It’s mathematically impossible to vote between more than two candidates.

Fair Voting Between More Than Two Candidates Is Mathematically Impossible

Fair Voting Between More Than Two Candidates Is Mathematically Impossible

The state of the universe.
Oct. 28 2016 5:52 AM

An Election Season Reminder That Voting Is Mathematically Flawed

There is no fair way of assessing a populations’ preferences when there are more than two candidates.

A man casts his ballot at polling station during New Jersey's primary elections on June 7, 2016 in Hoboken, New Jersey.
A man casts his ballot at polling station during New Jersey's primary elections on June 7 in Hoboken, New Jersey.

Eduardo Munoz Alvarez/Getty Images

If you had told me when I moved to Utah in 2013 that it would be a swing state in the next presidential election, I would have laughed and laughed. But in the midst of this not-so-funny election season, Utah is indeed making news as independent candidate Evan McMullin’s presidential campaign has gained traction here. As of Thursday, election reporting juggernaut FiveThirtyEight is currently giving him a 19.6 percent chance of winning the state’s popular vote, and a sliver of a chance of throwing this election into chaos by stealing enough electoral votes that neither Hillary Clinton nor Donald Trump gets the requisite 270 to win the race outright.

This is not an article about the intricacies of the Electoral College or why you should or should not vote for a third-party candidate this year. I assume you’ve seen enough articles like that already—or heard John Oliver passionately argue against the idea. No, in this Octavia Butler dystopia of an election season, I want to encourage you to move your frustration beyond the two-party system and the Electoral College. Because those aren’t the only factors that are against us—the very fabric of mathematics itself falls down when it comes to trying to accurately assess voter preferences.


It’s not this election that’s rigged. It’s the entire idea of trying to vote.

If there are exactly two choices on a ballot and every voter votes for one of them, it’s easy to find the winner: Count the votes, and whoever has more triumphs. The winner must have more than half of the votes, so a majority of voters will have their preferred candidate win. That is about as fair as it gets.

But with three or more options, all bets are off. It’s not just a problem with the frequently used plurality-rule system—in which a candidate only needs more votes than others rather than a majority of votes to win. There is no system that can fairly decide the winner of an election with more than two candidates. It simply doesn't exist.

American voters are quite familiar with our system’s flaws vis-a-vis third parties. The pet examples are Ross Perot and Ralph Nader as “spoiler” candidates in the 1992/1996 and 2000 presidential elections—Nader in particular was blamed for Al Gore’s loss in 2000. The thinking goes that almost all Nader supporters would prefer Gore to George W. Bush, and in some states, notoriously Florida, the margin between Bush and Gore was smaller than Nader’s vote percentage. If Nader’s votes had gone to Gore, the outcome of the election would have been different. But because we do not have a preferential voting system, there was no way for Nader supporters to indicate that if their candidate lost, they would prefer Gore to Bush.


Preferential voting systems seek to solve the Nader problem by allowing voters to provide a ranked list of their preferences. There are many possible ways to allocate votes based on preferential ordering. One, called instant runoff voting, eliminates the candidate with the fewest first-place votes. Then the race is repeated for the remaining candidates. As a simplistic example, say 25 percent of voters had the preference rank ABC, 35 percent voted BCA, and 40 percent voted CAB. Candidate A got the fewest first-place votes, so we take A away entirely. Now 60 percent of people have the preference order BC over 40 percent for CB, so B wins. This is frustrating because more people ranked candidate C as their first choice than any other first choice. And going back to the original rankings shows that if C wasn’t going to win, more people would have preferred A to B. But the runoff system sticks us with candidate B.

Another way to score that election is called a Borda count, where candidates get more votes when they are placed higher by more people. In this example, we will give each candidate 3 points for every first-place vote, 2 points for every second-place vote, and 1 point for every third-place vote. Here, we’ll pretend we have 100 voters to make the numbers nice and round. A gets 75 points from the voters who ranked them ABC, 35 from the BCA voters, and 80 from the CAB voters for a grand total of 190 points. Candidate B gets 195 points and C gets 215, so C is elected. That seems fair, right?

But what if things were a bit more complicated—instead of 40 percent of voters having the preference CAB, 30 percent prefer CBA and 10 percent prefer CAB? Voters’ relative preference between B and C has not changed but now C gets 210 points and B gets 220, so B is elected instead of C.

Both of these methods of deciding the winner in this election seem flawed. Is this just because we haven’t figured out the right preferential voting system? That would be nice, but the answer instead is that we’re running into a mathematical obstacle called Arrow’s impossibility theorem. It states that any preferential voting system must fail some fairly basic test of fairness. The exact criteria are a bit technical, but the big sticking point is called the independence of irrelevant alternatives: If no one’s relative preference between two candidates changes, then the relative rank of the two candidates should not change. But in the Borda count example above, changing voters’ preferences between A and B changed the relative ranks of B and C even though C wasn’t involved in the preference changes. In 1951, Kenneth Arrow proved that problems like that are a feature of all preferential voting systems.


In my original example, 65 percent of people preferred candidate A to candidate B, 60 percent preferred candidate B to candidate C, and 75 percent preferred candidate C to candidate A. If you’ve played rock, paper, scissors, you understand the problem: No single move is always going to win, it all hinges on what you’re up against. (Mathematicians call preferences like this nontransitive.) No matter what system we use to choose between A, B, and C, a majority of voters will prefer a different candidate to the one who is chosen. If C is our choice, most people will wish B had been elected instead. If it’s B, most people will prefer A. If it’s A, most people will prefer C. In some elections, voters’ preferences will be better-behaved, with one candidate who beats every other candidate head-to-head. But whenever the voters’ preferences are non-transitive, there’s no way to be sure our choice of candidate doesn’t violate one of the fairness criteria. (For a more in-depth explanation and explicit examples of different voting systems and their foibles, I highly recommend “There is no such thing as public opinion,” a chapter in Jordan Ellenberg’s book How Not to Be Wrong.)

When I suggested this story to my editor, she asked if Arrow’s theorem is how we can end up with unpopular candidates winning major party nominations. After all, the primaries have a greater number of viable candidates than the general elections do, so they’re especially susceptible to problems arising from Arrow’s theorem. Frustratingly, even though we know this is a mathematical possibility, we don’t know how often real voters’ preferences are nontransitive and therefore problematic. Political polls don’t ask respondents to rank candidates, so we just don’t have the data. In 2012, researchers at Princeton University ran several polls to compare different voting methods on real data from voters and found that there was usually a winner who did satisfy the basic fairness criteria. As far as I can tell, there are no similar polls from this year’s primary season or other political contests that would allow us to assess this.

Of course, preferential voting is not our only option. In addition to the plurality system we use, there is approval voting where people can vote for, or approve, as many people as they want and the person who gets the most approvals wins. There are also systems where you rate candidates—5 stars for A, 4 for B, and 2 for C, perhaps. Arrow’s impossibility theorem does not apply to those systems, though they have their vulnerabilities as well, usually on the side of the voter. Someone could decide not to “approve” a candidate they thought was just fine because it could prevent their favorite candidate from winning; someone could rate all opposing candidates one out of five stars to boost their own candidate. All voting systems are subject to some sort of manipulation.

And maybe that’s the point. We have the voting system we have, and each of us works within that system to make our voice heard and elect the candidates we think are most deserving. Are there problems? Of course. But until we turn elections over to all-powerful, benevolent mind-reading robots, there will never be a voting system that perfectly reflects the will of the people, whatever that means.

Now take a deep breath and go vote!