3.1 The Allais paradox on the road to experimentation

Allais himself turned to an empirical perspective shortly after inventing the paradox. Having first circulated it in the spring of 1952 under circumstances we explain below, later in the same year he embarked on a ‘poll’ (sondage), as he put it, of real individuals. For a long time after this early study, he would allude to it without detailing its results, except to say that they supported his objections to the EU hypothesis. It was as late as 1979, in a collection prepared with Hagen, Expected Utility Hypotheses and the Allais Paradox, that he provided the following explanations (Allais Reference Allais, Allais and Hagen1979: 447–449).

In 1952, he had distributed a postal questionnaire to selected correspondents, and also presented it, in a shorter version, to the members of a seminar he was conducting in Paris. From the responses then received, he had selected 101 – 53 from the first source and 49 from the second – with a view to analysing them statistically. At the time, he had to content himself with simply publishing the questionnaire.Footnote ¹⁵ Being involved in other theoretical pursuits, he was unable to begin the statistical analysis until the 1970s, and this explains why he published his results so late – in the same 1979 collection in which one can read his apology.Footnote ¹⁶

By Allais’s own admission, he could only provide ‘selected findings’. His statistical analysis consisted in little more than statements of proportions and averages. Despite the decision theorists’ occasional requests, nothing else on the 1952 ‘poll’ filtered out from Allais’s writings after 1979.

Inspection of the questionnaire shows that it included the paradox and other counter-examples, but did not place any emphasis upon them. Its main purpose was numerically to estimate the utility functions of selected subjects. Being a ‘cardinalist’ in the tradition of early neoclassicals, Allais was hoping to recover utility functions that would measure the subjects’ preference intensities, and not simply represent their preference levels, and many of his questions were intended to fix numerical values for these intensities. For instance, he would ask: ‘Is your preference for an inheritance of 10 million rather than no inheritance stronger than your preference for an inheritance of 150 million rather than 10 million?’ (Question 651, in Allais Reference Allais, Allais and Hagen1979: 616). Unlike some VNM theorists, Allais denied that preference intensities could be read off from an estimate of the u function in the EU formula. Allais actually claimed that no sense could be made of an EU formula that did not take the form Σp _i v(x _i), where the utility function v is substituted for u and measures preference intensities in the certainty case. This claim also brought him into conflict with the other VNM theorists, for whom the EU formula involved no measurement of preference intensities at all. These controversies over the ‘cardinality’ of the EU formula are logically independent of the argument against VNM independence, and although they should be part of a fuller account of Allais’s position, they are too complex to be explored here alongside the ramifications of the paradox.Footnote ¹⁷

Given the overall orientation of the questionnaire, it comes as no surprise that the ‘selected findings’ devote so little space to the experimental assessment of the counter-examples. Allais gives just a dozen lines to them in a note to the main text (Allais and Hagen Reference Allais and Hagen1979: 636). Regarding the paradox itself, he mentions 46% violations of the EU hypothesis. This is a lower proportion than for other counter-examples included in the questionnaire, but he unfortunately does not pursue the interesting comparison.

The ‘poll’ has remained famous because of its early date, which is enough to secure it a place in the history of the experimental work on decision, but it is hard to see in it a prototype of what this field would later become. Apart from the sketchy statistical treatment, the sample did not obey any rigorous principle of selection, the two groups of respondents did not answer exactly the same questions, and the questions themselves were too remote from real choices, except for the few based on the counter-examples.Footnote ¹⁸

The first rigorous experiments inspired by these counter-examples did not take place until the early 1960s, and they became more varied only during the 1970s. This strikes one as being an extended delay – the more so since, both shortly after and shortly before the ‘poll’, a significant number of experimental studies had come out on decision making, both under certainty and uncertainty.Footnote ¹⁹ Admittedly, these studies were primarily conducted by psychologists, who studied decision rules without attending to their axiomatic decomposition, unlike the economists and statisticians who were Allais’s prime public. Relatedly, the prevailing statistical methodology favoured the measurement of numerical magnitudes, especially of utility functions, even when the purpose was to confirm or refute an abstract hypothesis that could have been checked on merely qualitative implications. These contextual facts may explain why Edwards (Reference Edwards1954: 394) – perhaps the most prominent decision psychologist of the time – dismissed Allais’s questionnaire as being unusable.Footnote ²⁰ Such direct counter-examples as the paradox, which merely involved conflicting preference statements, long remained the preserve of properly trained theorists. They dissected them according to their particular goals, paying little attention to their empirical relevance. This applies to Savage (Reference Savage1954 [1972]: 101–104), Luce and Raiffa (Reference Luce and Raiffa1957: 25), and at greater length Raiffa (Reference Raiffa1968: 80–86), all of whom tried to reinstate the EU hypothesis against Allais. Savage’s reconstruction of the paradox is the topic of Appendix 3.

Five remarkable studies – by MacCrimmon (Reference MacCrimmon, Borch and Mossin1968), Moskowitz (Reference Moskowitz1974), Slovic and Tversky (Reference Slovic and Tversky1974), MacCrimmon and Larsson (Reference MacCrimmon, Larsson, Allais and Hagen1979) and Kahneman and Tversky (Reference Kahneman and Tversky1979) – eventually gave the Allais paradox a genuine experimental status. The first three studies involve populations of 30 to 50 subjects, the last rather more (up to 72), and the penultimate one rather fewer (19).Footnote ²¹ In each case, paired choice problems are posed, using Allais’s exact set-up (Slovic and Tversky), or a variant that preserves the probabilities but changes the outcomes (Moskowitz), or a variant that changes both (Kahneman and Tversky), or an even more remote form (MacCrimmon). In the study that is the most thorough, despite having the smallest number of subjects, numerical probabilities and money outcomes give way to algebraic parameters and multiple sub-experiments result from taking values in the ranges of variations of these parameters (MacCrimmon and Larsson). Across the studies as a whole, the percentage of violations of the EU hypothesis varies from 27% to 61%. The latter percentage appears in Kahneman and Tversky, who artfully calibrate their numerical values; one may consider this less representative than the others, which are concentrated between 27% and 42% (see MacCrimmon and Larsson’s summary, Reference MacCrimmon, Larsson, Allais and Hagen1979: 366–367).Footnote ²²

3.2 Empirical regularities and empirical refutations connected with the Allais paradox

The multiple replications in the 1970s have conferred upon the paradox an empirical robustness that is difficult to challenge. Thanks to their parametric method, MacCrimmon and Larsson also discovered some non-obvious general characteristics. On the one hand, violations do not depend on the absolute value of monetary outcomes, but on their relative value, following a given rule. On the other, they still occur when the certain lottery – p ₁ in Allais – is replaced by uncertain lotteries of a relevant type. Curiously, one feature that jumps out at any casual observer long remained unelaborated. Subjects who contravene the EU hypothesis align themselves with one of two violations, neglecting the other; thus, in Allais, most of them select (p ₁, q ₂) instead of (q ₁, p ₂). Conlisk (Reference Conlisk1989) seems to have been the first to explore the significance of this feature.

MacCrimmon and Larsson (Reference MacCrimmon, Larsson, Allais and Hagen1979: 360) used the term common consequence to designate the parametric form subsuming Allais’s numbers, and by the same token the empirical regularity, or effect, covering the paradox.Footnote ²³ Today’s decision theory has kept the term but redefined the effect more generally and abstractly. In this new sense, well explained by Machina (Reference Machina, Stigum and Wenstop1983, Reference Machina1987), the effect occurs in the following pattern of choice:

Here x is a money amount, and M, M′ and Q are lotteries; x lies between the worst and best amounts of M; and M stochastically dominates M′.Footnote ²⁴ The subject makes the choices (p ₁, q ₂), which contravenes the EU hypothesis. Allais’s numbers are recovered by setting x = 1 million, M = 1 million with prob 1, M′ = 0 with prob 1, and Q = l as defined in last section. This takes M and p ₁ to be degenerate lotteries, which the common consequence effect per se does not require.

The paradox can be subsumed under another putative generality. For Kahneman and Tversky (Reference Kahneman and Tversky1979: 245), it follows from a certainty effect; that is, the salient feature of the paradox is that p ₁ is a degenerate lottery offering a satisfactory outcome. This diagnosis has obvious force, but needs balancing against the previous one. Unfortunately, the two effects compare poorly, because they are not subject to the same degree of theoretical elaboration. While the common consequence effect has received a precise definition, the certainty effect has basically remained at the intuitive level at which it was originally introduced. Here is the most technical statement we found in Kahneman and Tversky: ‘the overweighting of outcomes obtained through certainty relative to outcomes that are only probable’ (Tversky and Kahneman Reference Tversky and Kahneman1986: S265). This is too vague for making it possible to apply MacCrimmon and Larsson’s method of parametric variations. All the same, Kahneman and Tversky’s unsystematic variations on the certainty effect deliver substantial rates of EU violations, and without being able to say much more, decision theory has allowed the two effects to coexist. In sum, the Allais paradox, viewed as a merely empirical phenomenon, lives at the crossroads between the certainty effect and the common consequence effect. The decision theorists’ views can be more entrenched, however: for example, Machina (Reference Machina, Stigum and Wenstop1983, Reference Machina1987) assigns the Allais paradox to the common consequence effect without discussing the certainty effect, while Munier (Reference Munier1991, Reference Munier, Diemer, Lallement and Munier2011) privileges the latter. The recent experimental treatment of the paradox often focuses on the certainty effect: see Weber (Reference Weber2008) and Andreoni and Sprenger (Reference Andreoni and Sprenger2009).Footnote ²⁵

Although sometimes overshadowed by the fame of the paradox, Allais penned another brilliant counter-example in 1952–1953. The violation of the EU hypothesis in the latter arises when one pair of choices results from the other by a homothetic reduction of the probability values of the best outcomes. MacCrimmon and Larsson (Reference MacCrimmon, Larsson, Allais and Hagen1979: 350) also gave this phenomenon algebraic form and studied it experimentally using differing numerical values for the parameters. They use the term common ratio to refer to either the parametric form or the associated effect. Today’s decision theory – see again Machina (Reference Machina, Stigum and Wenstop1983, Reference Machina1987) – has taken up the expression, this time keeping the initial parametric form unchanged:

with p > q, x < y, and α (the homothetic factor) lying between 0 and 1. The subject makes the choices (p ₁, q ₂), which contravenes the EU hypothesis.Footnote ²⁶ The many studies bearing on this effect, including MacCrimmon and Larsson’s (Reference MacCrimmon, Larsson, Allais and Hagen1979: 359), show rates of violations clearly above those obtained for the common consequence effect. In general, these studies make the parametric restriction p = 1, which amounts to involving the certainty effect, and thus no doubt increases the number of violations. Both the common consequence effect and the common ratio effect have provided powerful heuristics for the theoretical development of non-EU theories, which we do not cover here (readers may consult the references in fn 3).

Related to the empirical phenomena discussed above is the observation that different numerical utility curves are obtained for the same subject, depending on the kind of choice data that the experimenter elicits from this subject. This non-uniqueness would not occur if the EU hypothesis regulated the choices. Allais noted the phenomenon while analysing some answers to his ‘poll’ (Reference MacCrimmon, Larsson, Allais and Hagen1979: 474), and Karmakar (Reference Karmakar1978) identified it more fully; it was repeatedly corroborated afterwards. Machina includes this utility evaluation effect in his classic list of effects that are damaging for the EU hypothesis.

The philosophy of empirical sciences treats an experiment as refuting its target hypothesis only if it has been properly replicated. The paradox met this condition in the 1970s and can thus be said to have refuted the EU hypothesis. As to turning the refutation against VNM independence specifically, this is another matter, a proper discussion of which exceeds the scope of the present paper. Logically, the experiments associated with the effects of this section are all ambiguous. One can attribute their results to a violation of VNM independence, a violation of the principle of compound lotteries, or both violations occurring simultaneously. Decision theorists here face an underdetermination problem in the sense classically explained by the philosopher of physics Duhem.Footnote ²⁷ A majority of them have persisted with Allais’s claim that the refutation of the EU hypothesis reduces to that of VNM independence; some have even extended this claim to weaker forms than (A3) that emerge when the principle of compound lotteries is duly formalized. This overall conviction relies on an admixture of arguments, some logically articulated, others rough and intuitive, and still others pragmatic. The most influential argument may well be the pragmatic one that alternatives to the EU hypothesis are mathematically easier to develop by retaining the principle of compound lotteries, or at least some form of it. This is typical of the hesitant way in which the empirical sciences resolve the Duhem underdetermination problem (more on this line in Mongin Reference Mongin, Brenner and Gayon2009).Footnote ²⁸

4.1 The 1952 Paris conference and the invention of the Allais paradox

From 12 to 17 May 1952, the Centre National de la Recherche Scientifique (CNRS) hosted a meeting in Paris of the leading representatives of the nascent discipline of decision theory; this is actually one of its founding moments. Dealing with the theory of risk in economics, the conference brought two groups together, one defending the EU hypothesis, the other disputing it. The former included nearly all the American participants, from which came the name ‘American School’ that Allais would use from then on to designate it, but drew supporters from elsewhere as well. De Finetti also supported the EU hypothesis and Frisch was mildly favourable to it. The latter group was reduced essentially to three Frenchmen, Allais, Massé and Morlat. It is noteworthy that they had been trained as economic engineers, whereas most of their opponents were academics and not all were economists. Guilbaud, a mathematician who had recently converted to social sciences, put the proceedings together for publication.Footnote ³⁰ He included both the papers and the ensuing discussions, making this volume a source of great documentary value.

On the pro-EU side, Savage’s (Reference Savage1952) and Samuelson’s (Reference Samuelson1952a) contributions attracted most attention, the former because his axiomatization of subjective expected utility (SEU) was of stunning novelty, and the latter because he provided a VNM axiomatization that properly emphasized the role of VNM independence and developed an energetic defence of the EU hypothesis on this basis. Samuelson had initially reacted negatively to von Neumann and Morgenstern’s 1947 appendix (von Neumann and Morgenstern Reference von Neumann and Morgenstern1944), but subsequently had become an enthusiastic convert to their approach, and his axiomatic work was meant to present it in the best possible way. We discuss (and actually rebut) Samuelson’s defence of VNM independence in Appendix 2.Footnote ³¹ Two other participants who might have fought on the pro-EU side kept a low profile at the conference, namely Friedman (despite his close association with Savage) and de Finetti (despite the clear connection between some of his pre-war work and the EU hypothesis).

The anti-EU minority was impressively active, bringing ever more objections to both the axioms and the hypothesis itself, sometimes in the form of counter-examples, sometimes arguing from alternative proposals. Massé and Morlat’s paper is impressive, and a short comment made by the latter perhaps even more so, but Allais had a more prominent role in the debates, and Frisch’s closing address registers this state of affairs by singling him out as the representative of the dissident position. He made no less than two presentations, the first extending general economic equilibrium to situations involving risk, the second devoted to the individual theory of choices involving risk – the one that concerns us here.Footnote ³² Moreover, he unceasingly tried to refocus the often diffuse discussion on the fundamental disagreement between him and the American school.Footnote ³³

Reading through the proceedings, one is surprised to find that they neither contain nor even mention the paradox or any of the other counter-examples that made Allais famous. By contrast, Massé and Morlat’s contribution contained one, and the aforementioned comment by Morlat another, which may be the better of the two (since it is difficult to find, we restate it in Appendix 1). Both already consist of paired choices that contradict the EU hypothesis and are offered as objections to VNM independence. The delayed appearance of the Allais paradox raises a curious historiographical problem, already discussed by Jallais and Pradier (Reference Jallais, Pradier, Fontaine and Leonard2005). The paradox first appeared in 1953 in Econométrie, which has wrongly led to an impression that it was presented at the conference; but it actually belongs to Allais’s memoir, which was for some reason included in Econométrie alongside the conference proceedings. It is known that Allais worked on the memoir during the summer of 1952, and this places an upper limit to the dating of the invention of the paradox. After carefully studying the whole sequence, Jallais and Pradier (Reference Jallais, Pradier, Fontaine and Leonard2005) settle on a lower limit for the circulation – if not the invention – of the paradox. In an encounter on the fringes of the colloquium that has remained famous among decision theorists, Allais presented Savage with one or several choice problems that trapped him into violating the EU hypothesis. Jallais and Pradier accept the widespread view that the paradox was presented during this conversation, and, exploiting more evidence, add that it is unlikely to have circulated earlier than this, despite Allais’s occasional suggestions to the contrary. As for the possible influence of Massé and Morlat’s counter-examples on his own, Jallais and Pradier neither confirm nor reject it.Footnote ³⁴ They deflate the issue by arguing that the two authors’ conference examples were themselves nothing exceptional at the time. Others were already known, and to invent more of them was a common mathematical exercise among the French (the Americans, for their part, seem to have first come across this genre while in Paris).

These scholarly points are important, because they help us recognize what Allais’s target might have been when he launched his counter-examples: they are like a last argumentative resort, for use once more traditional means of exposition are exhausted. During the colloquium, Allais had expressed himself at a higher level of abstraction and with more use of mathematics. Although little inclined to the axiomatic method himself, he was drawn to it by the need to engage his opponents on their own ground. But judging from the conference discussions, this highbrow strategy failed. Savage and Samuelson remained inflexible, and the less committed participants contented themselves with being polite. In writing the memoir, and then extracting from it an article that emphasized the counter-examples at the expense of pure theory, Allais presumably believed that he would at last shift the balance in his favour.Footnote ³⁵

4.2 Allais’s structured argument against the EU hypothesis

The Econometrica article, on which we now focus our comments, puts forward four theses intended as the steps of a demonstration. The logical structure is so clear that it is surprising that subsequent decision theorists have paid so little attention to it. The theses run as follows. (1) The EU hypothesis does not apply to ‘the real man’, and everyone today agrees on this, including in the American school. (2) The only remaining question is how the hypothesis relates to the ‘rational man’, and this cannot be settled by merely pointing out that it conforms to an axiom system, since the question would arise again in relation to this axiom system. (3) If one begins with an ‘abstract definition’ of individual rationality under risk, one cannot deduce the EU hypothesis from it, because the latter is far more constraining than any such definition would entail. (4) Another way of approaching individual rationality under risk is to study how rational persons act, but this ‘experimental definition of rationality’ is no more favourable to the EU hypothesis than is the preceding one. Allais does not state his definitive conclusion entirely clearly, and we can distinguish two possible variants for it. Either (5) there are circumstances in which individual rationality under risk is compatible with departing from EU (the weak conclusion), or (6) there are circumstances in which individual rationality under risk entails departing from EU (the strong conclusion).Footnote ³⁶

Step (1) already signals an open break with what would become the standard interpretation of the paradox. Why would Allais propose an empirical refutation of a hypothesis that he already took to be false? And one, moreover, that he thought everybody else took to be false? Strange as this seems in retrospect, Allais was only slightly exaggerating when he said that his colleagues shared his empirical dismissal of the EU hypothesis. With the single exception of Friedman, the conference participants were not prepared to fight over this ground. Since this is an important fact both in itself and for our contrast of interpretations, we adduce some evidence from the conference proceedings.

Marschak had set the stage of the conference by an early pronouncement contrasting the ‘descriptive’ with the ‘normative’ use of the EU hypothesis. Only in the latter use, he said, could the hypothesis be defended as such; in the former use, it was ‘too rigid’ and would have to enter a scheme of stochastic preference (Econométrie: 25–26). Later in the conference, Savage prefaced the formal statement of his axioms with the claim that they were ‘principles of conduct or decision’ that ‘nobody would like deliberately to contravene’. It followed that agreement with real behaviour was ‘sometimes’ possible (Econométrie: 29). This weak defence is compatible with the admission that the EU hypothesis is empirically refuted as a generality. In commenting on the paper, Allais was quick to exploit the loophole: ‘it is naturally not here a matter of the behaviour of real men since M. Savage freely concedes his theory is not applicable to them’ (Econométrie: 38). Savage did not complain, and the debate then moved on to the normative issues. Only Friedman made a sustained attempt at shifting it towards ‘empirical generalizations’ (Econométrie: 71), but from the terse comments he attracted, his paper does not seem to have been a hit.Footnote ³⁷ Then came Samuelson, trenchant as ever: ‘so far as explaining the behaviour of real men on our planet is concerned, the [EU] hypothesis is rather insignificant’ (Econométrie: 142). Having ‘given up the empirical validity of the theory’, he would defend ‘its normative interest’. Samuelson’s very clear statements channelled the discussion exactly as Allais wished, and the latter could emphatically comment: ‘There is a well-posed problem: should a rational man behave according to the [EU] hypothesis, or should he not? The problem lies here and nowhere else’ (Econométrie: 155).

These quotes from the conference proceedings buttress the contrast of interpretations on which the present paper revolves, and as a secondary purpose they help dispel a not uncommon misconception of the beginnings of decision theory. As the story goes, the early followers of von Neumann and Morgenstern privileged an empirical interpretation of the EU hypothesis and its underlying axioms, and it was not until they encountered Allais’s counter-examples, also viewed from the empirical perspective, that they shifted to a normative interpretation. This ‘normative retreat’ story has no clear grounding in von Neumann and Morgenstern, and directly contradicts Marschak’s and Samuelson’s very early claims that the theoretical merit of the EU hypothesis could only lie in the normative value of the underlying axioms. Our quotes further show that this story runs afoul of the best evidence one can gather on the collective state of mind in 1952. The thin textual basis for ‘normative retreat’ has to do with Savage, who first supported a bluntly empirical interpretation of the EU hypothesis in his 1948 paper with Friedman, and later moved to the normative interpretation that permeates his 1954 Foundations. However, our evidence casts light even on Savage’s attitudes. His conference paper already expresses doubts about the empirical prevalence of EU-conforming choice, and emphasizes rather the normative plausibility of his axiomatic conditions. If Savage did change his mind, this change was well advanced in 1952 and certainly did not have to await the Allais paradox to crystallize.Footnote ³⁸

We now return to the thread of Allais’s reasoning. Although obvious, the point made in step (2) was justified contextually. Von Neumann and Morgenstern (Reference von Neumann and Morgenstern1944: 9) had stressed the need to explore the formal meaning of ‘rational behavior’, and some of their followers, like Marschak (Reference Marschak1950: 111–112, 134–135), had taken up this theme less cautiously, by identifying ‘rational behavior’ with the axiomatic content of VNM theory. By contrast, step (3) is very informative. For Allais (Reference Allais1953b: 522), the ‘abstract definition’ of individual rationality has only three components: the ordering property of choices, the use of objective probabilities if they are available, and an ‘absolute preference axiom’, which in today’s words means that preferences respect first-order stochastic dominance comparisons.Footnote ³⁹ This purposefully short list is not enough to entail the EU hypothesis. In our chosen axiomatic decomposition, the first item corresponds to (A1), and the third is strictly weaker than (A3). Among von Neumann and Morgenstern’s followers, some ugly confusion had prevailed on this logical relation, but Allais made no mistake.Footnote ⁴⁰ Concerning the second item, Allais contents himself with the banal observation that to use objective probabilities is a condition for practical success and it is therefore rational to follow them if possible. This comment has no analogue in the formal treatment. Notice that frequentism is Allais’s conception of an objective probability.

Step (4) is where the Allais paradox belongs. It illustrates the ‘experimental definition’ of rationality at work, and, being allegedly decisive, closes the whole argument against the EU hypothesis. The need for this step follows from the fact that step (3) cannot be sufficient. EU theorists could retort to Allais that his ‘abstract definition’ begs the question of what rationality is, no more and no less than their axiom systems do. Allais would of course resist the objection, arguing that his definition is not as specific as their axiom systems; but since specificity is a matter of degree, this comparison does not settle the matter. We explain the presence of step (4), hence of the paradox, by the conjecture that Allais expected this objection and thus felt the need to say something more against the EU hypothesis. Having located the paradox in Allais’s structured argument, we proceed to discuss the normative force that Allais conferred on it.

5.1 Choice-relative normative senses and their limitations

Among the reasons for selecting (p ₁, q ₂), the most obvious one is the certainty of gain in p ₁. According to Allais (Reference Allais1953b: 527), a ‘very prudent’ person will tend to prefer p ₁, which guarantees a not inconsiderable amount, to q ₁, which can offer more but without an absolute guarantee. And this same person might well prefer q₂ to p₂ , on account of a bolder criterion of choice – in the first instance the mathematical expectation of gain – because in this case both options are similarly uncertain. Another less obvious reason matters, although Allais presents it in connection not with the paradox itself but with the counter-example that comes next in his article. This says that the chances of outcomes in a lottery should not be evaluated in isolation from each other, since they exist in relations ‘of complementarity (or of non-complementarity)’ (Allais Reference Allais1953b: 528). In terms of the lotteries in the paradox, the 11 chances of winning 100 million do not mean the same thing in p ₁, where the 89 remaining chances concern the same gain, as in p₂ , where they concern a zero gain. Similarly, the 11 chances of drawing lottery L do not mean the same thing in q ₁, where the remaining 89 chances bear on a positive gain, as in q₂ , where they bear on a zero gain. These comparisons require the principle of compound lotteries to hold.

Allais’s second argument seems less natural than the first, since what it compares are lotteries across choice problems (here p ₁ with p₂ ), as opposed to lotteries within the same choice problem as in the first (p ₁ with q ₁, or p ₂ with q ₂). Such reasoning is unlikely to enter individuals’ deliberations, unlike the observation that p ₁ differs from the other lotteries by being certain. However, reflective individuals may come across it when trying to rationalize their choices ex post. The main advantage of the second argument over the first is its higher generality. Subjective ‘complementarities’ or ‘non-complementarities’ can occur for various patterns of chances, in particular when chances do not coalesce into certainty and thus extreme prudence has nothing to recommend. The 1953 article privileges the first argument, and several of Allais’s later texts do not even mention the second, which seems like a weakness from his own perspective.Footnote ⁴¹ There is a striking parallel between the two arguments and the certainty and common consequence effects of the empirical literature. What this literature offers as an explanation of choice behaviour now appears from the normative angle of what may justify it.

Some decision theorists have endowed Allais’s paradox with normative force, these being exceptions to what in Section 3 we described as the received view. The most significant proposal made in this respect differs from Allais’s suggestions both in terms of content and level of technicality. It takes up a dynamic argument that EU theorists have long made in favour of VNM independence. Broadly speaking, these theorists reformulate static choices, as in the paradox and related counter-examples, as if they were multi-stage choices for which new principles – relative to intertemporal decision – can be defined formally and assessed normatively. Along this way, an equivalence emerges between VNM independence and a conjunction of such principles. Then, a normative argument for this condition follows from the claim that these equivalent principles are sound. EU theorists make this claim, and use the resulting argument to dismiss the paradox and other counterexamples, granting the assumption that they involve a violation of VNM independence rather than any other condition. Whereas EU theorists seemed to have taken the upper hand with this argument, Machina (Reference Machina1989, Reference Machina, Bacharach and Hurley1991) countered it by denying that the dynamic principles equivalent to VNM independence were all sound, and he used the Allais paradox to illustrate his point, thus offering a new normative defence of it. Some decision theorists have pursued the same line, while EU theorists have tried to reinforce their initial position.Footnote ⁴² This is an important debate, but it does not easily connect with Allais’s own defence of the paradox. Remarkably, at the Paris conference, Samuelson presented what may be the first historical defence of VNM independence by dynamic principles, but Allais brushed it away as being irrelevant (see Appendix 2 for details).Footnote ⁴³

We now return to Allais’s above two arguments, asking whether they are capable of underwriting a judgement of rationality in favour of the paradoxical choices. The notion of rationality applicable to choices, as distinguished from the agent who performs them, is consistency of choice. Rationality viewed in this way only imposes formal conditions upon choice, whereas the justifications above determine it substantially. A comparison might bring this point home. Standard economics, which epitomizes the notion of rationality just mentioned, treats the existence of a utility function and the property that choices maximize it as being desiderata of rationality, on the ground that choice inconsistencies arise if they are not satisfied. By contrast, it treats special properties of a utility function, such as monotonicity or concavity, as being neither rational nor irrational, because choice consistency says nothing here. The attitudes of letting the certainty of an outcome weigh heavily, or of paying attention to chance complementarities, are broadly analogous to the special properties of a utility function, i.e. neither rational nor irrational by the consistency-of-choice rationality criterion. Thus, the answer to the question that begins this paragraph must be negative. The same answer would actually follow from step (3) by a less roundabout argument. If the ‘abstract definition’ of rationality is too weak to entail the EU hypothesis, it should consistently not entail any violation of this hypothesis either.

To take stock, we have found genuine reasons for making the paradoxical choices, but are unable to endow them with full normative force. If we stopped the analysis here, only the weak conclusion (5) would emerge. One would be entitled to discard the objection of irrationality against the paradoxical choices, but not to turn it against some EU-respecting choices, as the strong conclusion (6) requires. Modest in itself, this upshot would stand out only by virtue of the historical circumstances. At the Paris conference, Samuelson had daringly claimed that the VNM axioms derived from the very notion of rationality under risk, and Savage had made related – though not so sweeping – statements. Against such strong contentions, Allais only needed to establish (5). Actually, step (3) would already fulfil the argumentative task if the audience were prepared to endorse Allais’s weak ‘abstract definition’ of rationality. Only against a recalcitrant audience does step (4) enter play, and this further limits the impact of the paradox. However, there is more to be said for its normative force.

5.2 An agent-relative normative sense and its promises

Reading between the lines of the 1953 article, we discern another argument, this one concerning the rationality of the agents themselves as against the rationality of their choices. Few – if any – readers of Allais have noted that he discusses his counter-examples in terms of agent-relative properties. These are: being ‘prudent’ or ‘very prudent’ (Allais Reference Allais1953b: 524–525, 527–529); being ‘very well aware of the probability calculus’ (Reference Allais1953b: 524); being ‘rational by general consent’ (Reference Allais1953b: 525, 527–529); being ‘both reasonable and prudent’ (Reference Allais1953b: 533); and of course, being ‘rational’ (throughout and in the title). What characterizes the ‘rational man’ in contradistinction to the ‘real man’ appears to be the ability to choose in an all-things-considered, fully reflective manner that eschews any later change of mind, even in the presence of an opponent (see Reference Allais1953b: 539, concerning Allais himself as an example). This ability should apply not in isolation, but across multiple choices (see Reference Allais1953b: 534 and elsewhere). Extrapolating from these hints, we propose to say that, for Allais, the ‘rational man’ strives to conform his particular choices to general rules of conduct that he is prepared to answer for.Footnote ⁴⁴

Such a conception fits in with rationality as consistency, provided the latter idea is appropriately broadened. Consistency in the standard economics sense is a property of choices alone, and thus independent of the agents’ motivations. However, in a different sense, which surfaces in various philosophical writings, it is the property that an agent displays of being able to subsume his or her actions – choices being a particular case – under general rules of conduct. Thus construed, consistency becomes a property of the agent’s motivations. This notion of rationality as internal consistency must now be related to the other agent-relative properties that the last paragraph mentioned. Let us consider these properties in turn.

It is easy to dispose of the property of being rational by common consent. One should clearly view this only as a (fallible) criterion for recognizing rational people, not as a component of rationality itself. Prudence raises a more subtle problem. According to the traditional definition, prudence means the ability to conduct the affairs of one’s life wisely; this would make it partly coincidental with instrumental rationality, and some philosophers including Kant have in fact equated the two ideas. However, on a more popular conception, prudence is the attitude that consists in forestalling possible harms by relevant precautionary measures. Viewed in this way it is a psychological property, which rational individuals may or may not exhibit. Allais’s own understanding of prudence is not entirely clear, but his conjunctive formula – ‘both reasonable and prudent’ – does suggest the latter rather than the former sense.Footnote ⁴⁵ By using a term of ordinary language, he might have created an ambiguity, but he at least avoided another one, since he thus made clear he did not mean ‘risk-aversion’ in the technical sense associated with VNM utility functions.Footnote ⁴⁶

The ability to make probabilistic judgements raises few substantial problems. If one adheres to the notion of rationality as reflective compliance with general rules of conduct, one has no reason to make probabilistic ability a necessary part of rationality. As to Allais’s own position, it is rather unclear. He uses a conjunctive expression again – ‘rational men and very well aware of the probability calculus’ (Reference Allais1953b: 524, our emphasis). This suggests he endorses the view that probabilistic ability is extraneous to rationality. However, as we have seen, his ‘abstract definition’ of rationality does refer to the probability calculus, at least in one interpretation (the objective one, which Allais favours). We have no easy solution to offer to this exegetical conundrum and simply gloss over it.Footnote ⁴⁷

At any rate, Allais needs to endow his individuals with some probabilistic ability, since the ‘complementarities’ or ‘non-complementarities’ he deems relevant to their choices can only be discovered by those who know how to operate with the multiplicative rule of probability. He also needs this ability to turn his counterexamples specifically against VNM independence. The nagging question of compound lotteries surfaces once again – although now in a normative context of discussion.

With the notion of agent-relative rationality to hand, we can reinforce the previous arguments on choice-relative rationality. Being detached from any personal features, these arguments provided only possible reasons for the paradoxical choices. But if the agents align their choices with general rules of conduct, and these rules agree with the arguments in question, then the possible reasons become actual and the normative status of the arguments is enhanced. The strong conclusion (6) is now within view. This needs a further condition, however: it must be the case that the paradoxical choices are uniquely compatible with the adopted rules of conduct. If the rules encompass both paradoxical and non-paradoxical choices, as in many non-EU construals of the ensuing literature, no argumentative progress towards (6) can take place. It is dubious that Allais went through all these points, but he was clearly tempted by the strong conclusion (6).Footnote ⁴⁸

5.3 The ‘experimental definition of rationality’

Whether one targets the strong conclusion or just the weak one, it is not enough to say abstractly what rational individuals are; one also needs to be able to spot them. Allais’s expression, ‘the experimental definition of rationality’, raises this problem.Footnote ⁴⁹ We now propose several solutions to it, each taking the form of a possible experimental procedure.

• Procedure (P1) involves setting particular rationality criteria in advance, using them to select the participating subjects, and then running the experiment. The objective is to find out what choices these presumably rational subjects make.

• Procedure (P2) does not select subjects in advance, but devises an experimental stage that makes selection possible. The objective is again to find out who the rational subjects are and what choices they make, but this time the selection is endogenous to the experiment.

• Procedure (P3) runs like (P2), and pursues the same objective, while also trying to learn more about what rationality substantially constitutes; for the data may raise questions that point beyond the conception of rationality as consistency, which still underlies (P2). Hence, this time it is rationality itself that the experiment contributes to determining.

(P1) is Allais’s own procedure. By appealing to his intuition about the ‘rational man’, he preselected the recipients of his questions, first in informal asides at the Paris colloquium, and then more systematically with the famous ‘poll’. Those whom he approached were primarily decision theorists, and as such capable of formulating rules of conduct for risky choices. Presumably, Allais expected nearly as much from his École des Mines students, another sophisticated audience. But it is of course dubious to correlate rationality with theoretical reputations and even just the passing of exams, and Allais’s poll has been mocked – not without reason – for manifesting the prejudices of a meritocratic polytechnicien.

(P2), an immediate improvement, uses the experiment itself to select rational individuals. Since they are disposed to relate any particular choice they make to the rules they endorse, it is sufficient to put this disposition to the test. The protocols can be summarized as follows (more detail in the next section). They will combine stages of choices among lotteries, with reflective stages in which subjects evaluate the rules the experimenter submits to them for consideration. Subjects who endorse rules that run counter to these choices, or disapprove of rules favouring them, and nonetheless persist in making such choices, exclude themselves as being irrational.

(P3) is linked to (P2), but extends its explanatory aim. Here the notion of rationality not only serves to identify the choice-making subjects, but itself becomes an object of study. The protocols of (P2) may introduce several operative notions of consistency, and one might then try to investigate their coincidence or lack of same.Footnote ⁵⁰ This inquiry remains within the purview of rationality as consistency, but the data produced by implementing (P2) may suggest more remote questions. For instance, the experiment might employ diverse stimuli for rational reflection – the experimentalist might intervene in person, or not, subjects might run through their reflections in isolation, or collectively, and so on. If these stimuli operate differently, a comparison is in order. It may also be that definite psychological characteristics emerge among the subpopulation of rational subjects – e.g. they might display more or less prudence than average. Such correlations would enrich the understanding of rationality with substantial, not merely formal, elements. All these questions belong more to the cognitive sciences than to decision theory as traditionally conceived. (P2) is already an ambitious and provocative procedure by ordinary standards, and the next section will focus on it, leaving (P3) as a mere sketch of an alternative perspective.

6.1 Early attempts at experimenting on rationality

MacCrimmon’s study (Reference MacCrimmon, Borch and Mossin1968) calls for special attention, because it influenced the others and made the connection with Allais’s normative perspective.Footnote ⁵² This study proposes nothing less than probing ‘the normative implications of the postulates’ of decision theory (Reference MacCrimmon, Borch and Mossin1968: 4). Five postulates are dealt with, each one paired with a specific experiment; among them, (A3) has a variant of the Allais paradox as its experimental counterpart. The chosen variant is questionable, however, as MacCrimmon’s later paper with Larsson would indirectly recognize. Instead of considering only probability values, MacCrimmon assumes states of the world that realize them in particular ways, a translation that changes the meaning of (A3). This is the same confusion of frameworks we already noticed in early discussions of the EU hypothesis (see also Appendix 3).

This weakness compromises MacCrimmon’s results, but not his general approach, which was to pay attention to the subjects’ rationality within the experiment. In the case of the Allais paradox, subjects had first to choose between (analogues of) p ₁ and q ₁, and p ₂ and q ₂; they then received two ‘prepared answers’ in written form, each stating both a pair of choices and reasons justifying this pair. Allegedly, these answers came from other subjects in a comparable experiment. One of them had (q ₁, q ₂) and a defence paraphrasing VNM independence (or rather what takes its place here), the other had (p ₁, q ₂) with an argument drawing on the certainty effect, and subjects had to state in writing which one they selected and why they did so. This whole process was repeated twice over with minor variations. Finally, the experimenter conducted face-to-face interviews to check whether subjects had understood the experiment properly, and make them evaluate the competing justifications (as well as others that might have occurred to them).

This protocol corresponds to MacCrimmon’s aim of probing ‘the normative implications of the postulates’ if one grants the Allaisian conception that the judgement of rational individuals can serve as a touchstone for the rationality of choices. By prompting a deliberation process among subjects, it operationalizes the idea of the rational individual within the experiment, and thus belongs to the (P2) type of our taxonomy. However, MacCrimmon does not implement (P2) properly. When stating his results, he emphasizes that the interview stage witnessed a shift towards conformity with (A3), but he does not say how many of the subjects who manifest the shift are rational in the desired sense of achieving consistency between their choices and reasons. That is, MacCrimmon uses his protocol to enhance the rationality of his subjects, but finally refrains from turning it into a selection device, as (P2) more specifically requires. The other papers of the sequence show the same vacillation except, to an extent, for the one by MacCrimmon and Larsson (Reference MacCrimmon, Larsson, Allais and Hagen1979), which is the last and most thorough of them all.Footnote ⁵³

The interview plays an equivocal role in MacCrimmon’s protocol. It verifies that the subjects have understood the ‘prepared answers’, possibly identifies their other reasons, and collects their considered final positions. But clearly, the subjects might take as a persuasive argument something which the experimentalist thinks of being only as a semantic elucidation. Struck by this objection, Slovic and Tversky (Reference Slovic and Tversky1974) replaced the interview with another round of choices, while preserving a general ternary structure: a stage of initial choices, a stage of presenting reasons, and a final stage that encapsulates the normatively relevant information. The authors also modified MacCrimmon’s first two stages by returning to a formulation of the paradox in terms of probability values alone, a change that could shift the third stage in the paradoxical direction. Indeed, at this stage, they found that 17 out of 29 subjects chose in the Allais way.Footnote ⁵⁴

Not stopping there, Slovic and Tversky thought up another experimental sequence consisting of two stages only: the first devoted to evaluations, and the second to choices. This experiment uses the same choice problems and the same statement of reasons as before, but different subjects. In the first stage, Slovic and Tversky introduced the reasons to the subjects, and then asked them to evaluate those reasons on a numerical scale. The second stage consisted of choices. The choices turned out to be severely discrepant with the numerical marks given to the reasons (the reasons tending to favour Allais, the choices clearly conforming to EU). Slovic and Tversky drew no conclusions from the study overall because, quite obviously, they did not know how to reconcile this experiment with the previous one. The scepticism they finally expressed connects with the shift that Tversky would soon make, giving up any further experimental implementation of rationality, and turning instead to purely empirical research on behaviour. This was a pity, since the conflict between the two experiments called for performing yet others (admittedly, more in line with (P3) than (P2) according to our taxonomy).

Likewise starting out from MacCrimmon, Moskowitz (Reference Moskowitz1974) had also arrived at a ternary structure, i.e. a stage of presenting reasons placed between two stages of choice. His study includes two notable innovations. First, he compares different visual modes of presentation for identical choice problems, thus foreshadowing the ‘framing effects’ of later decision psychologists.Footnote ⁵⁵ Second, he includes in the intermediate stage a free discussion among the subjects. At the final stage, subjects moved towards the EU hypothesis, but a significant proportion remained faithful to the paradoxical choices.

The last study, by MacCrimmon and Larsson (Reference MacCrimmon, Larsson, Allais and Hagen1979), which we brought to the fore in Section 3, needs to be reconsidered here, since it takes up and actually pushes much further MacCrimmon’s initial project of testing normative statements of decision theory.Footnote ⁵⁶ Here the protocol returns to a binary structure: a stage of choices followed by a stage presenting the reasons for choice; perhaps as a result of intervening criticisms, the authors eschew the discussion with the subjects. They do not have a second round of choices either, and this may be explained by the sheer complexity of the first two stages, each of which is much richer than in any previous study. In the first stage, MacCrimmon and Larsson test their parametric representation of the common consequence and common relation effect, which leads them to investigate whole ranges of numerically defined choice problems. In the second stage, they introduce an extremely well-stocked inventory of reasons for the possible choices. For the first time, they present these as general rules of conduct, the previous practice having merely involved ad hoc rationalizations for the numerically defined choice problems (as in MacCrimmon’s ‘prepared answers’). Rules are generally formulated in the imperative mood. Some reproduce axioms from decision theory (like VNM independence in several variants), others are rules of thumb contradicting the axioms (like the one about preferring certainty in some circumstances), others reproduce weak necessary conditions for the axioms, and there are yet others relating to utility representations. Once subjects have been introduced to the rules, they are requested to mark them from 0 (complete disagreement) to 10 (complete agreement). This use of a numerical scale accords with Slovic and Tversky’s work, but is more extensive here.

We have already covered the results of MacCrimmmon and Larsson’s first stage. Concerning those of the second stage, the average marks show a lower agreement with EU-style rules than with the Allais-style rule of preferring certainty. This holds for both the common ratio effect (MacCrimmmon and Larsson Reference MacCrimmon, Larsson, Allais and Hagen1979: 358) and the common consequence effect (Reference MacCrimmon, Larsson, Allais and Hagen1979: 368). However, the value of this finding is diminished by the observation that the average marks are rather high (above 5), suggesting a bias towards acceptance. The most interesting results exploit all the information by cross-referencing choices and marks. When dealing with the common ratio effect, MacCrimmon and Larsson divide their 19 subjects into three groups: those whose choices are entirely consistent with EU-style rules (3), those whose choices are only partly consistent with them (4), and those whose choices are clearly inconsistent with them (12). The average mark is very slightly higher for EU-style rules in the first group, and notably higher for the Allais-style rule in the third group. These data indicate some form of consistency between choices and approved reasons, but the data for the common consequence effect are unfortunately chaotic. As MacCrimmon and Larsson discover, subjects whose choices are entirely consistent with EU-style rules (10 out of 19) rank them on average lower than they do the Allais-style rule.

With this cross-comparison of the two stages of the experiment, MacCrimmon and Larsson come close to an endogenous selection of rational subjects, as recommended by (P2), but stop short of this objective. Thanks to the number of choice problems offered at the first stage, they are able, for each effect, to screen out subjects who choose irregularly (e.g. the 4 subjects of the second group of the common ratio effect). Quite rightly, they limit their tentative normative conclusions to regular subjects (e.g. the 3 + 12 of the first and third groups of the same effect). However, we have just seen that, in respect to these, they produce only averages, and this is clearly the wrong kind of data. Only individual data would make it possible to screen out those regular subjects who approve rules that clash with their choices or disapprove rules that agree with these choices – that is, irrational subjects according to (P2). This is the more disappointing since their small population of subjects – 19 as against 38 in MacCrimmon (Reference MacCrimmon, Borch and Mossin1968) and 29 and 39 in Slovic and Tversky’s (Reference Slovic and Tversky1974) two experiments – made the collection of individual data a relatively easy task.

It is thus hardly surprising that MacCrimmon and Larsson can only offer a very broad qualitative conclusion: ‘many careful, intelligent decision makers do seem to violate some axioms of expected utility theory, even upon reflection on their choices’ (Reference MacCrimmon, Larsson, Allais and Hagen1979: 403). Logically, the strong conclusion (6) against the EU only needs an existential claim; in Allais’s terms cited above, ‘certain implications can to the contrary be entirely irrational in some psychological situations’. The problem was to establish this claim beyond doubt, which Allais had not yet managed to do. MacCrimmon and Larsson fill the gap differently from Slovic and Tversky (in their first experiment). The latter have EU-contradicting answers in their second round of choices, which is arguably more rational than the first; the former had EU-contradicting pronouncements, and this carries special weight among rational men, in the (P2) sense of rationality. The two sets of results support each other, which is satisfactory from the viewpoint of establishing the target claim. However, given the high sophistication of MacCrimmon and Larsson’s experiment, one would have expected more. In particular, by using their data to the full, they would have been able to quantify the proportion of rational subjects, as well as the sub-proportions of EU and Allais supporters among them.

MacCrimmon and Larsson offer interesting material for researchers who would like to pursue the study along (P3) lines. Given the large number of rules considered, it is possible to observe the subjects’ patterns of endorsement and investigate how these take into account the logical interrelations of the rules. This criterion of internal consistency could be investigated both for itself and in relation to the external criterion of consistency between endorsed rules and choices, which is the one considered in (P2). The comparison might even be extended to a third notion of consistency that underlies MacCrimmon and Larsson’s results, i.e. regularity, in the sense of making choices that are always pro or contra a given rule or set of rules (we have just seen how they use this notion in reporting on common ratio experiments). Such investigations would drag the experimental treatment of choice in the direction of today’s cognitive sciences.

6.2 Some lessons for today’s experimental economics

Table 1 summarizes the methodological features of the four reported studies and introduces a proposal of our own, which is meant to improve on the experimental implementations of rationality that guided the studies. The main technical innovation of this proposal is to combine Slovic and Tversky’s ternary structure (choices followed by a presentation of reasons followed by choices) with MacCrimmon and Larsson’s restatement of reasons in terms of general rules of conduct. At the same time, we would recommend pursuing the strategy of selecting rational subjects within the experiments more consistently than was done in any of the previous studies. Once again, this requires collecting individual data (very much like the panel data of micro-econometrics).

Table 1. Summary of the methodological features of the four reported studies

With the proposed ternary structure, the consistency between choices and endorsed rules will be assessed on two different sets of choice data. It is appropriate to give more weight to choices at the last round since the reflective stage influences them. With this in mind, we submit some guidelines to select subjects. Two kinds of experimental sequences would suggest an ‘irrational man’ classification:

• initial choices – endorsed rules of conduct in agreement with choices – final choices disagreeing with these reasons (hence also with the initial choices);

• initial choices – endorsed rules of conduct in disagreement with choices – final choices disagreeing with these reasons (for instance similar to the initial choices).

Conversely, two kinds of experimental sequences would suggest a ‘rational man’ classification:

• initial choices – endorsed rules of conduct in agreement with choices – final choices in agreement with these reasons (hence also with the initial choices);

• initial choices – endorsed rules of conduct in disagreement with choices – final choices agreeing with these reasons (hence not with the initial choices).

Subjects exhibiting the last sequence convey their effort at rationality by revising their initial position, but those exhibiting the penultimate sequence may be suspected of being simply inert. However, the second round of choices involves choice problems that differ from those of the first round, which means that subjects who consistently apply the same rule of conduct in the two rounds must have made an effort of thought. At least, they are less likely to be inert than if they were simply asked whether, on reflection, they approve or disapprove of their previous choices.Footnote ⁵⁷ While the proposed screening is no doubt imperfect, it suggests that one can nonetheless implement (P2). This would also require increasing the number of subjects compared with the initial studies. We only know of one current attempt to explore the avenue opened up here.Footnote ⁵⁸