The Majority That Goes in a Circle

Take three voters and three options, and let each voter rank them. The first prefers A to B to C. The second prefers B to C to A. The third prefers C to A to B.

Now ask the majority. Between A and B, the first and third voters prefer A, so A wins two to one. Between B and C, the first and second prefer B, so B wins two to one. Between C and A, the second and third prefer C, so C wins two to one.

The majority prefers A to B, B to C, and C to A. There is no top. Whichever option you announce as the will of the people, a majority prefers something else to it.

This was published in 1785 by the Marquis de Condorcet, in an essay on applying probability to decisions taken by majority vote, and it is worth being exact about what it shows. It does not show that voters are confused. Each of the three has a perfectly orderly ranking, the kind economists call transitive: if you prefer A to B and B to C, you prefer A to C. Every individual passes that test. The majority fails it. The incoherence is not in anybody; it appears when you add them up.

The obvious defence is that the example is contrived. It is small, but it is not contrived. Each voter's ranking is a rotation of the others', and three blocs, each with a different favourite and a different order for the rest, are all it takes. Nothing about it requires bad faith or confusion.

Condorcet had a rival proposal in view. Jean-Charles de Borda, a naval officer and mathematician at the Academy of Sciences, had suggested scoring rankings by position: with three options, two points for a first place, one for a second, none for a last. Borda's method always produces a winner, which is its attraction. Condorcet's objection was that it can produce the wrong one.

Here is the kind of case he meant. Five voters. Three rank A, then B, then C. Two rank B, then C, then A. A is the first choice of an outright majority. Now score it. A gets two points from each of three voters and nothing from the other two, six in all. B gets one point from each of the three and two from each of the two, seven. Borda's method elects B, over an option that more than half of the electorate put first.

Condorcet's own criterion was different: the winner should be the option that beats every other in a head-to-head majority vote. That is now called a Condorcet winner, and it has a strong claim wherever it exists. The paradox is the discovery that it need not exist.

So the two men had found, between them, the shape of the whole subject. One method always gives an answer and can override a majority. The other respects the majority whenever the majority has a coherent view, and sometimes it has none.

The practical consequence is that with three or more options, the counting method is not a neutral device for reading off what voters want. It contributes to the result. Burlington, Vermont, gave a clean demonstration in 2009, when it elected a mayor using ranked ballots counted by instant runoff. Kurt Wright led on first preferences. Bob Kiss won the runoff count. Andy Montroll, eliminated before the final round, would have beaten each of them in a head-to-head vote, as the published ballots showed. Three defensible methods, three different winners, from one pile of paper. The city voted to repeal the system the following year.

A cycle also hands a quiet kind of power to whoever controls the agenda. Suppose a committee votes on options two at a time, the winner going forward to meet the next. In a three-way cycle the option introduced last always wins, because it meets whichever option beat the other one, and it beats that. A chair who knows the preferences and sets the order can choose the outcome while every individual vote is perfectly fair. Legislatures that take amendments one at a time are running this procedure, and the order of business is contested accordingly.

Condorcet did not live to see much of this taken seriously. He sat in the revolutionary assemblies, fell foul of the Terror, went into hiding, and died in prison in 1794, days after his arrest.

What he left is a sentence that is still uncomfortable. The will of the majority sounds like a fact waiting to be measured. With two options it is one. With three, it may be something the counting procedure creates.