What are the disadvantages?

Opponents argue that because it is more complex, ranked-choice voting depresses turnout and leads to more errors because it can be confusing. In a rural state like Maine, where half of the communities count ballots by hand, ranked voting could also lead to more errors by the people doing the counting. Opponents argue further that the voting method can skew the results toward someone not favored by the majority, and that it can be costly.

Why did Maine voters approve ranked-choice voting?

It was close — 52 percent in favor to 48 percent against. But a slim majority seemed to want change, and ranked-choice voting had been successful in two mayoral races in Portland. Last year’s measure was put into motion in 2014, but the campaign coincided with rising voter anger over Mr. LePage, who has repeatedly embarrassed the state with contentious comments.

Mr. LePage was twice elected in three-way races with less than a majority of the vote, which may have given voters an added incentive to approve this measure. The state’s Democratic Party supported it, though ranked-choice voting does not favor one party or another. It also earned the support of many independents and the League of Women Voters of Maine.

How has it worked elsewhere?

Ranked-choice voting is used in Ireland and Australia in national elections, and it is used to pick the Oscar nominees for best picture. But only 11 American cities, all of them liberal, have instituted it, so their experience may be too limited to draw firm conclusions.

Here in the United States, it generally appears to be working. But there is some question about whether it decreases turnout. Some studies say it has; others say that when compared with the current system in most cities, it has not. A study of the 2014 mayoral primary in Minneapolis found that voters who were more affluent and white turned out at a higher rate and completed their ballots more accurately than minorities and those in low-income areas.

Howard Dean, a Democrat and former governor of Vermont, favors ranked voting, saying that without a majority vote, “you can’t hold the powerful accountable.” But Gov. Jerry Brown of California, also a Democrat, vetoed a bill in September that would have expanded it in a number of jurisdictions. He called ranked voting “overly complicated and confusing.”

Ranked voting has certainly led to some unexpected results. In the mayor’s race in Oakland, Calif., in 2010, Jean Quan, who came in second in the initial round of counting, had strategically campaigned to be everyone’s second or third choice. In subsequent rounds, she surged to victory on the strength of those second- and third-place votes after other candidates were eliminated. Supporters of the candidate who had come in first in the initial round accused Ms. Quan of gaming the system.