The Cycle

The first plate of this volume laid down transitivity as one of the two conditions that turn a preference into a number, and defended it in the usual way — not as something people reliably do, but as something a rational person must do, on pain of the money pump, that perpetual machine which relieves a circular chooser of his wealth one small trade at a time. The defense is elegant, and it has the feel of a proof. But notice what it quietly concedes: that the rule against going in circles is enforced rather than observed, held up as a standard the chooser ought to meet rather than reported as a regularity he meets. This plate presses on that concession. It asks whether intransitivity — a preference that loops, A over B over C and back to A — is in fact the rare derangement the theory treats it as, or whether it is a common and even reasonable thing; and whether the money pump, the great argument that it cannot be tolerated, proves as much as it seems to. The answer to the first question was given, with disarming simplicity, by the mathematician Kenneth May, and it is that a looping preference is not the footprint of a confused mind but the natural consequence of the most ordinary feature of real choice: that the things we choose between differ along more than one dimension at once.

To see why multiplicity is the culprit, begin with what transitivity silently assumes. A transitive preference is one that can be laid out along a single line, every option somewhere on it, higher meaning better; and if A sits above B and B above C, then A perforce sits above C, because that is what a line is. The trouble is that almost nothing of consequence is chosen on a single line. A job is salary and location and prospects and the people; a house is price and space and light and the commute; a person to marry is — well, everything at once. When the options differ on several counts, the chooser must somehow gather the counts into one verdict, and the most natural way to do it, the way anyone reaches for, is something like a majority of the considerations: A is preferred to B if A wins on more of the dimensions that matter. This rule is not foolish. It is even-handed, it uses all the information, it treats each consideration as a vote. And it is exactly the rule that breaks transitivity.¹

May made the breakage concrete with three candidates and three considerations. Imagine choosing among three suitors, ranked on intelligence, on looks, and on wealth. The first is the cleverest, the second the handsomest, the third the richest, and each is middling on the count where another leads — so that, set side by side, the first beats the second on two counts of three, the second beats the third on two of three, and the third beats the first on two of three. Each pairing has a clear and defensible winner; each winner commands a majority of the considerations. Yet the three winners chain into a circle — first over second, second over third, third over first — with no option standing above the rest and no line on which to place them. May did not merely construct this on paper; he put choices of just this shape to a room of students and found the circular pattern appearing in their preferences exactly as the structure predicts, not from carelessness but from the conscientious weighing of competing virtues. The intransitivity was not noise to be averaged away. It was the signal of a mind doing precisely what the situation asked: honoring more than one good at once, and finding, as anyone must, that several goods do not always agree on an order.²

The shape of this will be familiar to anyone who has watched an assembly fail to make up its mind, for it is the same shape exactly. A century and a half before May, the Marquis de Condorcet had shown that a group voting by majority over three options can cycle — a majority preferring the first to the second, another the second to the third, another the third to the first — so that the will of the majority chases its own tail and no option can claim to be the people's choice. What May saw was that this was never a peculiarity of committees. The single chooser weighing several criteria is himself a committee, his considerations the members, his verdict their vote; and the cycle that haunts the assembly haunts him for the identical reason. The lesson runs deep and will return in the volumes on the aggregate and on social choice: whenever many things must be melded into one ranking — many voters into a public will, many criteria into a single preference, many attributes into a price — the melding can produce loops that none of the parts contain. Intransitivity is not a failure that befalls the multidimensional; it is a structural possibility built into the very act of collapsing many dimensions into one. To demand that a chooser never cycle is, at bottom, to demand that the several things he cares about always happen to line up — and they do not.³

So much for whether people cycle; they do, and reasonably. There remains the money pump, the argument that whatever people do, they ought not cycle, because a circular chooser can be bled. Look at it more closely than the first plate did. The pump works by leading its victim around his own loop: he holds C, and since he prefers B to C he pays a trifle to swap C for B, and since he prefers A to B he pays again to swap B for A, and since — fatally — he prefers C to A he pays a third time to swap A back for C, arriving where he began one coin poorer, and round again without end. The picture is vivid and the conclusion seems airtight. But it rests on a chooser who never lifts his eyes from the single trade in front of him. A person with the wit to have preferences at all has the wit to notice that he is being walked in a circle and emptied, and the moment he notices, he stops — he declines the next swap, or refuses to value the options pairwise when he can see the whole carousel at once. The money pump does not drain a thinking person; it drains an automaton that evaluates only locally and never asks where the sequence leads. What the argument truly establishes is narrower than it pretends: not that cyclic preferences are impossible to hold, nor that they ruin whoever holds them, but that they are in principle exploitable — that they expose their holder to a charge a clever counterparty could try to levy.⁴

And that reframes the whole question the plate began with. Is the rule against going in circles a fact about people, or a fee we charge them for breaking it? The honest answer is that it is neither cleanly, and a little of both. It is not a robust fact: people cycle whenever their criteria pull apart, which is often, and they are not deranged when they do. It is not quite a fee, either, for the fee is almost never collected — real markets are not staffed by tireless arbitrageurs walking us around our loops, and the thinking chooser sidesteps the pump the instant he sees it. What transitivity really is, this volume can now say plainly, is a coherence condition: the precise price of admission for being represented by a single number, a utility laid out along one line.⁵ If you insist that a person's worth-rankings collapse without remainder into one scale, then transitivity is exactly what you must assume, and its failure is exactly what you must forbid. But the insistence is the theorist's, not the chooser's. The chooser is not trying to be one-dimensional; he is trying to do justice to several things he cares about, and the loops are the visible trace of that effort, the places where the many goods he honors decline to fall into a single file. Like the reversals of the third plate and the menu-dependence of the eighth, the cycle is a seam — a spot where the clean machinery of preference, pressed against the multidimensional substance of an actual decision, shows the strain of all it has agreed to leave out.