NANCY — I have come up out of the tunnel, across France, and into Lorraine, to a conference on economics and philosophy, and the first thing worth saying about the host city is that it is the birthplace of the man who discovered that three of anything is trouble. Henri Poincaré was born here in 1854. The masthead of this blog has carried the words “Three Implies Chaos” since its first day, and this dispatch is filed, as it happens, from the place where the chaos was first found.
The previous post in this series was about a bottleneck. This one is about something more unsettling: a discovery, made twice, a century and a discipline apart, that orderly systems stop being orderly the moment you add a third moving part.
The third body
Two bodies moving under their mutual gravity are a solved problem, and have been since Newton. Each traces a conic section — for a planet around a star, Kepler’s ellipse — and the solution is closed-form and exact: tell me where the two bodies are now and how fast they are moving, and I will tell you where they will be at any moment in the future or the past, forever. The heavens, taken two at a time, are the most predictable thing there is.
Add a third body and that predictability does not merely get harder to compute. It ends. There is no closed-form solution to the three-body problem, and for two centuries this was taken to be a temporary embarrassment, a puzzle awaiting a cleverer mathematician. In 1885 King Oscar II of Sweden offered a prize for what amounted to a proof that the solar system is stable. Poincaré entered with a memoir on the three-body problem, and won.1
And then, with the prize-winning memoir already set in type, an error surfaced. Poincaré had believed he could show the orbits were stable. The mistake, when he ran it down, did not just spoil the proof; it reversed the finding. The corrected memoir — which he paid to reprint, at a cost greater than the prize itself — described something for which no one yet had a word: trajectories so intricately folded that the smallest uncertainty about where a body starts makes its long-run path impossible to know.2 The error had been concealing the discovery. Three bodies do not merely lack a tidy solution. Three bodies imply chaos. Historians of mathematics now read that corrected memoir as a founding document of the modern theory of dynamical systems; the field that a later generation would call chaos theory begins, in a real sense, with a printer’s proof that had to be pulled back. It is fitting, given where this series has been, that the same man would later be asked to audit the most notorious misuse of mathematics in French public life.3
The political three-body problem
Now leave the heavens and watch a legislature choose a policy. Suppose the choice can be laid out along a single dimension — spending, from less to more — and each member has an ideal point and prefers outcomes nearer to it. Then majority rule is beautifully behaved. There is a stable outcome, the ideal point of the median member, and it defeats every rival in a straight pairwise vote. The reason is that a line has an unambiguous middle, and the member at that middle commands a majority against all comers: any proposal to move off the median is opposed by that member together with everyone on the side it moves away from, and that bloc is a majority by the very definition of where the median sits. This is the median voter theorem, and it is the two-body problem of politics: solvable, predictable, at rest.
Add a second dimension — let the legislature decide not only how much to spend but how to divide it — and the median vanishes. Generically there is now no point that beats every other, no stable outcome, no core. And the instability is not mild. McKelvey and Schofield proved in the 1970s that when the core is empty, a determined agenda-setter can walk a majority from any outcome to any other by some finite sequence of pairwise votes.4 Whoever controls the order in which the alternatives come up controls where the process stops. This is not a theoretical curiosity. A rules committee, a chair deciding what reaches the floor, a leader sequencing the amendments — each holds a real version of exactly that lever. The heavens have sensitive dependence on initial conditions. The legislature has sensitive dependence on the agenda.
I want to be careful here, because the rhyme is seductive and it is only a rhyme. Poincaré’s chaos is a fact about a deterministic dynamical system, one in which nearby trajectories pull apart exponentially as time runs forward. The McKelvey–Schofield result is a fact about the geometry of majority preference, in which the relation “a majority prefers” simply has no summit and no resting place. The mechanisms are not the same, and neither result is a special case of the other. What is the same is the shape of the discovery: a system that behaves itself with two elements stops behaving itself when a third is added, and the loss of order is not a gap in our cleverness but a property of the thing itself.
What we impose
If both the heavens and the legislature are chaotic in principle, the obvious question is why neither looks chaotic in practice. The planets have held their courses for as long as anyone has watched them. Legislatures pass budgets, and mostly budgets that resemble last year’s. If order is not guaranteed, where is all this order coming from?
Poincaré had an answer, and it is the reason he belongs at a conference on philosophy quite as much as one on economics. Some of the order we believe we observe, he argued, is not read off the world but supplied by us — chosen, the way one chooses a geometry, because it is convenient, and then quietly mistaken for a discovery.5 Stability, on this view, is frequently something we manufacture and then forget that we manufactured.
The political form of that thought is exact and well worked out. A legislature is stable not because the chaos theorem is wrong but because a legislature is never a bare majority-rule machine. It has agendas, committees, calendars, a status quo that holds its place until displaced, rules about what may be voted on and when. Those institutions are the conventions that hold the chaos still; they manufacture a resting point that the preferences, left to themselves, do not contain. The result has a name — a structure-induced equilibrium — and the adjective carries the whole argument. The equilibrium is induced, by structure. It is not found in the people. It is also why the same legislators, governed by a different set of rules, will reliably arrive somewhere else: the chaos is genuine, and the rules are what reach into it and draw out a single point.
Which is the question this conference, and the rest of this trip, is really about. If stability is imposed rather than discovered, then not every kind of it is worth wanting; a structure can hold the chaos still in ways that are arbitrary, or self-serving, or simply unaccountable. The question worth asking is not only how collective choice is made stable but how it is made stable legitimately. That is the question Maggie and I first took up (at Norman Schofield’s suggestion in a bar in Barcelona, no less) in Social Choice and Legitimacy.6
At her keynote here in Nancy, Maggie is addressing a similar question: how can we evaluate any algorithm that might govern and shape our incentives (and, presumably, choices)? Teaser: Simpson’s paradox brings some friends to the party.
The three paradoxes she will set out have something unsettling in common: a screening institution can be tuned to the sharpest optimum its designer can find and still deliver an outcome that no one would call legitimate. The chaos theorem tells us order must be built rather than discovered. The keynote asks what happens when it is built well and the result is still wrong. I will file on it from here in a few days.
With that, I leave you with this.
Notes
1 Henri Poincaré (1854–1912) was born in Nancy and is often called the last of the universalists — a mathematician at home across the whole of the subject and much of physics besides. The prize was offered in 1885 by Oscar II, king of Sweden and Norway, and awarded to Poincaré in 1889. The episode of the error, discovered after the winning memoir had been set in type for the journal Acta Mathematica and corrected by Poincaré at his own expense — the reprinting costing him more than the prize had paid — is told in full in June Barrow-Green’s history, Poincaré and the Three Body Problem (1997).
2 What Poincaré had found is now called sensitive dependence on initial conditions: two starting points, however close together, eventually diverge, so that prediction beyond a certain horizon becomes impossible in practice even though the system is entirely deterministic. The same phenomenon was rediscovered, and given the name the butterfly effect, by the meteorologist Edward Lorenz in the 1960s.
3 In 1904, during the review of the case against Alfred Dreyfus, the Cour de cassation asked three mathematicians — Poincaré, Gaston Darboux and Paul Appell — to assess the pseudo-probabilistic “system” by which Alphonse Bertillon claimed to have proved Dreyfus the author of the incriminating document. Their report found Bertillon’s use of the probability calculus to be without scientific foundation. It is the same gesture that has run through this series from London onward: a discipline called in to audit a number that an institution had abused.
4 The one-dimensional result is Duncan Black’s median voter theorem (1948). The multidimensional picture is owed to Charles Plott (1967), who showed that a core survives only under a knife-edge symmetry condition, and to Richard McKelvey (1976) and Norman Schofield (1978), who showed that when no core exists the majority relation reaches everywhere. The political half of this story had its own post here last month, “You Can Get There From Here.”
5 The philosophical position is Poincaré’s conventionalism, set out in Science and Hypothesis (1902): certain apparent facts about the world — the choice of geometry chief among them — are conventions adopted for their convenience rather than truths discovered in nature. Its political counterpart is the theory of structure-induced equilibrium, named by Kenneth Shepsle in 1979: legislative institutions, from agenda rules to committee jurisdictions to the standing of the status quo, generate a stable outcome that the configuration of preferences, left alone, would not possess.
6 John W. Patty and Elizabeth Maggie Penn, Social Choice and Legitimacy: The Possibilities of Impossibility (Cambridge University Press, 2014). The book takes up how collective choices can carry legitimate authority given precisely the instability results described above — how, and whether, an induced order can also be a justified one.