Plate vi in print

The Paradox

Allais and Ellsberg; the self-evident axioms failing exactly where they seemed surest.

A rule that orders each gamble must also survive the comparison between them; the difficulty appears when both choices are kept in view.

— the argument of this plate, after Allais and Ellsberg

A person can prefer a secure fortune to a chance of a larger one without disturbing expected-utility theory. The curve of utility can make the possible gain too small, relative to the possible loss, to justify the wager. The sharper test comes when the same person faces a second comparison and makes a choice that no single curve can reconcile with the first. Allais designed such comparisons around certainty; Ellsberg found a related difficulty where the chances themselves were incompletely known. Their examples reach the assumptions that allow a pattern of choices to become a numerical scale. The question is whether the person keeps the relations among choices that the scale requires. To see what fails, one must keep the whole pattern in view. An isolated preference for safety proves very little. A carefully chosen pair can make the difficulty exact.

Allais's 1953 questionnaire placed two offers beside one another. Measure the prizes in millions of francs: A pays a hundred for certain; B pays five hundred with a ten-per-cent chance, a hundred with an eighty-nine-per-cent chance, and nothing with a one-per-cent chance. Now consider a second pair. C pays a hundred with an eleven-per-cent chance and nothing otherwise; D pays five hundred with a ten-per-cent chance and nothing otherwise. Imagine preferring A to B and D to C. In the first comparison, the security of a hundred seems worth preserving against the small chance of losing everything offered. In the second, both prizes are doubtful, and the larger prize seems worth the one-percentage-point reduction in the chance of winning. These are intelligible accounts of the two choices. Whether they can belong to one expected-utility ordering is a separate matter, settled by putting the comparisons together.1

1. Maurice Allais, “La psychologie de l'homme rationnel devant le risque: la théorie et l'expérience,” Journal de la Société de statistique de Paris 94 (1953), p. 59, questions 35–36. The original prizes are 100 and 500 million francs; the body retains those amounts and probabilities. The instructions at pp. 55–57 specify hypothetical, separate, once-only offers, net of tax and with unchanged purchasing power. The paired preference here is the pattern being tested, not a claim that every respondent or reader selects it.

Write the utility of receiving a hundred as u(100), and similarly for the other prizes, holding the person's background circumstances fixed. Preferring A requires the utility of a hundred to exceed the weighted utilities of B. Cancel the eighty-nine-per-cent chance of a hundred on both sides: eleven per cent of u(100) must exceed ten per cent of u(500) plus one per cent of u(0). But preferring D to C requires exactly the reverse inequality. Curving the utility function more sharply cannot satisfy both. The source of the conflict can also be seen without symbols. Split A into an eighty-nine-per-cent chance of a hundred and an eleven-per-cent chance of a hundred. A and B then share the first branch. Replace its payment with zero in both options and the pair becomes C and D. The common-consequence restriction preserves the ordering through that replacement; the specified choices reverse it. The exceptional attraction of certainty has affected a comparison the expected-utility arithmetic requires to remain unchanged.2

2. For A ≻ B, the necessary inequality is .11u(100) > .10u(500) + .01u(0). For D ≻ C it is .10u(500) + .01u(0) > .11u(100). These cannot both hold for the same outcome utilities. Allais's theoretical discussion is at pp. 47–55. The cancellation applies to any curvature, not just a linear utility of money; changing the common branch is the operative comparison.
⁂

Ellsberg's urn changes the setting from stated chances to incomplete knowledge. It holds thirty red balls and sixty balls divided between black and yellow in an unknown proportion. One ball will be drawn at random. First choose between a hundred dollars if it is red and a hundred dollars if it is black, with nothing otherwise. Then choose between a hundred dollars if it is red or yellow and a hundred dollars if it is black or yellow. A person who chooses red in the first pair and black-or-yellow in the second selects a known chance each time: one-third in the first case, two-thirds in the second. The alternatives depend on the undisclosed division of the other sixty balls. Ellsberg described this pattern in his informal questioning, while also reporting people who chose differently. Its importance does not depend on assigning it to everyone. It supplies a particular pattern against which the proposed extraction of beliefs from choices can be tested.3

3. Daniel Ellsberg, “Risk, Ambiguity, and the Savage Axioms,” Quarterly Journal of Economics 75 (1961), pp. 653–656, with notes 4–6. The article identifies the responses as nonexperimental observations and calls for controlled testing. The distinct two-urn example appears in The Unknown; the single-urn comparison here isolates the common-event restriction.

Suppose one set of subjective probabilities governs both choices, and the utility of the hundred-dollar prize is the same whichever color produces it. Strictly preferring red to black then requires the assigned probability of red to exceed that of black. Strictly preferring black-or-yellow to red-or-yellow requires the probability of black plus yellow to exceed that of red plus yellow. Subtract the common yellow probability and black must exceed red. The two requirements conflict. The second pair differs from the first only by awarding a hundred on yellow in both options; the sure-thing principle requires that common change to leave their order intact. Again, greater curvature in the utility of money cannot reconcile the pattern, since each bet has the same two monetary consequences. The conclusion is exact but bounded. These choices cannot be represented by that single additive probability distribution and state-independent expected utility. They do not establish that the chooser has no beliefs, cannot estimate any chance, or must refuse to act.4

4. Ellsberg, §II, pp. 647–656, especially the sure-thing discussion at pp. 648–650 and the proof in note 5 at p. 655. With the two utility payoffs normalized to 1 and 0, the required inequalities are pR > pB and pB + pY > pR + pY. A single preference for red could instead be represented by many probability assignments. It is the combined pattern, under unchanged information and outcome values, that creates the contradiction.

The descriptive and the normative questions now separate. As a description, the representation must answer to what people choose under the specified conditions. As a proposed discipline, it asks whether those people would endorse the restriction once its implications were made clear. Ellsberg reported several reactions: some respondents adhered to the axioms, some reconsidered their initial answers, and some wished to retain the offending preferences after reflection. Those observations leave a disagreement to examine. One person may regard the common-event argument as a reason to revise a choice; another may regard the quality of the available information as a reason to seek a different account of choice. Allais likewise refused to settle the dispute by defining rationality as obedience to the axioms under dispute. A formal contradiction within a representation identifies what cannot be held together. Deciding which commitment should change requires reasons beyond the fact that the representation has an elegant theorem.5

5. Allais, §I.9, pp. 54–55, distinguishes alternative grounds for assessing rationality from defining it by the proposed axioms. Ellsberg, §I, pp. 645–646, and the end of §II, pp. 655–656 with note 6, distinguish prediction, reflective endorsement and correction. These are the authors' stated disputes and reported reactions; they do not establish population-wide frequencies or settle which normative criterion ought to govern every decision.

The paradoxes leave utility with a more demanding task. A scale recovered from choice must specify the comparisons over which it works and the assumptions that permit the recovery. A scale offered as a guide must explain why its restrictions deserve allegiance when a person feels their force differently in the case before him. Neither task is answered by treating every departure as noise or every departure as wisdom. The relation between a judgment and its circumstances has become part of the inquiry. The next plate, The Frame, moves that inquiry one step earlier. It asks how the same consequences can be understood as different prospects when they are described from different points of view. The difficulty will then concern the formation of the alternatives themselves, before a stable ordering has even been asked to compare them.