The Science of Voting

by Brian Dunning
October 25, 2011

In the 1969 film Putney Swope, members of the board of executives were prohibited from voting for themselves, so they all voted for the one board member they were sure nobody else would vote for. Ergo, this free, democratic election produced a chairman that no voter wanted.

In a perfect democracy, everyone gets an equal opportunity to vote, and equal representation. Therefore, we hold elections to let everyone have their say, to either vote representatives into office, or to enact certain laws. It's a fine idea, and most countries do their level best to implement such systems. Some voters take advantage of it, and some choose apathy and don't vote. Some try to anticipate what other voters might do, and cast a vote in an unexpected direction not to vote for a candidate, but to affect another candidate's chances. This is what went wrong in Putney Swope: each voter cast a throwaway vote hoping to improve his own chances. In most elections, everyone has the right to do any of these things; the election is theirs, and theirs to decide. But what many of them might not know is that virtually any electoral process is flawed. Some outcomes are surprising. There are a number of different circumstances in which the candidate most desired does not win.

Democratic voting is only simple if there are just two candidates, or if it's a Yes or No vote. In those cases, any attempt to vote tactically or to create a voting block — casting votes that don't represent your preference — work against you. What we're talking about today are elections where there are three or more candidates. And the idea that all the various systems for running such elections are flawed (subject to results that do not represent the group's preference) is not just a whim or a crazy opinion of mine. It's proven by Arrow's Impossibility Theorem, named for the economist Kenneth Arrow, winner of the 1972 Prize in Economic Sciences in Memory of Alfred Nobel and the 2004 National Medal of Science. He proved it in 1951 with his Ph.D. thesis at Columbia University.

Arrow's theorem can be simplified into one clear statement: that no fair rank-based voting system exists when there are three or more candidates. To this I ask: What do you mean by fair? That's the key to Arrow's theorem. It holds true, depending on a rigid definition of fair that must satisfy three criteria:

1. If every individual prefers X to Y, then the group prefers X to Y.

2. If every voter's preference of X over Y stays the same, then the group's preference of X to Y stays the same, even if other preferences change: such as Y to Z, or Z to X.

3. There can be no dictator, as Arrow called him; a single voter with the power to dictate the group's preference.

Arrow's theorem applies to election systems that require voters to rank the candidates. This is the case with most voting systems worldwide. Typically, when you vote, you mark an X in the box for one candidate. That's a ranking; you've ranked that candidate first. Arrow's theorem applies to these simple ranking systems, but its richest mathematical complexities come from systems with three or more candidates and the voters rank all candidates in order of preference. This isn't used in many real-world elections, but it's the theoretical basis for social choice theory.

Update: Since this episode came out, Ranked Choice Voting (RCV) has become extremely popular. As a rank-based voting system, RCV also fails Arrow's Impossibility Theorem, though its built-in runoff can improve its outcomes. But as it can sometimes produce an unfavored candidate (dramatically illustrated in the popular Radiolab episode), it's clear that RCV retains the same fundamental flaws that the economists have pointed out, and can still fail to produce the Condorcet winner (see below). Being popular on the Internet does not equal being mathematically sound. See the references at the bottom for more information. —BD

The ideal outcome in any election is to choose what's called the Condorcet winner. A Condorcet winner, named after the French mathematician and political scientist, is the candidate who would beat all other candidates in a simple two-man majority race. There isn't always a Condorcet winner in every election, but there usually is. The most common voting system is plurality voting, where the candidate with the largest number of votes wins. However, there are numerous situations in which the winner of a plurality vote is not the Condorcet winner who should have been elected. This is most often seen in a vote-splitting situation, where there are two similar candidates and one oddball candidate. One of the similar candidates is often the Condorcet winner; but because of their similarity, party votes are often split between them, and the oddball candidate wins. This is the most obvious failure of election systems, and it's exactly what Kenneth Arrow was talking about.

Experiments conducted by Don Saari, professor of mathematics and economics at the University of California Irvine, highlighted these failures of existing plurality systems. In one test, he had voters rank beer, wine, and milk in their order of preference. They all did so. But then there are different things you can do with those results. Saari made it a simple plurality election, giving a vote to each first-ranked item, and milk won; but he found that a majority of the voters would have preferred either beer or wine to it. The net result was that Saari proved that some system could be designed to select any desired outcome you want, given the same set of votes to work from.

This inherent tendency for voting to fail is called the voting paradox, also described by Condorcet. Yet, it's the system that virtually all nations rely upon for most or all of their elections. Something's broken somewhere.

Perhaps the most interesting solution to the election problem is called lottery voting, or the random ballot. In this type of election, every voter would cast his vote; but the winner would be determined not by a count, but by a single ballot selected at random from the total. Although at first it sounds outrageous, it actually has serious benefits. First, it renders irrelevant all attempts at strategic voting. Each voter knows that if his ballot is selected, the candidate he wrote will be the winner. There is no upside at all to trying to split the vote, or to vote for your second choice knowing he has a better chance of winning. Lottery voting is one of the only systems in which the voter's best strategy is always to vote for his most preferred candidate.

In conventional election systems, oddball candidates that may only have 5% of the vote do not have any realistic chance of winning the election, particularly in an electoral college system like United States Presidential elections where they will always receive 0 electoral votes. But in lottery voting, these candidates would actually have a 5% chance of winning.

Another interesting alternative is range voting. The best example of range voting is the scoring at athletic events where each judge holds up a scorecard. Every voter gives a number to each candidate, say 0 to 10, and each candidate's total scores are averaged. Range voting has a lot of benefits that are attractive to voters. You can give everyone a 0, or you can give everyone a 10. You can express your thoughts about one or more candidates without wasting your vote, and still be able to give a high score to your preferred candidate. If there's a candidate you're not familiar with, you don't have to give any number to them, and you will not affect their average.

Significantly, every candidate has a realistic chance of winning in range voting. Voters don't need to vote for a potentially less-favored candidate simply because he has a better chance to win than their favorite candidate. For this reason, range voting is often touted by supporters of minority political parties. In fact, range voting often confers upon minority candidates a benefit called the nursery effect. In experiments, it turns out that people will give either an honest or a median vote to candidates who are relatively unknown; while at the same time being more likely to give the lowest votes to an opposing-party candidate that they don't like, but who may be far more experienced than the unknown candidate. Thus range voting can give bestow unearned support onto minority candidates.

But some in the major political parties advocate for it as well. A favorite example of US Democrats is the 2000 Presidential election, won by the Republicans after recounts of the Florida vote. Had range voting been in effect, Florida Democrats could have expressed their support for both Al Gore and Ralph Nader, giving Nader the Florida support he'd earned but without taking any crucial support away from Gore, that many say cost him the election. On the flip side, Republicans who gave their vote to Ross Perot in 1992 need not have been forced to choose between him and George H.W. Bush: They could have given both high scores, and Bush probably would have beaten Bill Clinton. Range voting eliminates both strategic voting and vote splitting, and in simulations, always elects the Condorcet winner.

But going back to Arrow's theorem, this should be an impossibility. Range voting appears to be fair to all the voters and to all the candidates. Therefore, it would seem to be a violation of Arrow's theorem. But it turns out that it's not. The reason is because range voting is a numerical rating system, whereas Arrow's theorem applies to rank orderings. When range voting, you can give the same score to multiple candidates; you do not have to rank them in order. There is no requirement to always prefer one candidate over the other, so it's a different type of system than those to which Arrow's theorem applies.

A variant of range voting is called approval voting, where you vote either for or against each candidate, but you can vote for as many as you like; even all or none. It's basically range voting with only two choices, 0 or 1. Approval voting also avoids the pitfalls of Arrow's theorem because it does not require ranking. It's simpler than range voting; and in both real-world and simulation examples, it selects the Condorcet winner virtually every time that one exists, in contrast to plurality voting which fails frequently.

Repairing our election methodology is, undoubtedly, a process fraught with political pitfalls, but I'll leave those to the political bloggers to hash out. The science says to listen a little more closely to Kenneth Arrow, and abandon rank-based voting systems. Change to any system that allows voters to express their preferences without requiring them to force-rank one candidate as the best, and the winner most approved of by the group as a whole will be elected far more often than not.

Correction: An earlier version of this erroneously referred to the Prize in Economic Sciences in Memory of Alfred Nobel as the Nobel Prize in Economics. —BD

By Brian Dunning
Follow @BrianDunning