II. SAVAGE’S APPROACH

As author of the first comprehensive presentation of what came to be known as the subjective-Bayesian viewpoint, in the late 1950s and 1960s Leonard Jimmie Savage (1917–1971) was widely regarded as its champion.Footnote ⁵ Among decision theorists the axiomatic set-up of Savage’s Foundations was immediately understood as providing the missing link between von Neumann and Morgenstern’s expected utility theory—where probabilities are objectively given—and a treatment of uncertainty built on rationality principles. This was a long-awaited development by an entire community of scholars fostering a view of the rationality of agents as free from psychological assumptions, that is, not based on introspection but behaviorally founded, derived from observed choices as in the revealed preference approach (Giocoli Reference Giocoli2003, pp. 388–393).

Savage’s main goal was to suggest an economic approach to statistics, to characterize statistics as a behavioral discipline, urging statisticians to embrace an economic way of reasoning when thinking about data. To do this, Savage (Reference Savage1954, p. 7) started off by building up “a highly idealized theory of the behavior of a ‘rational’ person with respect to decisions.” As is well known, Savage’s main result was to demonstrate that an agent maximizing (subjective) expected utility is someone who can be thought of as if obeying a few simple axioms on the set of actions—which Savage called “acts”—she can take, plus certain technical conditions. Together with completeness and transitivity, preferences over acts are assumed to conform to an independence axiom, dubbed the “sure-thing principle” by Savage.Footnote ⁶ These axioms delimit individual preferences over acts whose consequences are dependent on the possible realization of events—in principle, even singular events with no objective probability attached. Savage then showed that subjective probability distributions over events, satisfying the usual laws of probabilities, can be elicited from choices over acts. This made it possible to enlarge the domain of decision theory from risky situations such as lotteries to apparently every kind of uncertainty.

Already in 1951, on the basis of Savage’s (Reference Savage1950) initial presentation of his ideas, Kenneth Arrow had claimed that a new consensus was emerging about the behavioral significance of a distinction between risk and uncertainty. This was the view worked out at the Cowles Commission for Research in Economics in those years, where developing a “genuine science of economic behavior” implied the acknowledgment that the problem of uncertainty should be addressed with renewed mathematical rigor (Cowles 1950–1951): theoretical research in decision-making under uncertainty became representative of a new methodological approach at Cowles, namely, abstract model-building (Herfeld Reference Herfeld2018, pp. 29–30). At Cowles scholars working on a varieties of areas, most notably general equilibrium theory, were moving beyond calculus towards new mathematical technics, concentrating on existence proofs and representation theorems (Düppe and Weintraub Reference Düppe and Weintraub2014).

As part of this methodological turn away from empirically oriented analysis and to normativism, Savage’s axiomatization of decision-making showed the crucial fact that “Knight’s uncertainties seem to have surprisingly many of the properties of ordinary probabilities” (Arrow Reference Arrow1951, p. 417).Footnote ⁷ In their celebrated volume on games and decision theory, Duncan Luce and Howard Raiffa (Reference Luce and Raiffa1957, p. 304) opened the chapter on individual decision-making under uncertainty with an endorsement of Savage’s representation of the decision problem in terms of “states of nature,” “acts,” and “consequences.” They praised the “elegant” feature of his approach that “no concept of objective probability is assumed; rather, a subjective probability measure arises as a consequence of his axioms.” Savage (Reference Savage and Neyman1961, p. 576) himself stated that, in the subjectivist viewpoint, probability simply is “an index, in an operational sense …, of a person’s opinion about an event.”Footnote ⁸

As already noted, not even Maurice Allais’s (Reference Allais1953) questioning about the descriptive validity of the theory had a negative impact on the diffusion of his theoretical corpus, since Savage (Reference Savage1954) had objected to Allais’s critique that the normative content of the theory was untouched with respect to the choices of a rational individual. In response to Allais’s (Reference Allais1953) critique of expected utility—and his own failure, as an experimental subject, to adhere to it when tested—Savage (Reference Savage1954, pp. 102–104) argued that theories of rational behavior have a normative status that is unquestionable even in the light of adverse evidence, which can only indicate “irrational” choices. As a matter of fact, his own experience as a violator was that a rational decision-maker who instinctively violates the theory when confronting Allais’s example will reverse her choice after “thorough deliberation.”Footnote ⁹

In the early 1960s, then, the historical background of the subjective-Bayesian revolution is one that sees Savage’s approach as amply accepted among decision theorists and favorably imported in economics to deal with microeconomic issues in general. Savage could then stay focused on what had actually been his main aim in the Foundations, that of showing how the subjective probability approach could strengthen Wald’s (Reference Wald1945) viewpoint that every statistical problem should be seen as a decision problem, to be solved by the statistician as if she were a rational decision-maker acting under uncertainty, the uncertainty related to the unknown probability distribution underlying the available data. On realizing that instead of providing stronger, behaviorally founded bases for traditional inference, he had made it necessary to abandon standard inference techniques, Savage concentrated on the spread of new statistical tools and effective methods for inference (Giocoli Reference Giocoli2013).Footnote ¹⁰

For instance, his contribution to the Fourth Berkeley Symposium on Mathematical Statistics and Probability was a sort of manifesto for a paradigmatic shift in statistics: Savage objected to the way the traditional view of frequency probability had oriented statistical theory and concentrated on “another view [that] seems now to be entering upon the scene,” the one that “may be called Bayesian or neo-Bayesian” (Savage Reference Savage and Neyman1961, p. 575). As a matter of fact, most of his efforts were directed at an unreceptive audience of frequentists such as George Barnard, Ronald Fisher, Jerzy Neyman, and Egon Pearson, who endorsed a probability approach to statistical inference that was alien to a subjective view of the probability involved.Footnote ¹¹

Therefore, the majority of Savage’s papers after the Foundations of Statistics were devoted to the illustration and defense of the new viewpoint in a quite hostile statistical arena (Lindley Reference Lindley1980).Footnote ¹² Since a relevant question in what follows is why Savage did not address in his publications the critical arguments made by a few critics among decision theorists, it can be argued that his resolve to stay focused on the diffusion of his creed among statisticians contributes to explaining Savage’s reluctance to get involved in a foundational debate.

III. FRIENDLY FIRE IN ECONOMICS

The symposium on “Decision under Uncertainty,” published in 1961 in the Quarterly Journal of Economics (QJE), made apparent both the status already gained by Savage’s Foundations as the mainstream treatment of decision-making and a quest for its fundamental reorientation. Decision theorist Daniel Ellsberg (Reference Ellsberg1961) and economist William Fellner (Reference Fellner1961) examined Savage’s subjective probability approach from the viewpoint of adherents to a well-structured theoretical corpus. Specifically, Ellsberg (Reference Ellsberg1961, p. 657) argued that “the Savage axioms, and the general ‘Bayesian’ approach, are unquestionably appropriate when a subject is willing to base his decisions on a definite and precise choice of a particular distribution,” since in such a situation “his uncertainty … is unequivocally in the form of ‘risk.’” But he also questioned Savage’s approach on the grounds that it was not rich enough to deal with all instances of actual decision environments. Both Ellsberg and Fellner found that the reduction of decision-making under uncertainty into the framework von Neumann and Morgenstern (Reference von Neumann and Morgenstern1947) had devised for risk was unwarranted: they disputed that all uncertainties could be reduced to risks.Footnote ¹³

On the basis of observed violations by a number of colleagues he tested in the late 1950s at the RAND Corporation and Harvard economics department, Ellsberg, a Junior Fellow of the Harvard Society and RAND analyst since 1959, criticized Savage by claiming that vagueness about probabilities could lead individuals to violate the axioms of consistent behavior upon which Bayesian decision theory was based. Savage’s sure-thing principle, in particular, was shown to be violated by individuals acting in what Ellsberg called “ambiguous” environments. Ellsberg’s (Reference Ellsberg1961, p. 646) claim was that, in a series of examples concerning choices from urns containing colored balls, he was able to identify “a class of choice-situations in which many otherwise reasonable people neither wish nor tend to conform to the Savage postulates, nor to the other axiom sets that have been devised.” What is more, Ellsberg suggested that, since in many cases the violations were deliberate—that is, individuals were not inclined to change their choices even after being asked to thoroughly reconsider the significance of the theory—the use of sharp probability priors could be inappropriate even from a normative point of view. In current economics literature these results are well known as the Ellsberg Paradox. Footnote ¹⁴

Fellner’s (Reference Fellner1961) argument was motivated on theoretical grounds. Fellner, a professor of economics at Yale University whose book on oligopolistic competition was one of the first to acknowledge the relevance of game theory, but who had been critical of the way von Neumann and Morgenstern dealt with the beliefs of the entrepreneur (Fellner Reference Fellner1949, p. 37–41), objected to the reasonableness of Savage’s assumption. He argued that the observable decision weights that an individual attaches to prospective events—namely, the probabilities attached to an event if the individual actually bets on the event—may differ from the theoretical probabilities she would attach in Savage’s set-up. When the individual regards some of her beliefs as “shaky judgements,” Fellner maintained, subjective probabilities may be distorted, meaning that they would not obey the usual laws of probability, with the distortion depending on the perceived instability of certain judgments. Fellner concluded that this was to be expected in particular when events are unique, that is, when they do not belong to any “standard process” with objective characteristics. In these cases, “we should not regard it as irrational if a person develops a reaction to uncertainty rather than is guided exclusively by mathematical expectations” (Fellner Reference Fellner1961, p. 685). The similarity with Ellsberg was apparent, since Ellsberg had distinguished unambiguous from ambiguous urns, claiming that people usually prefer to bet on an unambiguous urn with respect to an ambiguous one even though the subjective probability attributed to them by Savage’s approach was supposed to be the same.Footnote ¹⁵

The “Decision under Uncertainty” QJE symposium had Howard Raiffa arguing in favor of Savage’s approach. Raiffa, a mathematical statistician at Harvard University who was developing new mathematical tools for applied business decisions from a subjective viewpoint (Raiffa and Schlaifer Reference Raiffa and Schlaifer1961), reaffirmed his faith in Savage’s axioms as normative guides.Footnote ¹⁶ Referring to experimental sessions he conducted using adaptations of Ellsberg’s example, Raiffa (Reference Raiffa1961, pp. 692–693) admitted that “if certain uncertainties in the problem were in a cloudy or fuzzy form, then very often there was a shifting of gears and no effort at all was made to think deliberately and reflectively about the problem,” confirming the experience of Ellsberg and Fellner. However, the message he drew was different from theirs. Ellsberg’s choice examples simply showed, in his view, that “there is a need to teach people how to cope with uncertainty in a purposive and reflective manner.” Raiffa reported that experimental subjects who were initially inclined to violate the axioms behaved consistently with them when a different way of looking at Ellsberg’s problems was suggested to them.

It is worth noting that the Allais-Savage debate was not referred to by any of those participating in this new round of debate: Allais’s (Reference Allais1953) criticism focused on utility axioms in an unambiguous context with objective probabilities—a situation of risk—while the concern now was the limitations of probability axioms in contexts in which only subjective probabilities were available—a situation of uncertainty. But Raiffa implicitly assumed Savage’s solution to Allais as the conventional way to deal with hypothetical examples: in the experimental sessions he conducted, Raiffa presented to his subjects Ellsberg’s urn problems in a rearranged table form, similarly to what Savage had done (Reference Savage1954, p. 103) when examining Allais’s example.Footnote ¹⁷

As is well known, the mainstream endorsed Raiffa’s rejection of Ellsberg, whose paradoxical results had to wait more than twenty years to be addressed consistently in decision theory (Schmeidler Reference Schmeidler1989). But Savage was not involved in the defense of his approach against Ellsberg and Fellner. Quite probably this was because, as noted above and as the correspondence we shall examine later confirms, he was consumed by the debate in what he perceived as the much more relevant statistical arena. As a matter of fact, in Savage’s published works only a few words are dedicated to Ellsberg’s and Fellner’s criticism. Savage’s annotated bibliography prepared for the second edition of the Foundations of Statistics presents the 1961 “Decision under Uncertainty” symposium as “an account of an important line of dissent from the theory of personal probability and utility” (Savage Reference Savage1970, p. 25).Footnote ¹⁸ Savage (Reference Savage1970, p. 24) also acknowledges that de Finetti and Savage (Reference de Finetti, Savage and Savage1962) “discuss a large number of doubts and difficulties about the actual application of personal probability.” In fact, the notion of interval-valued, as opposed to sharp, probability prior is examined at length in de Finetti and Savage’s Reference de Finetti, Savage and Savage1962 paper. But neither Ellsberg nor Fellner is mentioned in that paper.

Before turning to what the correspondence shows, it is worth recalling that Savage had already faced the issue of vagueness, in both his Foundations and later contributions. Indeed, we shall see that the private discussion with his critics is reminiscent of a distinction between sure and unsure opinions Savage (Reference Savage1954, p. 58) had examined in his Foundations, concluding: “The notion of ‘sure’ and ‘unsure’ … is vague, and my complaint is precisely that neither the theory of personal probability, as it is developed in this book, nor any other device known to me render the notion less vague.” Later, in what can be seen as an unspoken reference to Ellsberg, Savage (Reference Savage, Machol and Gray1962b, p. 165) remarked: “some people see the vagueness phenomenon as an objection [to precise personal probabilities]; I see it as a truth, sometimes unpleasant but not to be escaped by a new theory.” And when discussing certain “difficulties” in the theory of personal probability, he argued: “Some have tried to reflect the phenomenon of vagueness within the theory, while others believe that, though vagueness must somehow be reckoned with, its nature defies formalization” (Savage Reference Savage1967a, p. 308).

In view of Savage’s commitment to the mathematical rigor sponsored at Cowles—namely, to provide a set of rules constraining the decision-maker’s choices—the issue of whether the critics would be able to propose a formalized version of their suggestions is crucial in the correspondence to which we now turn. As a matter of fact, only critiques amenable to formalization, showing themselves operationally founded, would have caught Savage’s attention. He had indeed insisted from the outset that his theory was not about “what to believe in the face of inconclusive evidence, but … what action to decide upon such circumstances” (Savage Reference Savage1954, p. 2).

IV. ARCHIVAL EVIDENCE: FIRST PART

This section presents the private correspondence between Savage and Ellsberg and Fellner, two professed subjective-Bayesians who ended up as critics of the strict version of the approach. The correspondence is illustrative of Savage’s interest in the topic of vagueness and imprecision in probability priors as a possible generalization of his viewpoint. The main theme that emerges is that Savage did not formulate an outright rejection of the criticism, unlike what he had done when he had to confront Allais’s (Reference Allais1953) critiques. One may be tempted to argue that the fact that Savage did not publicly reply to Ellsberg’s counterexamples should be interpreted as showing that he did not attribute to Ellsberg any significant advance as compared with what he had judged as Allais’s inability to disqualify the normative status of the theory. But the exchange with Ellsberg and Fellner shows instead a more open attitude towards the alleged violations. Rather than a clear-cut denial of interest toward the normative impact of Ellsberg’s suggested violations, Savage seems to show an unwillingness to endorse a less firm subjectivist perspective while still fighting for the acceptance of the core of his viewpoint in the statistical arena.

Ellsberg was explicit about his attempt to influence the subjectivist developments. In his doctoral thesis at Harvard—unpublished for many years but circulated among major contributors to decision theory, including Savage and de Finetti—Ellsberg ([1962] Reference Ellsberg2001, p. xlix) acknowledged that “nearly every page of this study testifies to my intellectual debt to L. J. Savage,” to whom he was grateful “for encouraging me to believe that the arguments and counterexamples presented here deserved serious consideration.” Ellsberg’s claim, never objected to by Savage—neither in his published writings nor in the available archived correspondence—is that Savage was among deliberate violators of his own axioms when tested in February 1958 (Ellsberg Reference Ellsberg1961, p. 654).

However, Ellsberg ([1962] Reference Ellsberg2001, p. 165) defined himself a “less exacting ‘neo-Bayesian’” in the footsteps of Good (Reference Good1952), and presented his comments on Savage’s Foundations as “mainly critical, and by some standards heretical with respect to ‘Bayesian’ principles.” Ellsberg made it clear that the thesis was addressed mostly to “the currently convinced” of the subjective-Bayesian view, and claimed: “very simply: I wish to change their minds … I hope … to persuade them there are more ways of being reasonable under uncertainty than they currently imagine” (Ellsberg [1962] Reference Ellsberg2001, p. liii).Footnote ¹⁹ While sending him the thesis manuscript a few days before its defense, in his accompanying letter Ellsberg confesses to Savage: “I see from a copy of your letter to Fellner that I haven’t convinced you yet. That is just as well, since the enclosed manuscript (which is really a 400-page letter to you, designed to change your mind) would hardly have been worth the trouble” (D. Ellsberg to L. J. Savage, May 21, 1962, LJS Papers, 11, 260).

The correspondence collected at Yale University contains only a few letters (ranging from 1961 to 1963) and is surely not exhaustive about the debate among the two, who personally met a few times in the meantime.Footnote ²⁰ The correspondence had started on October 1961 with Savage’s acknowledgment of receipt of the QJE paper sent to him by Ellsberg and his suggestion to forward it to de Finetti, as he often did with correspondents. But there is no evidence that the comments promised by Savage to Ellsberg—in the hope they would arrive “soon enough to affect your thesis, insofar you find them pertinent” (L. J. Savage to D. Ellsberg, February 18, 1962, LJS Papers, 11, 260)—were ever written.Footnote ²¹

However, even in the absence of a proper answer to Ellsberg, the exchange with de Finetti is illustrative of Savage’s attitude.Footnote ²² Early in 1962 de Finetti asks Savage: “Have you read D. Ellsberg’s note (Quarterly J. of Econ., 75, 4, Nov. Reference Ellsberg1961) that claims that you were ‘inconsistent’ in answering to one of his questions concerning issues such as Smith’s?” (B. de Finetti to L. J. Savage, March 8, 1962, LJS Papers, 8, 194, author’s translation from the original Italian). In his reply, Savage states: “I have not only read Ellsberg’s paper but had a very thorough visit with him here in Ann Arbor. He is intelligent, steeped in the material, but quite blind about certain aspects of it. I feel that there may be a grain of truth in what he is trying to say, but find it very difficult to clear my own head on the subject” (L. J. Savage to B. de Finetti, March 16, 1962, LJS Papers, 8, 194).

We shall see that the reference made by Ellsberg to a letter to Fellner helps in understanding Savage’s reaction. But it is enlightening that after receiving the thesis, Savage urged Ellsberg to send his manuscript to de Finetti, as he had already done with the QJE article. Later in 1962, de Finetti provided his comments in a letter to Ellsberg, forwarded to Savage. De Finetti’s reaction to Ellsberg’s thesis is closely adherent to the subjective-Bayesian canon, which is thoroughly defended. But after pointing out that overall his position was “essentially the same as Raiffa’s,” de Finetti comments on the maximization of expected utility that he used to attribute to Daniel Bernoulli:

Notwithstanding the unsystematic content of the correspondence with Ellsberg, two things are worth noting before moving on to that with Fellner. First, Savage did not take the opportunity given to him by de Finetti to deny the normative significance of his own violations, as actual decision-maker, of the axioms: he did not object, not even in private, to being classified as a deliberate violator of his own axioms by Ellsberg. While of course this cannot but be speculative, we may take this episode as signaling that Savage was more intrigued by the significance of the empirical violation put forward by Ellsberg than he was with respect to Allais’s. Second, while de Finetti used the descriptive/normative distinction in his comments on Ellsberg’s thesis to reject his counterexamples as normatively irrelevant, he contemplated the issue of normative violation. Under the strict condition that a normative counterargument could be taken into account only if “mathematically disproving” the subjective-Bayesian approach, de Finetti admitted that a normative argument against the subjective approach could be of interest.

As already mentioned, there is evidence of Savage’s own reaction to Ellsberg in his exchange with Fellner. Late in 1961, Fellner had asked Savage to comment on his idea of probability distortions in the face of uncertainty, emphasizing the similarity with Ellsberg’s experimental results. Apologizing for not being ready yet to offer a detailed comment on his QJE paper because of other obligations, Savage replied to Fellner in the letter mentioned by Ellsberg:

Savage had taken months to reply to Fellner, pointing to the fact we have already mentioned that his research was devoted to developments in statistical practice rather than foundational issues. Indeed, while trying to convince statisticians that they should use a prior probability in every examination of data, it would have been counterproductive for his approach to admit that this prior can be vague in certain instances of relevance. But Fellner immediately insisted on his point, namely, the cognitive unease of dealing with uncertainty issues as if they were risky ones:

Even this time, Savage was not eager to answer soon. The idea to abandon, or even relax, his main assumption that precise subjective probabilities can be determined and used in every decision problem, while still trying to convince statisticians to follow him in statistical practice, was certainly not on his agenda. But in a reply dated June 1963, the issue raised by Fellner in his letter—and in particular a summary of his argument Fellner had enclosed in it—is re-examined:

And when Fellner pointed at the sure-thing axiom as the arbitrary assumption, Savage replied: “If I understand correctly, the essential point here is whether the original imperfect system exemplified by F. of S. admits mathematical improvement or only improvement by informal commentary. I have expressed a pessimistic opinion about the possibility of the latter, but I trust I would not be blind to a better system should one come along” (L. J. Savage to W. J. Fellner, July 1, 1963, LJS Papers, 11, 267).

It must be stressed, then, that the issue explicitly became whether or not normative progress was possible. Although it appears that Savage no longer showed interest in Fellner’s attempt to discuss a generalization of his viewpoint allowing for vagueness,Footnote ²³ the correspondence shows that Savage may have accepted a constructive criticism of his theory, possibly generating a new normative standard, but with the caveat that the new theory would be able to provide theoretical continuity in terms of the way his one was formulated, namely, its axiomatic structure.

In conclusion, the early 1960s exchange with Ellsberg and Fellner signals a positive attitude by Savage (and de Finetti) towards the significance of their criticism. This is not surprising on descriptive grounds: indeed, in those years both Savage and de Finetti had been insisting that theirs was a normative viewpoint, and that descriptive violations were plainly admitted. As already remarked, the axiomatic structure devised by Savage in his Foundations was explicitly intended to provide a normative guide to the formation of consistent beliefs by statisticians. Axioms are instrumental, Savage (Reference Savage1954, p. 20) claimed, “to police my own decision for consistency, and, where possible, to make complicated decisions depend on simpler ones.”Footnote ²⁴

But the correspondence also shows that, while they kept objecting to the possible normative relevance of the specific arguments raised by their critics, Savage and de Finetti considered the normative issue worth discussing. They both showed a more open attitude than they were used to doing in their published works. We have seen, on the one hand, de Finetti asking for a negative argument of mathematical kind, which he considered as a prerequisite to openly consider Ellsberg’s point as of normative value. On the other hand, in his questions to Fellner for axioms apt to replace the sure-thing principle, Savage admitted that a constructive argument would be of interest, insofar as “mathematical improvement” was in principle acceptable.

To be sure, both Savage and de Finetti insisted that more realistic hypotheses should not be detrimental to the normative appeal of the entire axiomatic structure of the subjective approach to decision-making: in particular, Savage would not abandon a normative proposition such as the sure-thing principle—which may appear “unsatisfactory without being arbitrary”—in the absence of an alternative proposition endowed with similar plausibility. But it is remarkable that in the discussion with Ellsberg and Fellner, a possible generalization of the subjective-Bayesian approach—officially denied in print while still fighting for acceptance among statisticians—was contemplated. And, indeed, this is the route followed in decision theory in the 1980s after the experimental evidence confirming Ellsberg’s results became overwhelming and started being interpreted as of normative value.Footnote ²⁵

V. FRIENDLY FIRE IN STATISTICS

As seen in the previous section, both Ellsberg and Fellner aimed at a normative criticism of Savage’s viewpoint, one that could not simply be classified as a descriptive violation of rational choice theory. The attention they received from Savage in private exchange was substantial but ineffective: Savage never really tackled the issue they raised in his publications. Although his counterexamples proved to be puzzling for Savage, Ellsberg’s aim “to change” Savage’s mind failed: the correspondence reveals that neither he nor Fellner was able to provide the requested alternative axiomatic set-up.

This section presents excerpts from the archival evidence available on a distinct but related criticism, suggested by Cedric Smith (Reference Smith1961) with his study on upper and lower probability values in a Bayesian framework. Mostly through an analysis of the thoughts shared with de Finetti while examining Smith’s point, Savage’s understanding of vagueness and possibly imprecise probability priors can be detailed as subtler than usually understood, confirming the evidence already examined on the Ellsberg Paradox.

As we have seen, a core aspect of Savage’s subjective research program was the possibility to provide a behavioral justification of the representation of degrees of belief as probabilities, that is, an operationally founded—in Percy Bridgman’s (Reference Bridgman1927) sense—subjective probability. This step toward behaviorism, refraining from introspection, had been proposed by Frank Ramsey and de Finetti in terms of betting quotients elicited from choices, plus a condition of consistency, usually interpreted as the impossibility to have a “Dutch Book” made against a rational individual. A crucial argument against accepting the vagueness of beliefs Ellsberg and Fellner wanted to allow, in order to deal with what they considered a kind of uncertainty irreducible to risk, was that it would be incompatible with this elicitation procedure.Footnote ²⁶

In their criticism, both Ellsberg and Fellner referred to this aspect. Ellsberg, in particular, devoted an entire section of his doctoral thesis to de Finetti’s “condition of coherence.” For purposes of measurement, de Finetti ([1937] Reference de Finetti, Kyburg and Smokler1964) had regarded probabilities as “prices” and assumed that the highest price an individual is ready to pay for betting in favor of an event—her probability for the realization of the event as a bettor—coincides with the lowest she is ready to accept for taking the other side of the bet—her probability as a bookie. In order to derive a sharp subjective prior satisfying the laws of probability, therefore, no inequality between upper and lower betting prices was allowed by de Finetti. Ellsberg accepted the subjectivists’ methodology of eliciting probability from choices, but he argued that he could see no rationale for the assumption that the decision-maker is always willing to take either side of a bet at the same price, as implied by de Finetti’s reasoning (Ellsberg [1962] Reference Ellsberg2001, p. 68). And in order to clarify his point, Ellsberg referred to the approach of deriving lower and upper probability priors from betting presented by Smith (Reference Smith1961).

Cedric Smith, a statistician at the Galton Laboratory of the University College, London, who specialized in genetics, was among the very few English statisticians endorsing Savage’s subjective-Bayesian perspective from the outset. But Smith’s Reference Smith1961 paper also belonged to the tradition of interval-valued probabilities, an approach that had been investigated by probabilists such as Bernard Koopman (Reference Koopman1940) and Good (Reference Good1952), who had worked in the footsteps of John Maynard Keynes’s ([1921] Reference Keynes1973) notion of epistemic probability and his hint that probabilities might be ordered only partially.Footnote ²⁷ But while Koopman and Good interpreted probabilities simply as “intuitive judgements,” that is, judgments based on introspection prior to objective experience, Smith followed de Finetti in adopting a behavioral perspective. He accepted that subjective beliefs are meaningful only if measured by means of the elicitation of probabilities from betting quotients accepted by individual agents in controlled environments, as in de Finetti ([1937] Reference de Finetti, Kyburg and Smokler1964). But he applied this procedure to derive possibly imprecise beliefs. He showed how to elicit from personal betting quotients upper and lower probability values, and clarified the formal conditions under which the decision-maker can then be attributed an interval of precise initial probabilities. As a result, Smith showed that a decision-maker refusing to bet on either an event or its complement can do so consistently, in de Finetti’s sense.Footnote ²⁸

Ellsberg used Smith’s analysis to argue that the probabilities that can be derived from the choices of a deliberate violator of Savage axioms in his urn examples define an interval of probabilities but cannot be termed “irrational” by means of the Dutch Book argument. Indeed, consistency does not apply only to a set of “precise, definite beliefs,” since “beliefs that must be treated as ‘indefinite’ within limits can still be precise enough to determine decisions in betting, and susceptible of quantitative expression in terms of inequalities” (Ellsberg [1962] Reference Ellsberg2001, p. 88).

VI. ARCHIVAL EVIDENCE: SECOND PART

We have seen earlier that de Finetti mentioned Smith when asking Savage about Ellsberg in his letter of March 1962. And, indeed, in the early 1960s de Finetti and Savage were discussing how to react to Smith’s proposal to generalize the subjective-Bayesian approach in order to allow for interval-valued probability priors. The correspondence between them on this point—mostly related to the drafting of de Finetti and Savage (Reference de Finetti, Savage and Savage1962), a joint paper they were working on—reveals the extent to which interval-valued probabilities could be contemplated. It is also interesting because the tension already emerged in the discussion with Ellsberg and Fellner between what is descriptively versus normatively relevant is even more evident in light of Smith’s argument.

De Finetti and Savage’s (Reference de Finetti, Savage and Savage1962) paper, titled “Sul modo di scegliere le probabilità iniziali” (How to choose the initial probabilities), was intended for a statistical audience. It presents to statisticians a summary of why, in their view, no problem can be correctly stated in statistics without an evaluation of the “initial probabilities.” But it also reflects the two subjectivist scholars’ perplexity with respect to the statement they were publicly defending, namely, that the reference to intervals of prior probabilities may pose “more severe problems than they are intended to resolve” (Savage Reference Savage1962c, p. 150).Footnote ²⁹

Savage’s correspondence with de Finetti shows that it was on Savage’s request that Smith’s contribution was made a subject matter of their investigation. While commenting on the translation into English of a piece by de Finetti, Savage had written: “I wondered whether before concluding on page 65 [of your Lectures] you would want to touch on the allegorical idea of many opinions in one person à la Smith. Also, it is less allegorical and to that extent more satisfactory, to speak of the many systems of opinion not altogether incompatible with the vaguely defined system of a real person” (L. J. Savage to B. de Finetti, March 3, 1961, LJS Papers, 7, 192).

Savage seems to suggest that Smith’s interpretation provides a technical, rather than simply metaphorical, way to introduce the issue of vagueness, and that this makes it worth examining (see also Savage Reference Savage1962d). And, indeed, his concern about imprecise probability priors is reflected in the section of the paper dedicated to Smith that reproduces part of the correspondence between the two authors ensuing from a preliminary draft. In particular, in a letter to de Finetti, Savage is reported to claim:

The issue at stake may appear a descriptive one, but the section on Smith’s approach hints at normative issues. De Finetti and Savage admit that Smith’s “particularly elaborate analysis” of Koopman’s and Good’s idea “to make imprecision precise” provides a precise criterion to determine the two limiting probability values: “In Smith’s case … the objections about the precision of the extreme values do not hold, because he gives a criterion that is … surely ‘precise’” (de Finetti and Savage Reference de Finetti, Savage and Savage1962, p. 135). Moreover, de Finetti and Savage (Reference de Finetti, Savage and Savage1962, p. 141) accept Smith’s considerations about the meaning of being reluctant to betting, since these considerations can refer to “what can be said of a certain behavior when a person has an incomplete knowledge of the opinions justifying a decision.”

This reference to incomplete self-knowledge entails a concession to a possible normative justification of using interval-valued probabilities. In fact, it is illustrated through the analogy with the case of a group of decision-makers who have to make a collective decision on the basis of their sharp but not necessarily unanimous priors—indeed, typically different, since “personal” in the subjective approach. In this case, de Finetti and Savage acknowledge, there is a theoretical, not concrete, difficulty in determining a single (collective) prior.Footnote ³⁰ And this is dubbed as analogous to the situation of an individual who has “various souls leaning towards contrasting opinions,” possibly because she is in doubt about whether to rely on her own sharp prior or on “the ones she has been made aware of by consulting experts she considers highly” (de Finetti and Savage Reference de Finetti, Savage and Savage1962, p. 142).Footnote ³¹

The correspondence between de Finetti and Savage in the early 1960s is punctuated by scattered but recurrent reference to the issue. For instance, while summarizing the last changes about Smith he had made to the draft of their joint paper, de Finetti writes to Savage:

To this point, Savage replies with a clear reference to the issue of how to deal with probability priors when more than a single one is admitted:

The admission that a phenomenon that the subjective viewpoint contemplates as obvious for group decision-making—that the arbitrary opinions of different individuals may not coincide—may also hold true for a single individual represents a significant concession to Smith’s developments and to the other critics of a strict version of the subjective-Bayesian approach. Under the behavioral perspective suggested by Smith, Savage appears to allow vagueness of the beliefs of an individual agent even from a normative perspective, a concession that cannot be retrieved in his published works in English.

As a matter of fact, during discussion with de Finetti about a new joint work that never came out in print, Savage raised the issue at least once more:

Even though Savage’s philosophical papers of the late 1960s do not offer any further substantial concession on vagueness, the correspondence suggests that his final position may have been open to a formally consistent representation of it.Footnote ³²

VII. CONCLUDING REMARKS

Contemporary decision theory under uncertainty has built on the so-called Ellsberg Paradox to justify theories that reject, on normative grounds, one of the main tenets of the mainstream subjective-Bayesian approach, namely, that individuals have precise probabilistic beliefs over any source of uncertainty (Gilboa et al. Reference Gilboa, Postlewaite and Schmeidler2008). This point was originally made in the early 1960s, when Ellsberg and other critics such as Fellner and Smith argued that the subjective-Bayesian model was too restrictive in its assumptions. These critics considered themselves supporters of the subjective view of probability and decision-making, but they could not understand Savage’s insistence on the strict version he shared with de Finetti. Their attack was directed to Savage’s axiomatic justification of the new paradigm, which in the early 1960s was already regarded by the majority of decision theorists as so compelling that it could attain the status of the mainstream view in decision theory and economics.

To what extent was Savage ready to admit a permissive variation of his theory of decision-making, such as the one suggested in quite similar fashion by Ellsberg, Fellner, and Smith in the early 1960s? The abandonment of one of the building blocks of the subjective-Bayesian approach—the representation of degrees of belief through a sharp probability prior—was never endorsed in his published writings, although he admitted that he found this kind of representation wanting on descriptive grounds. However, the doubts he shared in private correspondence with his critics appear to be extremely relevant since they have normative content.

This paper has documented that in his correspondence with what can be identified as a small group of inside critics of the subjective-Bayesian approach—and at the same time with his co-author de Finetti about how to deal with them—Savage openly examined the issue of how rational choice might be modified when a single individual has less than full understanding of the decision context. The correspondence shows that he never equated Ellsberg’s counterexample with Allais’s, suggesting that he found the former more puzzling than the latter. And that when pressed by Fellner about the implausibility of the sure-thing principle, he suggested a methodological way forward, admitting that normative progress could be envisaged. Moreover, when facing Smith’s operational justification of the criticism, he did not regard as untenable a variant of the subjective approach allowing for interval-valued probability priors.

It can then be concluded that Savage’s reluctance to endorse the critical viewpoint underlying the Ellsberg Paradox was related to methodological caution rather than to rejection of its content. The new notion of mathematical rigor he had endorsed as part of the group of mathematical economists and statisticians he had worked with in the 1950s was crucial to him. His doubts were mostly based on the inability of his critics to provide an alternative theoretical set-up rather than on a clear-cut denial of the normative relevance of their argument. He may have been ready to endorse it had a consistent theoretical corpus and the appropriate axioms been made available by his critics. He died at the age of fifty-four, far too early to see such analytical progress come into reality.

ARCHIVAL COLLECTION

Leonard Jimmie Savage Papers, archived at the Manuscripts and Archives Department of the Yale University Library (New Haven), as MS695.