A ranking can tell you where to go without telling you how far apart the destinations lie. Put three possible dinners in order, and the first can be chosen from any menu that contains all three; remove it, and the second becomes the choice. Nothing in that operation requires the pleasure of the first to be twice the pleasure of the second. The distance between them has done no work. This was the opening through which the measurable util could leave the theory of ordinary consumer choice without taking the choice itself away. The early margin had spoken as though satisfaction came in quantities. The ordinal reconstruction asked which conclusions required those quantities, and which needed only the order they happened to describe.
The earlier Substitution and Order followed this retreat from the cardinal magnitude. A scale that labels three bundles one, two and three selects the same best bundle as a scale that labels them one, four and nine. The spacing has changed; the ordering has not. A strictly increasing transformation can repeat the operation throughout a represented preference system. Hicks and Allen placed this indeterminacy at the beginning of their 1934 reconstruction, crediting Pareto’s distinction between a scale of preferences and a particular utility function. An indifference map records the bundles ranked alike and the ordering between those sets. It does not, by itself, measure the intervals of feeling between them.
Much still has to be specified before this becomes a usable account of a person choosing. The alternatives must be described, the feasible set known, and the preference relation sufficiently coherent for the comparisons being made. A finite menu with a complete and transitive ranking has a best-ranked option, though there may be ties. A smooth indifference map over continuous quantities asks for further regularity. Its downward slope assumes that both goods are locally desired; its customary inward bow is a restriction on the pattern of trade-offs, not a gift bestowed by the word ordinal. The theory has shed the need to measure the spacing between ranks. It has not shed the need to state the properties of the ranking.
Consider the bread and wine of The Balance once more. On positive quantities, let one index be U = log b + log w, and another be V = b²w². The second is exp(2U), a strictly increasing transformation, so it orders every pair of these bundles in exactly the same way. Yet the marginal utility of bread in the first index, 1/b, falls as bread increases with wine held fixed; in the second, 2bw², it rises. If these slopes were read as measurements of sensation, the same person would appear to have acquired the opposite psychology without changing a single choice. What survives is the relative trade-off: in either representation the ratio of the bread margin to the wine margin is w/b. The price-taking choice depends on that ratio at an interior optimum, not on which arbitrary numerical garment we give the ranking.
The map therefore retains a great deal. Given an adequately specified ordering and a feasible budget, it can identify the best affordable set; with prices changed, it can identify the best set under the new constraint. It can compare what the person chooses with another available bundle without attaching a number of pleasures to the difference. But knowing that a representation would suffice is not knowing the representation. A theorist who draws a family of curves has supplied an object that the evidence must still support. Observing a purchase is not the same thing as observing the slope of every curve across every possible budget. The ordinal turn made the informational requirement smaller. It did not make the missing information appear.
Samuelson pushed the question back toward the transaction itself. His 1938 note proposed beginning with demands at given prices and expenditure, then imposing consistency restrictions on that behaviour, rather than beginning with an introspected magnitude of satisfaction. This changed the direction of inquiry. If a buyer chose one basket when another was available, the choice could supply evidence about their relative standing. But the restriction was doing substantive work: the account did not become a theory merely by renaming whatever happened a preference. Samuelson supplied postulates, including a consistency condition; an observed pattern might fail the proposed requirements. Later finite-data analysis would make the distinction between rationalizing choices and merely relabelling them especially sharp.
A purchase can restrict the rankings consistent with a life. It does not disclose the whole life’s ranking at once.
A small invented inconsistency is more instructive than an assurance that choices are consistent. Let basket A contain two units of bread and one of wine, and basket B one of bread and two of wine. At bread and wine prices of two and one, A costs five and B four. Suppose the buyer, with five to spend, chooses A. Now reverse the two prices: B costs five and A four, and the buyer chooses B. At each observation the rejected basket lay strictly inside the available budget. Under a stable preference ordering and local nonsatiation—the availability of something better arbitrarily near any bundle—each choice requires its selected basket to rank strictly above the other. The two strict rankings cannot coexist. Changing prices can certainly change purchases; the defect here is the particular pattern of mutual affordability and selection, not the fact that the baskets differ.
This distinction prevents a careless reading of the simplest revealed-preference sentence. Choosing A when B is affordable establishes, within a maximizing representation, that A is at least as good as B; it need not establish that A is strictly better. Indifference can permit either choice. Strict affordability, combined with local nonsatiation, provides the stronger conclusion used in the example: if the cheaper B were equally good, an affordable improvement close to B would contradict A’s optimality. The evidence is thus choices together with the alternatives and assumptions that make those choices informative. Misrecord a price, overlook a constraint, or combine observations across a change in the person’s preferences, and the apparent contradiction may concern the maintained description rather than some timeless irrationality of the chooser.
The positive result is equally exact. For a finite set of observations at positive prices, the generalized axiom of revealed preference rules out a revealed-preference chain that returns through a strict affordability reversal. Afriat’s theorem establishes that passing the condition is equivalent to admitting a locally nonsatiated utility representation that rationalizes the observations; one can, in fact, construct a continuous, monotone, concave representation. That is an existence result, not the discovery of a uniquely true psychological function. Several rankings can agree on everything the observed budgets reveal and disagree elsewhere. Additional observations may narrow the possibilities; some comparisons can be inferred, while others remain open. The model becomes testable because it forbids certain patterns, not because a finite shopping history dictates every possible future purchase.
The loss of a common ruler matters most when the inquiry leaves the single chooser. Suppose an allocation helps one person and harms another. Their separate rankings can identify that disagreement, but the numerical gain assigned by one index cannot simply be subtracted from the numerical loss assigned by the other. Rescale one person’s index and the proposed sum may reverse its verdict while both people’s choices remain unchanged. A social comparison requires something further: a principle of aggregation, an account of rights or needs, evidence about experienced welfare, or another declared basis for judging between persons. The ordinal account does not forbid such judgments. It prevents an arbitrary choice of labels from making them on our behalf. Even for one person, the claim that observed choice is the right measure of welfare needs a reason beyond the fact that a choice occurred.
The plate’s question can now be answered without making desire vanish. If we possess a ranking adequate to the situation, a measure of the intensity between its positions may be unnecessary for predicting the selected bundle. That shows a redundancy in this representation of choice. It does not establish that wanting has no causes, that experience carries no intensity, or that all purposes of explanation can be served by the same economical description. And the opening “if” remains substantial: an order assumed by a model is not an order completely recovered from a handful of acts. What the ordinal turn carries away is the obligation to pretend we have measured what the choice argument never needed to measure. What it leaves is a cleaner margin—an opportunity to substitute, a constraint, and a comparison—and a clearer boundary around what those things allow us to know.