples’ first-choice votes, the winner is selected through a series of elimination rounds: The last-place candidate is knocked off, and his or her votes are reallocated to the second choice candidate of those voters. The process continues until one of the candidates gets a majority.

Two independent candidates were the top choices for tens of thousands of voters, and those votes will be the ones in question as the count goes on. If Mr. Golden were to be a second or third choice of enough of those voters, he could ultimately win. In fact, The Bangor Daily News predicted such an outcome, based on an exit poll of 534 voters.

Maine’s Secretary of State, Matthew Dunlap, is responsible for putting the ranked-choice system into place, and his office is processing all the ballots in preparation for counting the second choices of some voters.

Advocates for ranked-choice voting — some of whom view Maine as setting an example that other states soon may follow — argue that it allows voters more freedom to express their preferences, since they can, for instance, rank a third-party candidate as their first choice, without worrying about throwing away their vote.

Mr. Poliquin, in his lawsuit, took a different view.

The complaint, in which he named himself and three voters from his district as plaintiffs, derided ranked-choice voting as an “exotic” system that undermines voting rights and violates the United States Constitution. The suit called for a permanent injunction to stop the counting of votes and to “invalidate the challenged law and vindicate Plaintiffs’ constitutional right to have federal election returns counted in accordance with traditional — and constitutional — procedures.”

After Mr. Poliquin filed his lawsuit, Mr. Golden filed a motion to intervene as a defendant in the case.

In a statement, Mr. Golden’s campaign manager, Jon Breed, essentially accused Mr. Poliquin of challenging the system after the election because he was afraid of losing.

“The candidates campaigned, and the people of Maine’s Second District voted in accordance with the law,” he said. “Any attempt by Bruce Poliquin to change the rules after votes have already been cast is an affront to the law and to the people of Maine.”

Maine voters approved ranked-choice voting in ballot measures in 2016 and 2018, and the voting method was used earlier this year in the primary elections for governor, state Legislature, United States Senate, and the House. It was also used this month in the general elections for one of the state’s federal Senate seats and for its other House seat, but, because the incumbents in those races got over 50 percent of votes in the first round, this is the only race in which votes are still being counted.

Mr. Dunlap’s communications director, Kristen Schulze Muszynski, said on Tuesday that the office was still processing ballots and was aiming to finish and declare a winner this week.

“If we receive a court order to halt the process we will review it with our legal advisers,” Ms. Muszynski said.

Mr. Poliquin represents Maine’s heavily rural Second Congressional District, which includes most of the state’s landmass and faces more economic challenges than the state’s southern, coastal areas. To be sure, there are numerous Republicans who hold major elected offices in New England other than the House — including, in Maine, Senator Susan Collins and the outgoing governor, Paul LePage.

Kate Taylor reported from Cambridge, and Liam Stack from New York.

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