The number arrives with conditions attached. To say that a person maximizes utility can mean that his choices possess a certain order, that his gambles obey a particular arithmetic, or that he ought to choose as though they did. These claims have different burdens. A numbering can accurately record an order without explaining its origin. A theorem can establish what follows from assumptions without showing that anyone meets them. A rule can expose a conflict without deciding which side of the conflict deserves allegiance. The axioms make those burdens visible if we resist letting their common name do all the work.
Begin with the modest task of arranging a finite set of alternatives. Completeness requires a comparison: one is at least as good as the other, or both are equally ranked. Transitivity requires the comparisons to fit together. If A is at least as good as B, and B as C, A cannot fall below C. A complete, transitive ranking of finitely many alternatives can be represented by numbers, with equal numbers for ties and higher numbers for higher positions. This is a useful representation, but the distances between those numbers are free to change. The numbering has not discovered how much better A is than B, or why the person prefers either.
An inability to make a comparison must not quietly be entered as indifference. Indifference says the alternatives occupy the same place in the ordering; incomparability says that ordering has not supplied their relative places. Nor is every hesitation evidence of a contradiction. The person may need information, may be comparing consequences differently, or may have changed his mind. A claim about stable preferences has to specify the alternatives and circumstances across which stability is expected. Otherwise, the model can be protected against every observation by inventing a new preference after each choice, and the impressive word representation has become little more than a transcription.
Lotteries demand more than this ordinal discipline. Continuity connects an intermediate alternative with suitable mixtures of better and worse alternatives. Mixture independence requires a ranking to survive when each option is combined with the same third lottery in the same proportion. Together with the ordering and mixture structure, such conditions support the expected-utility representation developed in The Wager. Independence here concerns probabilistic mixtures. It is not merely the claim that adding another item to a menu should leave the ranking of the original items unchanged. The distinction matters because the famous tests change the structure of gambles, not simply the number of dishes on a table.
Allais made the restriction confront certainty. His 1953 questionnaire offered a hundred million francs for certain against a gamble: a ten-per-cent chance of five hundred million, an eighty-nine-per-cent chance of a hundred million, and a one-per-cent chance of nothing. In a second pair, an eleven-per-cent chance of a hundred million competed with a ten-per-cent chance of five hundred million, with nothing otherwise in either case. Consider someone who chooses the sure hundred in the first pair and the chance of five hundred in the second. The first choice protects a fortune against a small chance of getting nothing; the second sacrifices one percentage point of winning for a much larger prize. Each has an intelligible motive. Their combination is the test.
Call the four prospects A, B, C and D in that order, and measure prizes in millions. Preferring A to B requires the utility of a hundred to exceed eighty-nine per cent of itself, plus ten per cent of the utility of five hundred and one per cent of the utility of nothing. Remove the common term. Eleven per cent of the utility of a hundred must exceed the other two terms. But preferring D to C requires precisely the reverse inequality. No additional curvature in the utility of money can reconcile them. An eighty-nine-per-cent branch paying a hundred in both of the first two prospects has become a branch paying nothing in both of the second two. Expected-utility arithmetic requires that common change to preserve their order.
The failure is narrower than the declaration that the chooser has no utility function. He could still give all four lotteries a complete and transitive ordering. What he cannot do, while retaining these two strict preferences, is represent that ordering by the expectation of one fixed utility assigned to their monetary consequences. An ordinal score for each whole lottery can record more patterns than the expected-utility formula permits. Allowing that flexibility saves the possibility of representation, but gives up part of the structure that allowed choices among mixtures to determine a cardinal scale. The lost restriction should be named, rather than mistaken for the disappearance of every order.
Ellsberg’s example tests what happens when the chances are incompletely known. An urn contains thirty red balls and sixty divided between black and yellow in an undisclosed proportion. A person first chooses between a hundred dollars on red and a hundred on black, with nothing otherwise. He then chooses between a hundred on red-or-yellow and a hundred on black-or-yellow. Preferring red in the first pair and black-or-yellow in the second selects the known chance each time: one-third, then two-thirds. The change between the pairs is a hundred-dollar payment on yellow, added to both options. The same event has been altered in the same way on both sides.
Suppose one additive probability distribution and the same value for the prize in every colour govern both choices. The first preference requires red to be more probable than black. The second requires black plus yellow to be more probable than red plus yellow; cancelling yellow requires black to be more probable than red. Those demands cannot both hold. This is the force of the sure-thing test. Greater concavity in the utility of money does not answer it, since every bet offers the same two monetary consequences. Nor does the contradiction show that the person has no beliefs about the urn. It identifies a particular way of recovering beliefs and values from choices that cannot represent the whole pattern.
An axiom tells us what cannot be kept together. It does not, by itself, tell us what to surrender.
As descriptions, the models must confront such patterns in the circumstances where they occur. A useful idealization may still explain other decisions, provided its departures and its domain are made clear. As standards, the axioms ask a different question: would the chooser endorse the restriction after seeing what his answers imply? Ellsberg reported disagreement even among reflective respondents, including people who revised their answers and people who wished to retain the offending preferences. Allais objected to settling that disagreement by defining rationality as obedience to the very axioms in dispute. The formal result can sharpen the discussion. It cannot replace it.
A defended standard must say why the feature it excludes should not matter. Perhaps a common consequence really supplies no reason to reverse a choice; perhaps uncertainty about the evidence is itself something a decision procedure should accommodate. The alternatives can be examined without declaring every hesitation irrational or treating every instinct as authoritative. What matters is whether the account survives the comparisons it asks us to accept. The final plate, The Worth, then takes a further step. Even a preference consistently represented, or reflectively endorsed, leaves the question of what that preference should be worth when another person’s life is placed beside it.