TOKYO — Earlier this week I wrote that there was a man I meant to come back to before I left Japan. This is me keeping the promise.
The meeting that Maggie and I came to Tokyo for is the gathering of the Society for Social Choice and Welfare, and at the turn of the century its president was Kotaro Suzumura, of Hitotsubashi University, across the city. If Kenjiro Nakamura was the man almost no one in my field can name, Suzumura is the reverse — a scholar so central that everyone in the room this week would know him on sight. He co-edited, with Kenneth Arrow and Amartya Sen, the multi-volume handbook that is the closest thing this field has to a constitution. His country named him to the Japan Academy and made him a Person of Cultural Merit. He died in 2020, at seventy-six, eminent in a way Nakamura never lived to be.1
Here is what makes him the right man to close the pair. Nakamura found where collective reason breaks. Suzumura found the conditions under which it holds together — and not only on paper. The memorial written by his closest collaborator says that Suzumura, more or less single-handedly, built the community of scholars that made Japan a power in this field at all. He found the conditions for coherence, and then he went and made the field cohere. The theorem and the life rhyme, and I want to earn that sentence before I use it again.
The least you can get away with
The question Suzumura answered is the exact inverse of Nakamura’s. Nakamura asked how many options it takes before majority rule must contradict itself. Suzumura asked the opposite. Given a tangle of judgments that is not yet a full ranking — partial, provisional, full of pairs you have never troubled to compare — how little does it need going for it before you can be certain it can be straightened into a complete, consistent ranking that never reverses a call you already made?
The classical answer was already on the books. In 1930 the Polish mathematician Edward Szpilrajn proved that any ranking which is merely transitive, however incomplete, can be extended to a full ordering that honors every judgment in it.2 You may leave as many pairs uncompared as you like; so long as what you have committed to is transitive, the gaps can always be filled. For four decades transitivity was taken to be the price of admission.
Suzumura’s 1976 result is that the price is far lower. He replaced transitivity with a condition now named for him and proved it is not merely sufficient but the weakest condition that works at all — necessary and sufficient, the exact floor. The condition is almost homely. Your judgments are Suzumura-consistent if, whenever you can get from one option to another by a chain of “at least as good as” steps, you never then turn around and call the starting point strictly better than the place you ended. That is the whole demand. You may be wildly incomplete. You may even fail transitivity here and there. What you may not do is close a loop with a strict preference inside it — and that loop is exactly the one a swindler needs to run a money pump on you, walking you around the circle and charging a toll at every corner.
Forbid that single loop, and everything else is negotiable. Your ranking can still be completed into something fully, conventionally rational. The one cycle that bleeds you is the only thing you are required to surrender.
A bargain, not a gift
It would be easy to read this as the happy ending the last dispatch lacked, and it is the closest thing the field has to one. But it is still a conservation result, which is what keeps it honest, and what makes it the true mirror of Nakamura rather than his refutation.
Look at what the theorem promises and what it quietly charges. It promises that a coherent completion exists. It does not promise the completion is unique, and in general it is not — to finish the ranking you must settle the pairs your judgments left open, and the theorem is serene about which way you settle them. The coherence is guaranteed; the content of the finished ranking is partly yours to impose. Suzumura tells you the door can be opened. He does not tell you which room you walk into.
So the impossibility does not vanish here any more than it did in Nakamura’s account. It thins to its irreducible minimum — the one forbidden loop — and then it relocates, out of the question “can this be made coherent?” and into the question “coherent in which of the many available ways?” Nakamura gave you two doors out of chaos and a count of the toll. Suzumura gives you one floor under coherence and a warning that the floor is all he guarantees. Between them they bracket the entire problem: one names the precise point at which order becomes impossible, the other the precise least you must surrender to keep it possible. (Ed: every dispatch this week has ended on a man and a number. I’m not saying it’s a formula. I’m saying I could set my watch by it.)
The man who lived his theorem
I said the theorem and the life rhyme, and I want to close on the rhyme, because it is the reason I came back to him. When Suzumura began, the work I have spent this week tracing to Tokyo — Inada’s conditions, the game theorists at the Tokyo Institute of Technology, Nakamura’s number — was a scattering of brilliant, half-connected results, admired abroad and under-cited, exactly the kind of tradition that quietly gets lost. Suzumura is the man who gathered it. He trained the students, founded the journal, edited the handbook, and built his country’s wing of this field into something no one could route around. The collaborator who knew him best put it without hedging: Japan became a power in this work, and it would not have been thinkable without him.
Sit with that for a second alongside the theorem. The man who pinned down the precise condition under which scattered, half-formed judgments can be drawn together into a coherent whole spent his working life drawing a scattered, half-formed field together into a coherent whole. He did not just prove the conditions for coherence. He went and supplied them.
So this is the pair, and I think it is why Tokyo, of all the places this trip could have taken me, is the one that earned two dispatches. Two men, a generation apart and a few miles apart, working the two faces of a single question. One found the integer at which reason falls into chaos and died before the field could thank him. The other found the slender condition that holds the chaos off, and lived to build the room I am sitting in this week. Between them they hold the whole thing open — the threshold and the floor, the period and the reprieve.
I leave Japan in a day, and I am glad I spent part of it on a man who, unlike the last one, got everything a scholar is supposed to get: the chairs, the prizes, the handbook with his name beside Arrow’s and Sen’s. He needs no rescuing from me. But the result I find most beautiful in all of his work is also the quietest thing he proved — the one that asks for so little and hands back so much — and it tends to disappear behind the eminence. So let me say it plainly, the way I said his predecessor’s name. The least you must give up to stay reasonable is the single loop that would let the world rob you blind. Kotaro Suzumura proved that, and then he lived as though he believed it.
With that, I leave you with this.
Notes
1 He was not only a social-choice theorist, which is part of why the eminence ran so deep. With Kazuharu Kiyono he proved the excess-entry theorem, a staple of industrial economics holding that free entry into a market can leave it with too many firms rather than too few; and a long strand of his work wrestled with the tension Sen exposed between individual rights and the Pareto principle. The consistency result is the one I keep returning to, but it sat inside a much larger body of work.
2 Szpilrajn later published under the name Edward Marczewski; the 1930 extension theorem is one of the few corners of mathematics where his earlier name is still the one you meet. His result is usually stated as the claim that every partial order can be extended to a linear one — the same idea, dressed for the order-theorists rather than the choice-theorists.