I just came across another apparently failed solution to the problem, which was used in Maine throughout much of the 1800s; here is the Maine high court’s account:
As originally established when Maine became a State in 1820, the Maine Constitution provided for the election of Senators, Representatives, and the Governor by a majority vote. Between 1830 and 1880, a number of elections yielded no candidate who achieved a majority vote. In those circumstances, the alternative means for election provided by the 1820 Constitution had to be utilized.For Representatives, that meant holding a series of new elections until a candidate won a majority.For Senators, that meant that the Representatives and the members of the Senate who had been elected by a majority would elect the winners by joint ballot.For Governor, it meant the House would select two of the candidates from those four candidates who received the highest number of votes and the Senate would elect the winner of those two.The result was widespread discontent — and, in 1879, threats of violence, which were quelled by the efforts of Joshua Chamberlain—caused by the expense and delay of holding repeat elections, by the election of candidates through legislative action rather than based on the will of the people, and by the claims of manipulation and allegations of self-dealing levied by opponents of the eventually-declared winners. In response to these challenges, by 1880, all three provisions had been amended to replace “majority” with “plurality” as is now found in the Constitution.
A more recent — and more defensible — statutory solution, rank-order voting, was just struck down by the Maine high court as violating this plurality requirement, though of course the Maine Constitution could be amended to remove that requirement.