Plate i in print

The Axioms

completeness and transitivity; the two conditions from which a number, and then the whole of demand, are made.

Let a person's preferences be complete and consistent; then there is a number fixed to each option, and to choose is only to take the largest number within reach.

— the premise of this plate, after Gerard Debreu, 1954

Rationality, in economics, is not a psychological achievement. It is a property of a list. A person is rational, the theory says, if the options before her can be arranged in an order — this one at least as good as that, that at least as good as the next — and if the order, once drawn up, does not contradict itself. There is no requirement that she be calm, or well-informed, or sensible by any standard outside her own; there is only the requirement that her wanting have a certain shape. From this slight and almost grammatical demand the entire apparatus of choice is built, and it repays the effort to see exactly how little is asked before seeing how much is got.

Two conditions do the work. The first is completeness: for any pair of options, the person can say that she wants the one at least as much as the other, or the other at least as much as the one, or that she is indifferent between them. No pair defeats her; there is always a verdict. The second is transitivity: if she wants A at least as much as B, and B at least as much as C, then she wants A at least as much as C. The ranking runs in one direction and does not loop. To these a third is sometimes added for tidiness — that each option is at least as good as itself — but it asserts nothing and may be passed over. Completeness and transitivity are the whole of it. Together they say that a person's wants form what a mathematician would call a total order: a single line on which every option has its place.1

1. A relation that is complete and transitive is a weak order; allowing for ties, it seats every option somewhere on one line. Add a continuity condition — that options near a given one are ranked near it — and a theorem of Gerard Debreu (1954) guarantees the order can be represented by a continuous numerical function. Drop continuity and the guarantee fails: lexicographic preferences, which rank by one attribute and break ties by a second, are complete and transitive yet admit no utility function at all.

What follows from so little is out of all proportion to it. If a person's preferences are complete and transitive — and, where the options form a continuum, vary without sudden jumps — then there exists a number attached to each option, a utility, with a single property: she prefers whichever option carries the larger number. The ranking has been turned into an arithmetic. And once it is an arithmetic, choice becomes the simplest operation in mathematics: from the options a budget allows, take the one whose number is highest. The chooser is a maximizer; choosing is maximizing; and the whole of consumer theory — demand that falls as price rises, quantities that move in signable directions with incomes and prices, a surplus that can be measured as an area beneath a curve — unfolds from the act of maximizing a function under a constraint. None of this required the utility to be a real magnitude of feeling. It required only that the ranking exist and behave. The number is a bookkeeping device for the order, nothing more, and the theory is scrupulous to claim nothing more for it.2

2. This is the ordinalist reading, the subject of the companion volume's third plate (№ 08, The Utility): the utility number carries no cardinal content, no quantity of satisfaction, only the order. Any renumbering that preserves the order — any increasing transformation — serves equally well. The function represents the ranking, and the ranking is the only fact.
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It is the modesty of the assumptions that wants examining, because it is partly a pose. Completeness, in particular, asks a great deal under the appearance of asking nothing. It requires that a person hold a settled verdict on every pair of options — not merely the pairs she has faced, but all of them; not merely the comparable, but the incommensurable; not merely apples against apples, but a year of her life against a sum of money, a friendship against a promotion, a thing she has never imagined against one she has. The theory does not permit her to answer I cannot rank these, or these are not the kind of things that lie along one line. Every such honest hesitation is ruled out in advance, defined away as a failure to be the chooser the theory describes. Economists have felt this discomfort for a long time, and some have worked out what becomes of the apparatus when completeness is dropped and preferences are allowed to be partial — to rank some pairs and fall silent on others. The structure survives, in a weakened form; but the silence the weaker theory permits is exactly the silence the standard theory cannot afford, because a gap in the ranking is a place where choice has nothing to maximize.3

3. On incomplete preferences — orderings that rank some pairs and leave others incomparable — see the line of work from Robert Aumann (1962) onward. Incompleteness is the formal home of noncomparability: the thought that some goods are not more, less, or equally valuable than others, but simply not set on a common scale. The standard theory excludes this not because it is false but because maximization needs a verdict everywhere.

Transitivity conceals a different sleight. It is rarely defended as something people reliably do; it is defended as something a rational person must do, on pain of absurdity. The argument is the famous one of the money pump. A person whose preferences run in a circle — who prefers A to B, B to C, and C back to A — can be relieved of his money without limit: he will pay a little to trade what he has for what he prefers, and a little again for the next, and then, preferring the first thing after all, a little again to trade back, arriving where he began some coins poorer and willing to go round once more. The prospect is meant to be so plainly intolerable that no one would own such preferences. But notice what the argument concedes. It does not show that people have transitive preferences; it shows that they would be punished for lacking them. Transitivity has quietly changed its standing, from a description of how people rank into a rule they ought to obey — a canon of rationality rather than a fact of psychology. The axiom that looked like an observation is a norm in observation's clothing, and a theory resting on it is, to that extent, an account of how an ideal agent should choose, not a report of how anyone does.4

4. The money-pump defense of transitivity is set out by Davidson, McKinsey & Suppes (1955) and recurs throughout decision theory. That a defense is needed at all — that transitivity must be argued for by exhibiting a penalty — is itself the tell. One does not argue that water is wet; one argues that a person ought not to hold preferences he can be pumped for. The fourth plate, The Cycle, takes up whether real preferences ever run in circles, and whether the penalty is as decisive as it sounds.

There is one further move, and it has done more than any other to shield the theory from its critics. Confronted with the charge that no one actually performs these rankings — that people do not consult an inner table of utilities before reaching for the cheaper tin — the theory's defenders need not deny it. They need only say that people behave as if they did. The billiard player computes no angles, yet the balls go where the equations say; the firm solves no calculus, yet survives only if it acts as though it had. So with the chooser: the claim is not that she maximizes consciously but that her behavior is as though she maximized, and the theory is to be judged by whether that as-though predicts what she does, not by whether it describes what passes through her mind. The defense is powerful and, within limits, fair. But it carries a cost the rest of this volume will collect. By severing the theory from any claim about the chooser's reasons, it converts rational choice into a black box that emits predictions, and stakes everything on those predictions holding. If the behavior itself misbehaves — if the same person, asked the same question two ways, answers two ways; if the set an option is drawn from changes which option is taken — the as-if has nothing to fall back on. It cannot retreat to the chooser's true preference, having renounced all knowledge of it. The plates that follow are the places where the behavior misbehaves, and the floor this plate has built begins, under their weight, to give.5

5. The as if methodology is Milton Friedman's (The Methodology of Positive Economics, 1953): a theory's assumptions need not be realistic, only its predictions accurate. The standing objection is that prediction without realism leaves the theory unable to explain — a black box that says what will be chosen but not why — and, worse, defenseless when the predictions fail, having forsworn any account of the mechanism it might otherwise repair.