Every Method Breaks Something

Condorcet found a bad case. The natural response is to look for a better method, one that always produces a sensible ranking and never misbehaves. For a century and a half people proposed methods, and other people found cases where they misbehaved.

In 1951 Kenneth Arrow turned the question round. Instead of testing methods one at a time, he wrote down what any reasonable method ought to do and asked whether anything could do all of it.

His conditions are modest, and that is the force of the result. Take any set of individual rankings of three or more options, any set at all, and produce a ranking for the group. If every single voter prefers A to B, the group should prefer A to B. The group's verdict on A against B should depend only on how voters rank A against B, and not on where they put some third option. And no single voter should be a dictator, whose ranking becomes the group's whatever anyone else thinks.

Arrow proved that no rule satisfies all of these at once. Any method that meets the first two, for every possible set of ballots, is a dictatorship.

The middle condition is the one that does the damage, and it has a clumsy name: independence of irrelevant alternatives. Its everyday form is the spoiler. A third candidate who cannot win enters a race, and the winner between the other two changes, although not one voter has changed their mind about those two. That is the group's verdict on A against B depending on C.

Plurality voting, where each voter marks one name and the most marks win, fails this condition constantly. In Florida in 2000 the two leading presidential candidates were separated by 537 votes, and a third candidate took about 97,000. Whether his presence changed the result depends on how his voters ranked the other two, which is precisely the information a single-mark ballot does not collect. Borda's points system fails it too, in a different way: moving C up or down a voter's ranking changes the gap between A's points and B's, so C can decide the contest between them. Methods built on head-to-head contests fail it whenever there is a cycle.

So far this sounds like a counsel of despair, and it is routinely quoted as one: Arrow proved that democracy is impossible, or that every voting system is as bad as every other. Neither follows.

First, the theorem is about what must happen on some pattern of ballots. It does not say how often those patterns turn up. Duncan Black had shown in 1948 that if the options can be laid out along a single line, left to right say, and every voter prefers options nearer their own favourite, then majority rule never cycles: the favourite of the median voter beats every rival. Real electorates are not that tidy, but they are often closer to it than the worst cases are.

Second, the theorem is about rankings in and a ranking out. Methods that ask for something else, a score for each candidate or approval of as many as you like, step outside its framework, and their advocates make much of that. They escape Arrow's theorem. They do not escape the next one.

In 1973 Allan Gibbard, and independently in 1975 Mark Satterthwaite, proved that any voting rule choosing among three or more possible outcomes, short of a dictatorship, will sometimes reward a voter for misrepresenting their preferences. Gibbard's version covers every kind of ballot. Tactical voting is not a defect of first past the post that a cleverer system would remove. It is a property of voting.

Third, and most usefully, the theorem changes what the argument about electoral systems is. It cannot be a search for the method with no flaws, because there is none. It has to be a choice about which flaw to accept: spoilers, or an override of an outright majority, or occasional cycles, or rewards for tactics, and in what kinds of election each one is likely to bite.

Arrow shared the Nobel prize in economics in 1972, for work that included this theorem. It is usually described as a negative result, and it is better described as a map. It tells you that every route across the territory crosses a river somewhere, and leaves you to decide which crossing you can live with.