Column: This is what democracy looks like

For the past 14 months, I’ve been following the campaign for the US presidential election with interest. As primary voting has pared down the roster of candidates, I’ve noticed a sudden rush of articles marveling at the influence of voters who made up their minds at the last minute, or imploring readers in exasperation to please consider voting for the candidate they’d actually like to see win.

In a race among many candidates, as this one was for the Democrats until about five minutes ago, it’s easy to see why voters might have trouble making up their minds—or why their votes might not reflect their true preferences. Of all the names on the ballot, you can vote for only one. If it seems from opinion polls and the votes that have been cast so far that your favorite candidate doesn’t have a chance and that your least-favorite candidate is in danger of winning, you might choose to vote for a candidate other than the one you most prefer, in the hope of securing a middling outcome over a terrible one. If the polling numbers are changing rapidly, then so might your choice of who to vote for.

But you’re not the only one who has an incentive to use such tactical thinking. Everyone else does too—including the people responding to the opinion polls that you used to decide which candidates have a chance of winning. With too many layers of gamesmanship, an election could produce a winner that satisfies no one, even when all voters act rationally in their own self-interest.

Miller’s Diary

Physics Today editor Johanna Miller reflects on the latest Search & Discovery section of the magazine, the editorial process, and life in general.

In 2018, in an effort to more accurately capture voters’ preferences, the state of Maine adopted ranked-choice voting in general elections for federal offices. Rather than having to pick a single candidate to vote for, voters can rank them all from most to least favorite. Votes are then counted through an instant-runoff process: The candidate with the fewest first-choice votes is withdrawn from the race, those votes are reallocated to the voters’ second-choice candidates, and so on until one candidate has a majority.

Several cities in the US and jurisdictions around the world also use instant-runoff voting for some elections, and there are calls to adopt it more broadly. But it’s not the only way to tally ranked-choice ballots. There’s also the Borda count, among others, in which candidates are awarded points according to their rank on each ballot (say, two points for first place, one for second, and none for third), and the candidate with the most points wins. Although not widely used in political elections, versions of the Borda count show up often in nonpolitical contexts, including the Eurovision Song Contest .

Which method of counting ranked ballots best reflects the will of the voters? It’s not clear. The PBS online show Infinite Series describes a five-way race in which four vote-counting methods yield four different winners. And they all fail to find the so-called Condorcet winner—the one candidate (who doesn’t always exist, but who does in this case) who can beat each of the other candidates in a two-way matchup.

Furthermore, each ranked-choice method has its own problems. The Borda count creates incentives for tactical voting: In a tight race between your first- and second-choice candidates, you might tip the election in favor of your favorite candidate by ranking your second-favorite candidate last. When told of his method’s vulnerability to manipulation, Jean-Charles de Borda replied, “My scheme is intended only for honest men.” (Good luck with that.)

In instant-runoff voting, on the other hand, you could imagine a race in which one candidate is everyone’s second choice, with the first-place votes divided evenly among three or four others. Intuitively, the universal runner-up might seem like a natural compromise candidate. But with no first-place votes at all, she (can I call her she?) is eliminated from the race in the first round of runoff voting.

Importantly, those difficulties are all limited to races with three or more candidates. In a two-way race, a ranked ballot is equivalent to a vote for a single candidate, and it’s simple to design a straightforward election in which the candidate who gets the most votes wins. (Or, if that’s too simple for you, you could divide the voters into groups called “states,” and you could process their votes through something called the “Electoral College.” I’m not saying you should. But you could.)

Which brings me to a 1973 theorem by (mathematically trained) philosopher Allan Gibbard. In a nutshell, Gibbard proved that it’s impossible to design a voting system for choosing among three or more candidates that doesn’t suffer from some problem or other. In most cases, that problem is vulnerability to manipulation by tactical voting: Some voters, under some circumstances, have the incentive to cast ballots that don’t reflect their sincere views. (The only nonmanipulable voting systems are so obviously undemocratic as to be nonstarters. For example, if you declare a predetermined candidate to be the winner no matter how anybody votes, then no one has any incentive to vote strategically. But I hope we can all agree that that’s not much of an election.)

Gibbard’s theorem builds on an earlier, and I think better-known, theorem by Nobel economics laureate Kenneth Arrow. The theorems are similar in flavor, in that they both show that no multicandidate election can satisfy certain seemingly desirable criteria. But there are a couple of important differences.

First, Arrow’s theorem considers only rank-order voting systems: Each voter submits a ballot ranking all the candidates, with no other information. Gibbard’s more general theorem covers all possible ways of deterministically choosing among candidates, including cardinal voting systems, in which a voter gives each candidate an independent numerical rating, like a Yelp review.

Second, whereas Gibbard showed that all election systems that are even minimally democratic are vulnerable to tactical voting, Arrow showed that they must violate a condition called independence of irrelevant alternatives , whose consequences for real-world voting are more subtle and difficult to intuit.

For both those reasons, I find Gibbard’s theorem a more powerful result than Arrow’s. If you’re interested in digging into the proofs, though, you should start with Arrow’s theorem. It’s accessible enough that I’ve seen it show up in math projects for undergraduates and even high school students, and Infinite Series presents a good sketch of it . I’m not aware of any similarly user-friendly treatments of Gibbard’s theorem, but there’s always his original paper .

The first-past-the-post system used in most US elections makes it particularly hard for third-party candidates to get off the ground, and most of our elections are quasi two-way races. So our presidential election seasons are often accompanied by laments that too many voters don’t like either of the major candidates and that there should be more options for those who don’t feel sufficiently spoken to by either the Democrats or the Republicans. (Although, in this year of all years: really?) Proponents of instant-runoff or another form of ranked-choice voting often tout their scheme of choice as the solution to the two-party system: By encouraging voters to rank their preferences honestly, they’ll make minor candidates more viable.

Compared with the existing system, ranked-choice voting schemes probably do reduce the incentive to think strategically at the ballot box. But they don’t eliminate it. As Gibbard showed, they can’t. Tactical voting will be with us always, whenever there are more than two candidates on the ballot.