Of all the ways to vote, the U.S. presidential election is the worst

We need weirder math in the voting booth

Everything I thought I knew about voting is wrong.

I trace the threads of my electoral education to elementary school—hunched over onto bent arms, breathing in the smell of wooden number two pencils and eraser dust, right arms outstretched, hands curled into fists. To vote no, we’d leave our thumbs hidden, wrapped around our fingers. To vote yes, we’d raise a thumb, waiting for a teacher to gently push it back down to signal that she’d counted the vote. I don’t remember if we were explicitly taught what voting meant, but like the Pledge of Allegiance, voting was part of the cultural norm. And whether it was by a show of hands, paper ballot, or secret thumb vote, there was only one constancy: plurality. Whoever gets the most votes wins, period.

There’s only one problem with plurality voting, the same method that Americans use to determine presidential elections. Mathematically speaking, it’s just plain terrible.

“The reason,” explained William S. Zwicker, a mathematician at Union College in New York, “is you can elect a minority candidate when there’s clear public opinion in the opposite direction.”

To understand how, here’s a quick thought experiment. Imagine that you and eight of your college buddies are planning a reunion trip. Your choices—based on who owns a timeshare—include the Caribbean destinations of Jamaica and Bermuda, or skiing in Colorado. Like any good children of a democratic republic, you decide to settle this choice with an American-style plurality vote. Two of you vote to go to Jamaica, but those two would happily go to the Bermuda instead. Similarly, three of you vote to go to Bermuda, but those three would happily go the Jamaica instead. After all, we’re talking about two tropical locales—and the alternative is a snowy slope.

Chart of countries ranked by preference of vacation destinaion
Same votes but different voting system can lead to much different outcomes. PopSci

Based on plurality voting you’d be heading to the chills of Colorado, even though 55 percent of you would clearly prefer a Caribbean destination. If you view elections as a contest—something where one wins and loses—that might seem ok. But if you view voting as a tool to capture voters’ preferences, plurality voting fails.

But under a different system, you could end up in Bermuda instead. And people are voting this way in the real world. In November, Maine passed a referendum switching from plurality voting to “ranked choice,” also known as instant run off voting, for its gubernatorial, congressional, and state legislative elections.

“I stumbled across rank choice voting after experiencing two governors elections in Maine in which there were three or more candidates, and those races devolved into conversations about vote splitting,” said Kyle Bailey. Bailey spearheaded the campaign to establish rank choice voting in the state. “Maine has some serious demographic and economic issues that were taking a backseat to conversations about ‘I don’t want to waste my vote.’”

In rank choice voting, you rank candidates in your order of preference: one, two, three…up to six in Maine’s case. If no candidate wins a majority (meaning they get more than 50 percent of the vote) they drop the candidate in last place out of the running. If you voted for that candidate, they tally a vote for your second choice instead.

Under our vacation example, we’d toss out the two votes for Jamaica. Instead we’d tally their second-choice preference—Bermuda, where they could get some similar sun and surf. With five votes for Bermuda and four for Colorado, you’d better break out the swimsuit.

“It works like a run off, without the cost and delay and lower voter participation that comes with having an actual runoff election,” said Bailey.

But not everyone is a fan of rank choice voting.

“The fundamental problem [with rank choice voting] is that its non-monotonic,” said Steven Brams, a professor of politics at New York University. “By raising somebody in your rankings you can hurt them rather than help them.”

Brams cites Burlington, Vermont’s 2009 mayoral election as an example of this scenario. In that election (which used rank choice) the more popular Democratic candidate lost to a less popular progressive candidate (this article details the specifics). Exactly why the progressive candidate won—and why that outcome didn’t make most residents happy—is difficult even for mathematicians to explain. Unsurprisingly, residents voted to return to plurality voting after the election.

And then, said Brams, there’s the issue of ranking itself.

“Voters have great difficulty ranking more than four or five candidates,” said Brams. “So what do you do in a presidential primary where there are 17 Republican candidates? I consider myself an educated voter and I could not have ranked 17 candidates. It’s forcing voters to do something that they have insufficient information to do.”

It’s not that Brams thinks plurality voting is better than ranked—he doesn’t. But he thinks there’s a simpler system out there that he argues works better than instant run-off voting. Brams champions something called approval voting. In fact, he literally wrote the book on it.

To understand approval voting, imagine your office of seven employees is ordering takeout together. You have four different choices. While in a plurality vote, you can only vote for one, and in a rank choice you rank your preferences, in an approval vote you check off all the options that you find acceptable.

Ballot approval voting style.
In approval voting, voters choose all of the options that they would prefer. PopSci

The approvals are tallied, and whichever one gets the most tallies wins. Because more people approve of barbecue than any of the other options, you’ll all be dining on ribs.

“It gives voters flexibility in expressing themselves,” said Brams. “If you don’t have a clear favorite then you can indicate that by approving of more than one candidate. Or if you do have a clear favorite but you know he or she is a fringe candidate, you can protect yourself by also approving of a more viable candidate, thereby have your cake and eat it too. This ability to express oneself better in a very simple way is the key feature of approval voting.”

But what happens if there’s a tie?

“The chance of a tie in a national election is infinitesimal,” says Brams. And rules—such as in the case of a tie, a run-off election between the two using plurality voting— could be constructed to keep things running smoothly.

The biggest criticism of approval voting is that it tends to elect centrists. “My response to that,” said Brams, “is that there are candidates like Ronald Reagan, who was identified as kind of right wing, who would have won under approval voting.”

There are other potential benefits too: more candidates in the fray. For example, in this election cycle Bernie Sanders made a decision not to run as an independent because he didn’t want to be a spoiler for Hilary Clinton. New York’s Mike Bloomberg made the same call. Approval voting opens up the opportunity for third, fourth, and fifth parties—and the people who support them—to have a voice.

In 2010, Zwicker participated in an unusual vote held in Normandy, France by the economist Jean-Franc¸ois Laslier. The group of 22 mathematicians, economists, and political theorists (all well-versed in voting theory) were there to vote on which of seventeen voting rules they believed to be the best. In first place was approval voting. Instant runoff voting (described as alternative vote) came in second. Nobody voted for plurality.

But those results shouldn’t come as a surprise: the group used approval voting to tally their choice.