Rationality under uncertainty: classic and current criticisms of the Bayesian viewpoint

Abstract

At least since Leonard Savage’s extension of von Neumann and Morgenstern’s expected utility, rational choice theory has been interpreted as a theory prescribing what individuals should do in any decision context, ranging from certainty to risk and uncertainty. After decades this received view, usually termed Bayesian, has been criticized for its normative content. This paper compares the current critique of the notion of Bayesian rationality, proposed by Itzhak Gilboa, with Daniel Ellsberg’s classic critique of Savage’s understanding of rationality. The paper argues that Ellsberg’s classic analysis of Savage’s theory totally anticipated today’s criticism.

KEYWORDS:

• Rationality
• probability
• risk
• uncertainty
• decision theory
• Ellsberg Paradox

JEL CODES:

• B21
• B41
• D80

Acknowledgements

A previous draft of this paper was presented at the ESHET Meeting in Antwerp (May 2017) and the STOREP Meeting in Piacenza (June 2007). Comments by George Bent, Nicola Giocoli, Massimo Marinacci, Enrico Petracca, Andrea Salanti, Erik Schokkaert and especially two anonymous referees are gratefully acknowledged. Usual caveats apply.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 The term Bayesian relates to the use of Bayes Theorem in the updating of probability priors, but the reference here mostly is to the assumption that individuals are supposed to be endowed with a single, definite probability prior in any possible instance.

2 Notable contributors include Eichberger and Kelsey (1999), Gilboa and Schmeidler (2001), Gilboa, Postlewaite and Schmeidler (2008, 2009, 2012), Gilboa and Marinacci (2016), Hansen (2014), Machina and Siniscalchi (2014), and Mukerji (2009).

3 In Ramsey’s (1964 [1931], 72)words, giving birth to the operational perspective characterizing the Bayesian approach, a degree of belief can be expressed as “the extent to which we are prepared to act on it” (see also de Finetti 1964 [1937], 102). Savage’s claim that preferences are defined by choices was formulated in term of acts, i.e., decisions whose consequences depend on the realization of exogenous events. That a decision-maker prefers act f to g, simply means that “if he were required to decide between f and g, no other act being available, he would decide on f” (Savage 1954, 17).

4 This is a form of radical behaviourism, in which the psychology of choice is summarized by the idea that mental states are distinguishable only in terms of behaviour. As in Samuelson’s (1938) revealed preference approach, Bayesian decision-making can thus be given an entirely behavioural foundation, freed from any process of introspection.

5 The tenets of Bayesianism are usually listed as follows: (1) a representation of the environment in terms of state space (all information is summarized in states of natures), (2) prior probabilities (degrees of belief are given by a single probability measure defined over the state space), (3) Bayesian updating (in view of new information posterior probabilities are derived from Bayes’s law), and (4) maximization of EU (individuals act as if they are maximizing a utility function). See Gilboa et al. (2012, 14–15).

6 On the history of the independence axiom see Fishburn and Wakker (1995).

7 See for instance Blume and Easley (2008), Binmore (2009), Eichberger and Kelsey (2009), and Machina and Siniscalchi (2014). Another prominent contributor to the literature, Peter Wakker, is less adamant about the actual normative significance of recent developments (Wakker 2008, 431).

8 Gilboa (2009, 133) claims: “Schmeidler’s intuition … has a behavioural manifestation in Ellsberg’s paradox. But Schmeidler did not start out by attempting to explain the experimental evidence.” See also Gilboa et al. (2012, 18–19), and Gilboa and Marinacci (2016, p. 398).

9 In a recent interview Gilboa claims: “My definition of rationality started with this: asking what do people have in mind when they refer to something as ‘rational.’ But the best definition I came up with is in terms of what most people would be willing to accept as their decision making models, as opposed to what they would like and could change” (Herfeld 2015, 128).

10 Gilboa’s claim is enlightening when he makes reference to the other sciences using the Bayesian approach: only economics—and neither statistics nor computer science nor philosophy—accepts Bayesianism as “the sole claimant to the throne of rationality” for the representation of uncertainty (Gilboa et al. 2009, 287).

11 The influence of these axiomatically based alternatives to strict Bayesianism has been huge, both in applications (e.g., Mukerji and Tallon 2004) and in refinements concerning operational decision rules (e.g., Klibanoff et al. 2005). However, as remarked by a referee, it is important to note that discordant views about the actual significance of the approach exist as well, even among main contributors to the topic. For instance, criticism of the normative achievements of the approach—with specific regard to updating and the failure of ambiguity aversion model to satisfy consequentialism, a theme not addressed in this paper—is presented in Al-Najjar and Weistein (2009). Experimental studies intended to test how violations of different axioms may justify the actual choices of individuals suggest mixed results. For instance, Halevy (2007) has found that violators of the sure-thing principle usually fail in tests of their compliance to the axiom of compound lotteries. Different elicitation methods have been proved to be crucial for the significance of collected evidence (Binmore et al. 2012). Outside the laboratory, when for instance observed decisions concerns investments, it appears even difficult to understand which alternative decision-makers perceive as ambiguous (Trautman and van de Kuilen 2015).

12 Similarly, in their survey of rationality issues, Blume and Easley (2008, 891) conclude: “Rationality does not mean expected utility. Expected utility is one small class of decision models for choice under uncertainty. Its dominance in application was understandable 30 years ago when few alternatives were on the table. Since then decision theorists have been creative in developing better-behaved alternatives, and equilibrium and game theorists have been clever in applying them.”

13 Herfeld (2018) provides evidence from archival material about the Cowles Commission in the late 1940s that Marschak and Tjalling Koopmans were responsible, as directors of the Commission, for its commitment to the axiomatization of economics. Their aim was to spread this viewpoint in branches, such as decision-making, that were traditionally seen as not easily separable from other behavioural sciences. Marschak (1946, 114) evidenced as a major methodological step taken by von Neumann and Morgenstern the separation of formal axiomatic structure and empirical interpretation and promoted the development of economics by means of the “tools of modern logic” in a series of Cowles Discussion Papers culminating in Marschak (1950, 1951) and.

14 It was decades before Savage’s position was questioned extensively. Machina (1987) reviews the wave of non-EU theories originated by the renewed interest in the Allais Paradox in the 1980s, Kelsey and Quiggin (1992) those related to the Ellsberg Paradox.

15 Savage’s “representation theorem” states that if an individual’s preferences satisfy his axioms, then these preferences can be represented by a utility function assigning a utility index to every consequence, and a probability function assigning a probability to every event. The crucial axiom to prove this result turns out to be the “sure-thing” principle: a preference between two acts is independent of the states in which the two yield identical consequences. For acts f and g, this requires f $\sim$ g ↔ f' $\sim$ g' whenever the state space can be partitioned into two parts, the irrelevant event and the relevant event, such that, on the irrelevant event, f $\sim$ g and f' $\sim$ g', and on relevant event, f $\sim$ f' and g $\sim$ g'. When probabilities are assigned to states, expected-utility maximization implies the sure-thing principle because the parts of the expected utilities of f and g or of f' and g' related to the irrelevant event cancel (Fishburn and Wakker 1995, 1136).

16 Ramsey (1964 [1931], 83) made this point arguing: “if anyone’s mental condition violated these laws [the laws of probability], his choice would depend on the precise form in which the options were offered him, which would be absurd. He could have a book made against him by a cunning better and would then stand to lose in any event.” See also de Finetti (1951, 223). In the literature this is usually referred as the Dutch Book principle (Gillies 2000).

17 As suggested by a referee, it must be remarked that Savage’s proposal was restricted to what he called “small worlds,” that is, decision contexts in which it is possible to take account in advance of all future contingencies. It is only in small worlds that it makes sense to insist on consistency. We shall not deal with this issue in this paper, but it is worth noting that this is a controversial issue among critics: for instance, Binmore (2009, 127) remarks that, although cited as the father of Bayesianism, when arguing that Bayesian decision theory is applicable only in worlds smaller than those in which it is usually applied, Savage “disavowed his creed before it was even born.”

18 Baumol (1951, 65) proposed a hypothetical example suggesting behavior that would contradict the EU hypothesis and yet would not be clearly “pathological,” arguing that he would not consider some of the results of the application of EU presented in Friedman and Savage (1948) as “introspectively obvious.” As shown by Moscati (2016), Samuelson’s (1950) criticism was also in the background.

19 Friedman and Savage (1952) is arguably the first paper of the period addressing the methodological issue of how to justify the claim that a proposition that cannot be empirically proved may still hold true, the question we saw Gilboa dealing with in the previous section. Friedman and Savage (1952, 469) argue that the “intuitive appeal” of the independence axiom—soon to be renamed as the sure-thing principle—is that if a “reader” considers the postulate in light of the illustration they provide in the paper “he will concede that the principle is not one he would deliberately violate.” That the emphasis placed on “deliberate” choice is a clear step forward from Friedman and Savage’s (1948) classic analysis is usually unnoticed, with the notable exception of Starmer (2005).

20 As is well-known, when he was presented Allais’s example at an informal meeting during a 1952 conference in Paris, Savage expressed preferences contradicting his axioms. Upon reflection, though, he claimed: “it seems to me that in reversing my preference … I have corrected an error” (Savage 1954, 103). For a historical reconstruction of the Paris conference see Jallais and Pradier (2005). Heukelom (2014) illustrates how Savage answered to Allais in the private exchange they had after the conference. Mongin (2014) explores the methodological features of the debate.

21 As noted by Mongin (2014), the delayed publication in English of his work contributed to the usual characterization of Allais as interested only in descriptive issues. But this characterization is inaccurate, since Allais aimed to question the normative adequacy of Savage’s theory as well. Allais’s (1979 [1953], 79–80) first detailed report of his objections to Savage at the Paris Conference hints at a normative viewpoint when he discusses an “experimental definition of rationality,” arguing that “rationality can be defined by having regard to the behaviour of persons who are commonly consider as rational.” Indeed, when one does not find “desirable to use an abstract definition of rationality, the only option is to rely on experience, and to observe the behaviour of men one has reason in other respects to believe act rationally.”

22 Papers by Brewer (1963) commenting on Fellner and Roberts on Ellsberg, with short replies (Fellner 1963; Ellsberg 1963), did not ignite a substantive theoretical debate. Apart from some notable exceptions, which we shall see later, even experimental analysis languished until the 1980s.

23 Ellsberg’s reply to Raiffa constituted part of his doctoral thesis, but, as we shall see in section 6, the thesis was not published at the time, and did not have an impact on following developments.

24 Traditionally, it has been assumed that Allais discussed risk, while Ellsberg uncertainty (Wakker 2008).

25 The only words dedicated to Ellsberg’s and Fellner’s criticism in Savage’s printed works appear in Savage’s (1970, 25) annotated bibliography prepared for the 2nd edition of the Foundations in which the two papers are presented as “an account of an important line of dissent from the theory of personal probability and utility.”

26 This is to be expected for the second urn, because their probabilities are objective in a frequentist sense. But is supposed to be true as well for the first one considering the conventional epistemic interpretation of probabilities, that is, for reasons either of symmetry or insufficient information (Hacking 1975).

27 This kind of choice is incompatible with the assignment of a single additive probability distribution: while the indifference of decision-makers between betting on the red (r) or blue (b) drawn from urn I or II examined separately means that their subjective probabilities p are such that p(r_I) = p(b_I) = 1/2 and p(r_II) = p(b_II) = 1/2, when they choose to bet on red (or black) from the second urn they reveal p(r_II) > p(r_I) (or p(b_II) > p(b_I)). It is following on this remark, that the non-additive probability approach of Schmeidler (1989) proposes an axiomatic system for subjective probabilities such that p(r_I) + p(b_I) < p(r_I $\cup$ b_I). In contemporary decision theory the individual who prefers to bet on urn II is said to show ambiguity aversion (Eichberger and Kelsey 2009; Machina and Siniscalchi 2014).

28 Ellsberg proposed also a second example, with a single urn whose composition is partly known and partly unknown, and found a similar attitude among people tested, with many decision-makers who refrained from betting on the drawing of balls belonging to the unknown part of the urn. Technically this second example is even more significant, as it offers a direct test of Savage’s sure-thing principle.

29 Ellsberg’s claim is that Savage failed his test in February 1958. The two remained in contact later and Savage was sent a draft of Ellsberg’s dissertation in Spring 1962. There is no evidence, either in published works or in the private exchanges archived at the Leonard Jimmie Savage Papers at Yale University, that Savage ever denied Ellsberg’s claim, not even in his exchange with de Finetti (see Zappia 2018).

30 Ellsberg suggests that a decision-maker should choose the action x associated with the highest value of the index ρE(x) + (1-ρ)min(x), where E(x), the expected utility of action x with respect to the “best guess” probability, coinciding with Savage’s prior probability. E(x) is weighted with respect to the degree of confidence, ρ, encapsulating the influence of the ambiguity surrounding the decision process. If no ambiguity is perceived the rule coincides with Savage’s. In case ambiguity is perceived (i.e., 0 < ρ < 1) the best guess distribution is weighted with Wald’s maxmin, assumed as the conventional decision rule for probabilistic ignorance. It must be noted that Eichberger and Kelsey’s (1999) axiomatization of a decision rule for ambiguity builds exactly on Ellsberg’s intuition.

31 MacCrimmon and Larson (1979, 382–384) summarized this first round of experimental studies—comparing individuals’ actual choices with their level of agreement with decision rules—with the claim that in “real situations of uncertainty,” such as stock market investments, deliberate Ellsberg-type violations occurred at such a high rate to demand further theoretical study on the non-additivity of beliefs. As noted by Mongin (2014, 766–768), before Kahneman and Tversky (1979) empirical studies were used to test reflective reasoning also in Allais, confirming that the issue of deliberate choices was intended as crucial under both risk and uncertainty.

32 As mentioned, Ellsberg did not refer to Allais in his paper and the discussion with Raiffa and Roberts contains no comparison between the paradoxes, testifying of the limited interest in Allais’s argument in the early 1960s. However, we shall see that the connection became apparent to Ellsberg, and worth dealing with, in his thesis.

33 Ellsberg, already at the RAND Corporation while working on the thesis, moved to the US Secretary of Defence in 1964, and never came back to academics. In the political arena he is better known for his releasing to the press of the Pentagon Papers in 1971. From then on he has been, and still is, an independent writer and political activist.

34 The correspondence included in the Savage Papers is illustrative of Savage’s sympathy for Ellsberg’s investigation. But Ellsberg’s inability to provide axiomatic justification for his result brought as a consequence Savage’s unwillingness to endorse a less firm subjectivist perspective while still fighting for the acceptance of the core of his viewpoint among statisticians (see Zappia 2018).

35 Ellsberg’s now much more elaborate positive analysis hinges on the formal structure guaranteed by Koopman’s and Good’s research on partially ordered probabilistic beliefs and constitutes the link with authors such as Levi (1980) and Gärdenfors and Sahlin (1982), who have worked on the philosophical background of representations of degrees of belief through a set of probability distributions.

36 Ellsberg’s suggestion for a comprehensive decision rule is intended to accommodate a wide variety of rational behaviour attributable to individuals who deliberately violate Savage’s axioms. Using the same notation of footnote 30, Ellsberg suggests now that a decision-maker should choose the action x associated with the highest value of the index ρE(x) + (1-ρ)[αmax(x) + (1-α)min(x)], where the best guess distribution is weighted with a mixture of max and min of outcomes with respect to optimism α (ranging from 0 to 1), as in Hurwicz (1951). It is worth noting that even this rule has been axiomatized in contemporary literature, as α-Maxmin of EU (Ghirardato et al. 2004).

37 Ellsberg (2001 [1962], 229) uses a similar argument against Raiffa, arguing that the way he offered the opportunity to choose mixed strategies in his replications of the urn examples banished ambiguity, in fact producing “crucially different” experiments from the ones he had suggested in his 1961 article.