The votes have been counted and President Trump has moved into the White House, but the campaign to upend the Electoral College is far from over.

Lawsuits aimed at striking down the winner-take-all system and giving electors more freedom to change their votes have been in the works since Mr. Trump won the Electoral College vote but not the popular vote on Nov. 8.

The idea is not to eliminate the Electoral College, which would require a constitutional amendment, but to require states to implement a system in which electors cast ballots based on the percentage of the popular vote.

“It’s crazy that our nation’s least-democratic election is the one for president,” said Lawrence Lessig, Harvard Law School professor and founder of Equal Citizens.

The group kicked off a project Sept. 14 aimed at filing lawsuits on behalf of a Republican voter in a blue state and a Democratic voter in a red state, with the goal of overturning the winner-take-all system in time for the 2020 presidential race.

The winner-take-all system is not mandated under the Constitution — Maine and Nebraska split their electoral votes — but Mr. Lessig said it has reduced the presidential race to a fight over a “handful of battleground states instead of a national referendum.”

“Proportional allocation of electors at the state level won’t eliminate the inequality produced by the Electoral College,” Mr. Lessig said, “but it would reduce it substantially. We believe the courts should step in to reduce the inequality of the current system as much as possible.”

Would such a change have thrown the presidential race to Democrat Hillary Clinton? Not necessarily. First, there are a number of ways to divvy up votes proportionally, including by congressional district, popular vote and proportional popular vote.

Maps devised by the website 270 to Win show that Mr. Trump still would have won under two congressional district systems, as well as the “proportional popular” method, in which two electoral votes are awarded to the popular vote winner and the rest are allocated based on the percentage of the vote.

Under the “popular vote by state” plan, Mr. Trump would have had the edge in electoral votes but would have fallen short of the 270 Electoral College votes required to capture the White House. That would have thrown the decision to the Republican-dominated House.

Virginia and Minnesota introduced bills this year that would have awarded electoral votes by congressional district. Although the measures failed, both were introduced by Republicans.

Support for dividing electoral votes on a percentage basis has shifted politically over the past 20 years, depending on whose ox is being gored.

After his party’s 2012 presidential election loss, Republican National Committee Chairman Reince Priebus endorsed the idea of changing the system, which presumably would have helped nominee Mitt Romney. Some Democrats accused Mr. Priebus of election-rigging.

Since Mrs. Clinton’s defeat last year, the idea has gained steam among Democrats. Mrs. Clinton last week called for eliminating the Electoral College and told CNN that “I think it needs to be eliminated. I’d like to see us move beyond it.”

Two of the past three presidents — George W. Bush and Mr. Trump — have won the presidency even though they lost the popular vote. Both were Republicans.

Equal Citizens spokesman Ken Scudder insisted that the Equal Votes project, which has raised $65,000 in the past week, “isn’t about relitigating 2016.” He said one benefit is that a proportional allocation system could improve voter turnout in solidly red or blue states.

“Right now, there is no reason for a Republican in California to vote in the presidential election,” Mr. Scudder said, “just like there’s no reason for a Democrat in Montana to vote for president.”

Meanwhile, Equal Citizens is representing three Colorado electors in a lawsuit against Secretary of State Wayne Williams for insisting that they cast their ballots for Mrs. Clinton at the Dec. 19 ceremony at the Statehouse in Denver.

“Our objective is to affirm a constitutional principle,” Michael Baca, a 2016 Democratic elector, said in a statement. “That principle is critical to our Framers’ design for electing the president. It is critically important that the courts clarify the rights of electors so that the uncertainty that surrounded the 2016 vote does not repeat itself.”

Mr. Baca was removed as an elector during the ceremony for casting his ballot for Ohio Gov. John Kasich as part of a coordinated national campaign to deny Mr. Trump the presidency.

The Colorado complaint seeks $1 in damages from Mr. Williams for enforcing a state law requiring “faithless electors” to cast their ballots for the presidential candidate who won the state.

The lawsuit said Mr. Williams “threatened and intimidated” the electors — Polly Baca, Michael Baca and Robert Nemanich — by saying he would replace them and refer them to prosecutors for perjury charges if they violated their oath to vote for the winner, Mrs. Clinton.

Mr. Baca was replaced and referred to the Colorado attorney general after he voted for Ohio Gov. John Kasich. The other two Colorado electors ultimately voted for Mrs. Clinton.

Lynn Bartels, communications director for Mr. Williams, said the Republican secretary of state had no choice based on court rulings in the weeks prior to the ceremony.

“Secretary Williams has repeatedly pointed out that everything he did in respect to electors was ordered by a judge, either at the district or federal court level,” Ms. Bartels said. “The secretary’s goal was protecting the rights of the 2.9 million Coloradans who voted in the 2016 election. Their decision: Colorado’s nine electoral college votes were to go to Democrat Hillary Clinton.”

Four Washington state electors voted for candidates other than Mrs. Clinton as part of the “Hamilton electors” strategy, which ultimately flopped. Only two Republican electors, both in Texas, cast their ballots against Mr. Trump.

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