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Ambiguity and the Bayesian Paradigm

  • Chapter
Readings in Formal Epistemology

Part of the book series: Springer Graduate Texts in Philosophy ((SGTP,volume 1))

Abstract

This is a survey of some of the recent decision-theoretic literature involving beliefs that cannot be quantified by a Bayesian prior. We discuss historical, philosophical, and axiomatic foundations of the Bayesian model, as well as of several alternative models recently proposed. The definition and comparison of ambiguity aversion and the updating of non-Bayesian beliefs are briefly discussed. Finally, several applications are mentioned to illustrate the way that ambiguity (or “Knightian uncertainty”) can change the way we think about economic problems.

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Notes

  1. 1.

    As Cyert and DeGroot (1974) write on p. 524 “To the Bayesian, all uncertainty can be represented by probability distributions.”

  2. 2.

    According to Pascal a pious life would ultimately induce faith. Importantly, Pascal did not assume that one can simply choose one’s beliefs.

  3. 3.

    Pascal did not finish his Pensées, which appeared in print in 1670, 8 years after his death. The text that was left is notoriously hard to read since he only sketches his thoughts (here we use the 1910 English edition of W. F. Trotter). Our rendering of his argument crucially relies on Hacking (1975)’s interpretation (see Hacking 1975, pp. 63–72; Gilboa 2009, pp. 38–40).

  4. 4.

    See Ore (1960).

  5. 5.

    Operationalism started with Bridgman (1927), after Ramsey’s articles of 1926a and 1926b.

  6. 6.

    Frisch (1926) was the first article we are aware of that adopted a similar approach in economic theory.

  7. 7.

    See, e.g., chapter 8 of Kreps (1988).

  8. 8.

    Today, the terms “ambiguity”, “uncertainty” (as opposed to “risk”), and “Knightian uncertainty” are used interchangeably to describe the case of unknown probabilities.

  9. 9.

    See Gilboa (2009) and Wakker (2010) for the analysis.

  10. 10.

    Bertrand’s (1907) early critique of the principle of indifference was made in the context of a continuous space. See also Gilboa (2009) and Gilboa et al. (2009).

  11. 11.

    It is not entirely clear how one can justify the Principle of Indifference even in cases of ignorance. For example, Kass and Wasserman (1996) p. 1347 discuss the partition paradox and lack of parametric invariance, two closely related issues that arise with Laplace’s Principle. Similar remarks from a Macroeconomics perspective can be found in Kocherlakota (2007) p. 357.

    Based on a result by Henri Poincaré, Machina (2004) suggests a justification of the Laplace’s Principle using a sequence of fine partitions of the state stace. This type of reasoning seems to underlie most convincing examples of random devices, such as tossing coins, spinning roulette wheels, and the like. It is tempting to suggest that this is the only compelling justification of the Principle of Indifference, and that this principle should not be invoked unless such a justification exists.

  12. 12.

    See Lewis (1980) and chapter 4 of van Frassen (1989) (and the references therein) for a discussion of the relations between “objectivity” and subjective probabilities from a philosophical standpoint.

  13. 13.

    Kass and Wasserman (1996), Bayarri and Berger (2004), and Berger (2004) discuss uninformative priors and related “objective” issues in Bayesian statistics (according to Efron (1986), some of these issues explain the relatively limited use of Bayesian methods in applied statistics).

  14. 14.

    See Pearl (1986) and the ensuing literature on Bayesian networks.

  15. 15.

    Throughout the section we use interchangeably the terms lotteries and simple probabilities.

  16. 16.

    Simple acts have the form \(f =\sum _{ i=1}^{n}p_{i}1_{E_{i}}\), where \(\left \{E_{i}\right \}_{i=1}^{n} \subseteq \Sigma \) is a partition of S and \(\left \{p_{i}\right \}_{i=1}^{n} \subseteq \Delta \left (X\right )\) is a collection of lotteries.

  17. 17.

    For this reason, mixing acts in this way is sometimes called “ex post randomization.” For recent models with ex ante randomization, see Epstein et al. (2007), Ergin and Sarver (2009), Seo (2009), and Saito (2015).

  18. 18.

    See Ghirardato et al. (2003) for a subjective underpinning of the AA setup.

  19. 19.

    See Gilboa (2009) for some more details on them.

  20. 20.

    Throughout the paper, cardinally unique means unique up to positive affine transformations.

  21. 21.

    See Fischhoff and Bruine De Bruin (1999) for experimental evidence on how people use 50–50 % statements in this sense.

  22. 22.

    We refer the interested reader to Denneberg (1994) and to Marinacci and Montrucchio (2004) for detailed expositions of Choquet integration.

  23. 23.

    Nakamura and Wakker’s papers use versions of the so-called tradeoff method (see Kobberling and Wakker (2003), for a detailed study of this method and its use in the establishment of axiomatic foundations for choice models).

  24. 24.

    See Marinacci and Montrucchio (2004) p. 73. They show on p. 78 that (21.9) can be derived from this result of Choquet through a suitable application of the Hahn-Banach Theorem.

  25. 25.

    Klibanoff (2001a,b) studied in detail the relations between randomization and ambiguity aversion.

  26. 26.

    For example, assume that there are three states of the world, and two acts offer the following expected utility profiles: f = (0, 10, 20) and g = (4, 10, 14). Assume that the DM is indifferent between f and g, that is, that she is willing to give up 1 unit of expected utility in state 3 in order to transfer 5 units from state 3 to state 1. Comonotonic independence would imply that the DM should also be indifferent between f and g when they are mixed with any other act comonotonic with both, such as f itself. However, while f clearly doesn’t offer a hedge against itself, mixing f with g can be viewed as reducing the volatility of the latter, resulting in a mix that is strictly better than f and g.

  27. 27.

    Schmeidler required that all three acts be pairwise comonotonic, whereas C-Independence does not restrict attention to comonotonic pairs (f, g). Thus, C-Independence is not, strictly speaking, weaker than Comonotonic Independence. However, in the presence of Schmeidler’s other axioms, Comonotonic Independence is equivalent to the version in which f and g are not required to be comonotonic.

  28. 28.

    See Ghirardato et al. (1998) for details.

  29. 29.

    See Ghirardato et al. (2004) for details.

  30. 30.

    Wakker (2010) also introduces the gain-loss asymmetry that is one of the hallmarks of Prospect Theory (Kahneman and Tversky 1979). The combination of gain-loss asymmtry with rank-dependent expected utility (Quiggin 1982; Yaari 1987) resulted in Cumulative Prospect Theory (CPT, Tversky and Kahneman 1992). When CPT is interpreted as dealing with ambiguity, it is equivalent to Choquet expected utility with the additional refinement of distinguishing gains from losses.

  31. 31.

    For a critical review of the maxmin and other non-Bayesian models, see Al-Najjar and Weinstein (2009) (see Mukerji 2009; Siniscalchi 2009b, for a discussion).

  32. 32.

    As Chateauneuf et al. (2005) show, this control prior exists because, under Axiom MC, the set C is weakly compact, a stronger compactness condition than the weak∗-compactness that C features in Theorem 4. Their results have been generalized to variational preferences by Maccheroni et al. (2006a).

  33. 33.

    In this regard, Arrow (1970) wrote that “the assumption of Monotone Continuity seems, I believe correctly, to be the harmless simplification almost inevitable in the formalization of any real-life problem.” See Kopylov (2010) for a recent version of Savage’s model under Monotone Continuity.

    In many applications, countable additivity of the measure(s) necessitates the restriction of the algebra of events to be a proper subset of 2S. Ignoring many events as “non-measurable” may appear as sweeping the continuity problem under the measurability rug. However, this approach may be more natural if one does not start with the state space S as primitive, but derives it as the semantic model of a syntactic system, where propositions are primitive.

  34. 34.

    A caveat: the unanimity rule ( 21.11) is slightly different from Bewley’s, who represents strict preference by unanimity of strict inequalities. This is generally not equivalent to representation of weak preference by unanimity of weak inequalities.

  35. 35.

    The name biseparable originates in Ghirardato and Marinacci (2001, 2002), which we will discuss later.

  36. 36.

    That is, if \(\succapprox ^{{\prime}}\subseteq \succapprox \) and \(\succapprox ^{{\prime}}\) satisfies independence, then \(\succapprox ^{{\prime}}\subseteq \succapprox ^{{\ast}}\).

  37. 37.

    This latter feature of \(\succapprox ^{{\ast}}\) relates this notion to an earlier one by Nehring (2001), as GMM discuss.

  38. 38.

    GMM also show the form that C takes for some CEU preferences that do not satisfy S.6.

  39. 39.

    Relations between ambiguity and games against nature are discussed in Hart et al. (1994), Maccheroni et al. (2006a,b), and Ozdenoren and Peck (2008).

  40. 40.

    Bayes (1763) himself writes in his Proposition 10 that “the chance that the probability of the event lies somewhere between …” (at the beginning of his essay, in Definition 6 Bayes says that “By chance I mean the same as probability”).

  41. 41.

    The function \(c: \Delta \left (\Sigma \right ) \rightarrow \left [0,\infty \right ]\) is grounded if its infimum value is zero.

  42. 42.

    This is so because one axiom relates preferences between mixtures with different coefficients α, β and the other – between mixtures with different constant acts x ∗, p.

  43. 43.

    Other sub-fields include choices from menus, decision under risk, minmax regret approaches, and others. On the first of these, see Limpan and Pesendorfer (2013).

  44. 44.

    Epstein (1999) takes the standard for ambiguity neutrality to be preferences that are probabilistically sophisticated in the sense of Machina and Schmeidler (1992). In his approach Theorem 18 below does not hold.

  45. 45.

    \(\mathop{\mathrm{dom}} c\) is the effective domain of the function c; i.e., \(\mathop{\mathrm{dom}} c = \left \{P \in \Delta \left (S\right ): c\left (p\right ) < +\infty \right \}\).

  46. 46.

    See Seidenfeld and Wasserman (1993) who study counter-intuitive updating phenomena in this context.

  47. 47.

    Mukerji and Tallon (2004) survey early works in this area.

  48. 48.

    This argument assumes that the decision maker starts with a risk-free portfolio. A trader who already holds an uncertain position may be satisfied with it with a small set of probabilities, but wish to trade in order to reduce uncertainty if the set of probabilities is larger.

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Acknowledgements

We thank Giulia Brancaccio, Simone Cerreia-Vioglio, Fabio Maccheroni, Andrew Postlewaite, Xiangyu Qu, and David Schmeidler for comments on earlier drafts of this survey. We are also grateful to many members of the “decision theory forum” for additional comments and references. Finally, we are indebted to Eddie Dekel for many comments and suggestions. Gilboa gratefully acknowledges the financial support of the Israel Science Foundation (grant 396/10) and of the European Reseach Council (advanced grant 269754), and Marinacci that of the European Reseach Council (advanced grant BRSCDP-TEA).

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Authors and Affiliations

  1. HEC, Paris, France

    Itzhak Gilboa

  2. Tel-Aviv University, Tel Aviv, Israel

    Itzhak Gilboa

  3. Università Bocconi, Milano, Italy

    Massimo Marinacci

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  1. (deceased), Dordrecht, The Netherlands

    Horacio Arló-Costa

  2. Center for Information and Bubble Studies, University of Copenhagen, Copenhagen, Denmark

    Vincent F. Hendricks

  3. University of Amsterdam, Amsterdam, The Netherlands

    Johan van Benthem

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Gilboa, I., Marinacci, M. (2016). Ambiguity and the Bayesian Paradigm. In: Arló-Costa, H., Hendricks, V., van Benthem, J. (eds) Readings in Formal Epistemology. Springer Graduate Texts in Philosophy, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-20451-2_21

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