The Voting Puzzle

Coming to grips with the mathematical obstacles to fair elections
May 2, 2016


Is there such a thing as a better method of choosing our leaders? It’s a question that pops up among the general public whenever we slog through a presidential election. But students ponder it every year in the perennially popular Math of Social Choice class, offered for non-math majors in the School of Arts and Sciences.

For starters, imagine that instead of casting a vote for Donald Trump, Ted Cruz or Marco Rubio, voters had gotten the chance to rank the Republican frontrunners in order of preference. Such a ballot might have revealed that even though Trump had the most first-place votes, the majority of voters ranked him as their least favorite candidate. It was true at the time: According to polling data, most Republicans would have picked Cruz or Rubio over Trump in a head-to-head matchup. Yet when only first-place votes count, Trump wins.

Seem unfair? That’s a typical presidential primary election for you.

In fact, the whole process of choosing a president is rife with what mathematicians consider inequity. “The U.S. election method is extremely messy and complicated,” says Christoph Börgers, a Tufts professor of mathematics who wrote the class’s textbook, Mathematics of Social Choice: Voting, Compensation and Division (SIAM, 2009). “It violates almost any fairness criterion you could think of.”

He’s not just talking about dodgy campaign finances or gerrymandering: It’s the fairness of voting methods themselves that concern him. Over the centuries, mathematicians have come up with ways to think about what would constitute fairness in an election procedure.

One of the many approaches is what’s called the majority criterion, which states that a candidate who gets more than half of the first-place votes should always be the winner. A reasonable requirement, right? Yet presidential voting can fail that test, as in 1876, when Samuel Tilden had 51 percent of the popular vote but still lost to Rutherford B. Hayes, thanks to the electoral college, or when Al Gore had more popular votes than George W. Bush in 2000.

The presidential election also violates the “one person, one vote” principle. “The effect of voter X changing his or her mind should be precisely the same as the effect of voter Y changing his or her mind in the same way,” says Börgers. Yet, he points out, “how important your vote is depends on which state you live in.” For example, if you lived in Massachusetts in 2000, your vote for Ralph Nader meant a lot less than if you lived in Florida, where it might have given the electoral college to Bush.

The Most Despised Candidate

But even if we decided to get rid of the electoral college and revamp the whole system, we’d still have problems. Over the years, there have been several different proposals for change; evaluating those options mathematically is a foundation of the class.

Some options would preserve our current plurality method, a “first past the post” way of tallying. However, that method only takes into account a voter’s top choice, not his or her least favorite choice. That’s why University of California mathematician Donald Saari calls plurality voting “the only procedure that will elect someone who’s despised by almost two-thirds of the voters.”

Then there is the “Borda count” method, today often used for sports awards, including Major League Baseball’s M.V.P. title and football’s Heisman Trophy. Candidates earn a certain number of points for each first-place vote they get, with fewer points for each second-place vote, and so on. The idea behind the Borda count method is to elect a good compromise candidate, with the notion that not being loathed by voters is just as important as being adored.

But Borda also violates the majority criterion. Even if more than half of voters thought Derek Jeter was the most valuable player, he could potentially lose in a Borda count. Perhaps that is why only a few governments, such as the Pacific island nation of Kiribati, use it in their government elections.

“Instant runoff” voting, on the other hand, does elect the majority winner. This is the method that’s used to choose the Best Picture winner for the Academy Awards. Each voter ranks the nominated films in order of preference. After the initial tally, the film with the least first-place votes is tossed out, and its votes are reassigned to each voter’s second choice. This goes on until one film gets a majority of votes, allowing the Academy to rest assured this year that Spotlight was indeed the film preferred by the widest consensus of voters.

Splitting the Vote

Unfortunately, though, instant runoff runs afoul of another fairness measure, put forward by the Marquis de Condorcet, an 18th-century French philosopher and a friend of Thomas Jefferson’s. According to the Condorcet criterion, if candidate A would beat every other candidate in a one-on-one contest, then A should win the election. But that wouldn’t necessarily happen in an instant runoff. And that’s just not kosher, at least according to the Condorcet approach.

“This is what splitting the vote is about,” says lecturer Linda Garant, who has taught many semesters of Math of Social Choice. “It happens all the time. And it’s what gets everybody up in arms. It makes people accuse other people of wasting their vote.”  

Advocates often propose changing voting methods for elections in the United States, and occasionally they get a few takers. Burlington, Vermont, tried instant runoff voting—like the Academy Awards—in its mayoral races, and in 2006, citizens were generally satisfied. But in the 2009 election, the new mayor didn’t have the most first-place votes, and wouldn’t have beat the others head to head. Soon after, disgruntled voters repealed the instant runoff method.

“If all of Burlington had taken our course, they would not have been shocked to see this happen,” says Garant.

It brings home the point that these sometimes centuries-old measures of fairness reviewed in class are not just fussy math exercises. “I’ll say to the students, Do you think that’s a reasonable criterion? Does this matter to you? And hopefully by the end of the course I’ve convinced them you can’t have all the criteria you want,” Garant says.

If you’re wondering what method would meet every fairness criterion, you can stop looking: Nobel Prize-winning economist Kenneth Arrow proved, in his impossibility theorem, that there isn’t one.

For his part, Börgers is a fan of a relatively young method that has an Internet following: the “Schulze Beatpath” method. “It’s a little complicated,” Börgers admits, but it meets several of the fairness criteria. Börgers has persuaded his colleagues in Tufts’ math department to use it for some of their own deliberations, such as deciding which fields they should prioritize when recruiting new assistant professors.

Of course, our presidential election system won’t be replaced any time soon, even though it probably should be. After all, “we can’t even get plurality voting working,” Garant says. “Look at the election with Gore and Bush and the hanging chads. We have to get the voting mechanism—the logistics of casting your vote—to be better before people will care about ranking their vote.” Schulze Beatpath? Maybe for Election 2070.

Julie Flaherty can be reached at

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