TOKYO — No guidebook leads with this, but Tokyo spent much of the twentieth century as one of the world’s quiet capitals of a strange and beautiful science: the mathematics of collective choice, the study of what a group of people actually do when they try to decide something together. The founding puzzle is older than the United States Constitution. In 1785 the Marquis de Condorcet noticed that a room of perfectly reasonable people could vote themselves in a circle — a majority preferring A to B, another preferring B to C, and a third, maddeningly, preferring C back to A — and that the circle was no one’s error but a standing property of majority rule itself. Arrow proved in 1951 that it could not be designed away. Everyone since has been mapping its edges: when the circle opens, and when it stays shut.
A startling share of that mapping was done here, and the histories written in English have never quite said so. They run through Rochester and Cambridge and Paris; they tend to step around Tokyo. But it was Ken-Ichi Inada who worked out, in the 1960s, the exact conditions on people’s preferences under which majority rule stays orderly. It was a game-theory group at the Tokyo Institute of Technology, gathered around Mitsuo Suzuki, that turned out a generation of social-choice theorists. And across the city at Hitotsubashi, Kotaro Suzumura was assembling the modern account of when a society’s choices deserve to be called rational at all — a man I mean to come back to before I leave Japan.
Out of that Tokyo Institute of Technology group came the one figure in the whole tradition who stands closest to the nerve of this blog. His name was Kenjiro Nakamura, and he found the exact number — not a condition, not a tendency, a number — at which Condorcet’s circle becomes impossible to avoid. The line at the top of this page comes from somewhere else: from a paper published in 1975 by two mathematicians studying something with no obvious bearing on voting at all, the way a simple rule applied over and over can stop settling down and start behaving wildly. They gave that behavior its name. They called it chaos, and they showed that the surest fingerprint of it is an orbit of period three. Their paper was titled “Period Three Implies Chaos.” The masthead of this blog is those same four words with the period walked around to the back — Three Implies Chaos. Period. — so that a word which once named a cycle of length three now means nothing but full stop, no argument. Nakamura’s theorem is that title spoken in the language of votes: three implies chaos there too.
The last of these dispatches got itself lost inside a clock; this one is about a man who is, in a way I find hard to shake, lost inside a number. Nakamura was born in 1947 and died in 1979, at the age of thirty-two, and most of the people who use his number every week could not tell you his first name.1 He is one of the foundations this blog stands on and by no means the only one — Condorcet is the ancestor, and the McKelvey–Schofield chaos theorem, which I will come to, is the structure later raised on top of him — but he is the one whose entire monument is a single integer, and the one whose city this is.
The number, and why it begins at three
So here is the number, since the rest of this is in its debt. Take any way a group might reach a decision — simple majority, two-thirds, a council with a chair who breaks ties, anything at all — and write it down not as a procedure but as a list: every coalition that, under that rule, is powerful enough to get its way. Call them the winning coalitions. Nakamura’s question was almost rude in its plainness. How many winning coalitions do you have to gather before you can find a bunch of them that share no single member in common?
That count is his number, and his theorem is that the count is an exact ceiling on how far the rule can be trusted. A rule can take any slate of options and reliably name a best one — no cycle, no A-beats-B-beats-C-beats-A spinning forever — for precisely as long as the number of options stays below its Nakamura number, and not one option longer.
Now apply it to majority rule. Its Nakamura number is three. Majority rule can therefore handle two options with perfect composure — which is what every up-or-down vote ever taken quietly relies upon — and at three options it can come apart. That three is the masthead made literal, and the small marvel it hides is that it is also Li and Yorke’s three, reached from the opposite end of mathematics. A map needs an orbit of period three before it tips into chaos; a vote needs a third option. Two different machines, the same trapdoor, the same number.
Why three and not two? The bare floor for any rule is two, but two is the number of a rule that has already surrendered. It means you can find two winning coalitions with nobody in common — that the rule stands ready to crown two opposed camps in the same instant — and a rule that will do that has contradicted itself before a single ballot is cast. Ask only that it not do that, the modest decency we can call properness,2 and the number is forced up to at least three. Below three there is nothing to find: two options is a bilateral choice, a yes or a no, and a yes-or-no cannot cycle. The pathology is not hiding inside small problems and swelling as they grow. It does not exist until the third option walks in, and it needs that third option to exist at all. This is why the result belongs to collective choice and not to exchange between two parties. Two parties can, in principle, always just trade. Three is where trading becomes governing, and governing is where the trouble lives.
The only two doors
A smaller theorem would stop at the diagnosis. Nakamura’s keeps going and tells you, with a completeness that still feels faintly unfair, every way out there is. There are two. There are only two.
The first door is to keep the agenda below the number — for majority rule, to let no more than two options compete at once. That sounds like a constraint until you notice it is most of how real institutions run. Line the whole question up along a single dimension, vote it pairwise, two at a time, and the middle option wins and goes on winning; the median holds. You have not defeated the impossibility. You have agreed in advance never to ask the question that would summon it.
The second door is to push the number up to infinity, and here the theorem is brutally specific about the only way it can be done: there must be a player who sits inside every winning coalition without exception. A vetoer. Someone past whom nothing can pass. Install one, and the number becomes unbounded and the cycles vanish for any slate of options, however vast — because you have stopped doing collective choice and started doing something with a sovereign in it. Rationality over an unlimited agenda is on the menu at exactly one price, and the price is a person who can stop anything. (Ed: you have just described every committee you have ever served on.)
So the catalogue of escapes is the whole moral. You may shrink the world until the contradiction no longer fits inside it, or you may pile up enough power in one place that a single will overrides the rest before they can disagree. The impossibility is conserved either way. It is the standing creed of this blog, and Nakamura proved the cleanest form of it I know — not merely that the trap exists, but that the doors out of it are counted, and that there are two.
The same discovery, twice
Here is the part that has been sitting on me since I began planning this dispatch. Nakamura was not working alone, though he may well have felt that he was. In the same few years, the same truth was being pried out of the opposite end of the problem by people who, as far as I can tell, never spoke with him. Richard McKelvey showed in 1976 that once a question is allowed more than one dimension, majority rule does not merely fail to settle it — it can be marched anywhere, an agenda-setter able to walk a group from any outcome to any other through nothing but a sequence of honest majority votes. Norman Schofield, in a working paper by 1976 and in print by 1978, charted the geometry beneath that failure, the dense lattice of cycles that floods a space of more than one dimension. The field came to file the two of them together, as the McKelvey–Schofield chaos theorem. And in those very same years, in a province of mathematics with no apparent connection to any of it, Li and Yorke were handing the word chaos itself to the systems that never settle. The decade did not coordinate. It kept arriving, independently, at the same idea — and more than once at the same number.
What took me an embarrassingly long time to see is that the chaos theorem and the Nakamura number are one discovery in two suits of clothes — the first geometric, the second combinatorial. McKelvey and Schofield show you the cycles filling the room; Nakamura hands you the exact integer past which the room stops being safe. And it was Schofield himself, by the early 1980s, who made the seam between them explicit, taking Nakamura’s number as the precise gauge of when a space of choices is small enough to hold still and when it is large enough to fly apart.
Nakamura did not live to see any of that. He died in 1979, the vetoers paper barely off the press. His selected work was gathered and published in 1981 by his teacher, and the field went on building on the number while the man it was named for receded into it.
The line home
I owe you a disclosure about the second half of that story. Norman Schofield was my teacher. The book Maggie and I wrote about social choice and legitimacy is dedicated, in part, to him. So when I draw the line from Li and Yorke’s title, through Nakamura’s number, through the chaos Norman spent his career mapping, to the four words at the top of this page, I am not tracing an abstract genealogy. I am standing at the end of it. The masthead I write beneath every week names a phenomenon that a man I never met pinned to an exact number, in the last full year of a life that ran only thirty-two.
I came to his city for a conference and a talk about something else, and I have spent the week thinking instead about a man I never met, whose number is in everything I do and whose name is in almost none of it. The discipline kept the integer and mislaid the man. I cannot hand him back his first name, but I can set it down once, plainly, in the city that is his: Kenjiro Nakamura.
With that, I leave you with this.
Notes
1 He was not a one-result man, which is part of what makes the afterlife of the single number a little poignant. His collected papers, edited after his death by his teacher Mitsuo Suzuki, range across coalition-stability notions borrowed from Luce and, in work with Mamoru Kaneko, a way of building a social welfare function out of a bargaining solution without any interpersonal comparison of utility. The number is what survived into common use, which is its own small lesson about what a field elects to keep.
2 The unconditional floor for the Nakamura number is two; the floor of three holds once the rule is required to be proper — never to count two coalitions with no member in common as both winning. An improper rule can contradict itself on a single yes-or-no question, so properness is close to the least one can ask of anything deserving the name of a collective decision, and it is exactly the condition that lifts the number from two to three.