Late last year, in the messy aftermath of Maine’s 2nd Congressional District election, Nathan Tefft, then the chair of economics at Bates College, refuted a claim by two-term U.S. Rep. Bruce Poliquin that ballots in the ranked-choice votes election he had just lost had been tabulated using artificial intelligence or passed through an inscrutable “black box.” In a traditional election, Poliquin maintained, he would have beaten state Senator Jared Golden. Instead, the results showed him losing by 1 percentage point. Tefft ran the publicly available numbers through Python, a data analysis program, and got the exact same results reported by the state.

“Yeah, it’s just math,” he told Maine Public Radio. “There’s no sort of statistical analysis. There’s no prediction involved, which is necessarily a part of artificial intelligence, by the way.”

Tefft, to the delight of Democrats, took Poliquin’s claim at face value. But Poliquin wasn’t literally claiming that the voting system had been taken over by robots or some such thing. He was saying, in effect, that it wasn’t fair. And in mathematical terms, that’s an irrefutable possibility.

Ranked-choice is one of dozens of possible selection systems that the social scientist Kenneth Arrow considered before coming to the conclusion in 1951 that none of them are immune to outcomes that appear to defy common sense. These paradoxes even happen in the best-regarded systems. For an idea of how that works, consider that a candidate who would beat every other candidate in individual head-to-head elections, known as the “Condorcet winner,” might lose an election when facing all of them at once. Similarly a group of diners forced to choose a single entree might pick salmon over chicken and chicken over beef, but then pick beef over salmon without any of them changing their minds.

It’s a different majority in each case, but that doesn’t make the paradox any less grating.

In Maine, discussion has been limited to ranked-choice and plurality voting; the new and the old. And maybe that’s why it’s such a contentious subject.

In a ranked-choice election, voters mark candidates in order of preference — their most prefered candidate as “first choice,” the next “second choice,” and so on. First choice votes are tallied and if no candidate wins a majority, the candidate with the fewest first-choice votes is eliminated and their second-choice votes are distributed to the remaining candidates. This continues until someone has a majority.

Supporters say this allows voters to cast ballots for the candidate they truly want to win and eliminates the phenonmenon of having to hold one’s nose and vote for a less appealing candidate who is thought to stand a chance. (To show the breadth of opinion about how such things should work, one popular idea in selection theory holds that if 49 percent of people strongly favor choice A?and 51 percent weakly favor choice B, choice A should win.) Opponents of ranked-choice believe the simplicity of plurality voting keeps everyone on equal footing, and that ranked-choice actually disenfranchises voters and leads to mathematically unacceptable results.

Last month, the conservative Maine Heritage Policy Center used Maine’s 2018 2nd District race in a report titled “A False Majority: The Failed Experiment of Ranked-Choice Voting” to show how ranked-choice failed to live up to its basic premise of electing the candidate favored by a majority of voters.

The 2nd District race was the first ranked-choice contest in Maine to go beyond the first round. Two independent candidates were eliminated and their second and third choice votes were reallocated to the frontrunners. Golden won with an official share of 50.5 percent to Poliquin’s 49.5 percent. But the percentages referred only to the ballots in the final count. As it turned out, enough ballots were “exhausted” between the first and second round of counting — meaning those voters only ranked candidates who were eliminated — that Golden’s share could have been accurately described as 49.2 percent of the total ballots cast on Election Day.

“In examining 96 ranked-choice voting races from across the country where additional rounds of tabulation were necessary to declare a winner,” co-authors, Adam Crepeau and Liam Sigaud wrote, “the Maine Heritage Policy Center concludes that the eventual winner failed to receive a true majority 61 percent of the time.”

“That’s kind of the only way it could really work,” Kristen Muszynski, spokeswoman for the Maine Secretary of State’s Office, told me when I?asked her about the conflicting interpretations of a majority. “There has to be a winner, so you’re winnowing it down right from the beginning … if everybody ranked all the way through, it would be the total ballots counted.”

The MHPC report describes the instant-run-off style of ranked-choice voting entails “a much more complicated — and somewhat artificial — decision” because the outcome of the first round is not known before the votes for the next round have been cast. The problem with ranking, according to the report, is that it requires voters to retain large amounts of information about candidates’ various positions. “The fact is that most Maine voters, like most voters in any election, do not follow political races closely enough to meaningfully rank candidates in contests with more than three or four candidates,” Crepeau and Sigaud wrote. “Giving knowledgeable voters more electoral influence may be defensible as a matter of political philosophy, but it is surely not the intent behind Maine’s adoption of ranked-choice voting.”

I put the question of the “false majority” to Matt Dube, an assistant professor in computer information systems at the University of Maine Augusta.

“I can see arguments on both sides of it,” he said. “But in theory, people who are not willing to rank that person are probably not going to be motivated to go out and vote for that person in a run-off either.” Dube specializes in electoral maps, so he’s familiar with applications of science to politics. “From that sort of anecdotal perspective,” he said, “it really wouldn’t matter functionally, [though] I can see the mathematical difference in the procedure.”

Muszynski expressed a similar feeling. “It’s basically the same thing,” she said. “You didn’t vote for the person who won.”

And what about the difficulty of ranking multiple candidates? Haven’t we been ranking things since the beginning of time? Kati Corlew, an assistant professor of psychology at UMA, said the confusion over ranked-choice voting probably has more to do with unfamiliarity.

“My guess is that the largest component in any perception of unfairness comes from the simple fact of this being a change to a new system,” she said. “I think when there’s a change to the rules, there’s a question in people’s minds of, ‘Is this a moral change, is this an ethical change?’”

The MHPC report describes a variety of mistakes voters can make if they misunderstand the ballot instructions or apply a faulty strategy to their choices. Giving multiple candidates the same rank (overvoting) or skipping over ranks (undervoting) can limit the life of the ballot through rounds of elimination. A voter who gives the same candidate multiple ranks (perhaps hoping to send more power their candidate’s way or thinking they will be able to re-enter the race in a later round as a wild card — they can’t), would similarly have screwed up.

The effect of exhausted ballots on the amount needed for a majority is shown, though never acknowleged, in the Secretary of State’s own video “explainer” on ranked-choice voting, in which a simulated four-way race ends with the winner receiving 14 of 30 ballots cast at the beginning of the election.

Jesse Clark, a Ph.D. candidate in the Political Science Department at MIT, recently conducted an online experiment involving a hypothetical congressional race based on Maine’s 2nd District. Half of the participants were assigned ranked-choice ballots. The other half got plurality ballots.

“People weren’t as satisfied with ranked-choice voting, he said. “They thought it was more confusing and that it would hurt their candidates.” Clark said he doesn’t know exactly why, but he suggested a few possibilities, including the “winner effect,” in which people are more likely to trust a system that produces the results they want. “That would be my guess as to why you’re seeing Republicans in Maine less satisfied and less confident with ranked-choice voting than Democrats right now, because Poliquin lost.

“The other explanation is: people know exactly what they’re looking at. ‘Hey that’s ranked-choice voting. I’ve seen that in the news. This weird state in the north has done it, and it’s a Democrat thing; I don’t like it.’”

So far, ranked-choice voting in Maine unquestionably has been a Democrat thing. It was first proposed in Maine in 2001 but really gained traction among left-leaning voters after the 2010 gubernatorial election, when Republican Paul LePage was elected with 37.6 percent of the popular vote. In the four-way race, LePage narrowly defeated Independent Eliot Cutler, who, in a twist on the usual spoiler narrative, got more votes than Democratic nominee Libby Mitchell. (Shawn Moody, a conservative Independent got 5 percent of the vote.) Democrats and left-leaning independents each blamed the other, or themselves, for LePage getting elected. Cutler returned in 2014, and though he only got 8.4 percent of the popular vote, the race was close enough that Democrats blamed him again for the re-election of LePage, who this time took a more convincing plurality of 48.2 percent. An analysis by the Bangor Daily News concluded that LePage probably would have won anyway.

In 2016, Maine voters narrowly approved a ballot measure to adopt ranked-choice voting for gubernatorial, legislative and congressional elections. Over the next two years, the system was rejected by legislators, went before the state Supreme Judicial Court and to another referendum. In 2018, it was rolled out for state primaries and federal congressional elections alongside traditional plurality ballots for other contests. The 2nd District race last fall was the first time a major ranked-choice race went to elimination rounds.

Poliquin, in an attempt to hold onto his seat, sued unsuccessfully over the constitutionality of ranked-choice voting and at one point demanded that he be declared the winner because he had won a plurality of first-place votes.

“Nobody’s tested it experimentally, whether a first-place vote is the same as a plurality vote,” Clark said. “When you see that claim, they’re assuming that they implicitly are, but nobody’s actually tested it. Are people really behaving the same way? We don’t know.”

Likewise, he said, Democrats and Cutler supporters in 2010 each assumed their candidate would have won if the other had not run. But this, he said, presumes that voters and candidates would have behaved the same way.

Maine is on track to become the first state to approve ranked-choice voting for presidential elections, and for now, it looks like ranked-choice is here to stay. But how fair is it? And is it even possible to know, in objective terms, whether one voting system reflects the desire of voters better than another?

Mathematicians add it up

In the summer of 2010, a group of mathematicians got together at the Chateau du Baffy in Normandy, France, a 15-minute drive from the D-Day beaches, to elect “the best voting procedure.” The list of contenders included some 20 systems resembling, in name, a catalog of card games or chess openings: Kemeny, Top Cycle, Majority Judgment, Range Voting, Borda, Black, Uncovered Set, Lexmin, Fishburn, etc. Plurality was represented, as was the “alternative vote,” which we know as ranked-choice.

Remarkably, as some of the participants pointed out in notes from the event, there was no debate over which voting system was best suited for choosing the best voting system. All easily agreed to use “approval voting,” in which voters mark any of the choices they approve of, and the winner is the one with the most votes. Approval voting also won the elections, which is not as suspicious as it sounds.

The participants all were familiar with the various systems and arrived with personal biases on the question of which produced the “outcome with the greatest aggregate utility,” but they agreed that approval voting, despite some flaws, was useful and simple to understand. Accordingly, it got a 68-percent approval rating. Three rule sets failed to get any approval votes. Jean-Francois Laslier, who ran the mini-election, highlighted one of them when he wrote about the workshop, adopting the delivery of an Academy Award ceremony for his title: “And the Loser Is … Plurality Voting.”

The Alternative Vote (ranked-choice) got a 45.5-percent approval rating, coming in second place.

In politics, math only goes so far. Experts can demonstrate that some voting systems are inherently fairer than others, but there’s still the matter of voter confidence. Historically, Jesse Clark said, that dips after a change is introduced to an established voting system. Voters also initially tend to back off, but they eventually return.

“Right now we’re in that slump where people are still learning about the system, but chances are they’ll bounce back to where they were before,” he said. “I’d take any study that comes out soon, including my own, with a big grain of salt.”